For convenience, we list the program that we have written to (try to) compute the inverse of the map δ : n ↦ σ(n)/n, which is the most essential part of our cyclic extension method. We have also included a short program that tries to extend a given group using the cyclic extension method.
max_denom = 10^30
max_denom_factors = 9
max_single_exp = 7
max_total_factors = 12
max_exp = 5
max_exp_sec = 2
max_depth = 10
def inverse_delta(q, M = [], depth = 0):
if depth > max_depth:
raise ArithmeticError('Maximum depth exceeded')
if depth<=1:
eexp = max_exp
else:
eexp = max_exp_sec
max_sum_exp = eexp
this_sum = 0
a = q.numerator()
b = q.denominator()
if b==1:
if a==1:
r = 1
elif a==2:
r = 6
elif a==3:
r = 120
elif a==4:
r = 30240
elif a==5:
r = 14182439040
elif a==6:
r = 154345556085770649600
else:
raise ArithmeticError('Large multiperfect number encountered')
for elt in M:
if elt.divides(r):
raise ArithmeticError('Forbidden prime encountered')
return r
if b > max_denom:
raise ArithmeticError('Denominator too large')
for elt in M:
if elt.divides(b):
raise ArithmeticError('Forbidden prime encountered')
sb = sigma(b)
if sb == a:
return b
if sb > a:
raise ArithmeticError('No solution found')
N = copy(M)
L = [] ; Lval = []
for p,e in b.factor():
if e > max_single_exp:
raise ArithmeticError('Prime exponent too large')
L.append(0)
Lval.append(p)
N.append(p)
if len(N) > max_total_factors:
raise ArithmeticError('Too many prime factors')
if len(L) > max_denom_factors:
raise ArithmeticError('Too many prime factors')
N.sort()
do_loop = true
while do_loop:
new_q = 1
for i in range(len(L)):
new_q *= Lval[i]^L[i]
new_b = new_q * b
new_q = a * new_q / sigma(new_b)
try:
c = inverse_delta(new_q, N, depth+1)
return (c * new_b)
except ArithmeticError:
pass
# update exponents for next case
loop = true ; j = 0
while (loop == true):
try:
L[j] += 1
except IndexError:
print (L, j, a, b)
raise IndexError
this_sum += 1
if (L[j] > eexp) or (this_sum > max_sum_exp):
this_sum -= L[j]
L[j] = 0
j += 1
if j >= len(L):
loop = false
do_loop = false
else:
loop = false
raise ArithmeticError('No solution found')
def try_to_extend_group_to_leinster_group(G):
if G.IsNilpotent():
if not G.IsCyclic():
return []
else:
if not G.CommutatorFactorGroup().IsCyclic():
return []
L = G.NormalSubgroups()
sum = 0
for gr in L:
sum += gr.Size()
sum = Rational(sum) / Rational(G.Size())
L = G.CompositionSeries()
M = []
for i in range(1, len(L)):
gr = L[i] / L[i+1]
if gr.IsCyclic():
M.extend(Integer(gr.Size()).prime_divisors())
try:
n = inverse_delta(2/sum, M)
return n
except ArithmeticError:
return 0
D2 x C3 = D(2) x C(3) D6 x C5 = D(2 * 3) x C(5) D114 x C4753 = D(2 * 3 * 19) x C(7^2 * 97) D270 x C17 = D(2 * 3^3 * 5) x C(17) D1170 x C29 = D(2 * 3^2 * 5 * 13) x C(29) D1638 x C17 = D(2 * 3^2 * 7 * 13) x C(17) D2310 x C769 = D(2 * 3 * 5 * 7 * 11) x C(769) D4050 x C7801 = D(2 * 3^4 * 5^2) x C(29 * 269) D8910 x C89 = D(2 * 3^4 * 5 * 11) x C(89) D22230 x C113 = D(2 * 3^2 * 5 * 13 * 19) x C(113) D27930 x C97 = D(2 * 3 * 5 * 7^2 * 19) x C(97) D67410 x C7489 = D(2 * 3^2 * 5 * 7 * 107) x C(7489) D87750 x C449 = D(2 * 3^3 * 5^3 * 13) x C(449) D106134 x C204853 = D(2 * 3 * 7^2 * 19^2) x C(11^2 * 1693) D127710 x C257 = D(2 * 3^3 * 5 * 11 * 43) x C(257) D181350 x C449 = D(2 * 3^2 * 5^2 * 13 * 31) x C(449) D212850 x C6449 = D(2 * 3^2 * 5^2 * 11 * 43) x C(6449) D217854 x C41 = D(2 * 3^2 * 7^2 * 13 * 19) x C(41) D418770 x C1409 = D(2 * 3^4 * 5 * 11 * 47) x C(1409) D525690 x C353 = D(2 * 3^4 * 5 * 11 * 59) x C(353) D774774 x C257 = D(2 * 3^2 * 7 * 11 * 13 * 43) x C(257) D1404810 x C8513 = D(2 * 3^3 * 5 * 11^2 * 43) x C(8513) D1659042 x C881 = D(2 * 3^4 * 7^2 * 11 * 19) x C(881) D2008890 x C1009 = D(2 * 3^2 * 5 * 13 * 17 * 101) x C(1009) D2709210 x C193 = D(2 * 3 * 5 * 7^2 * 19 * 97) x C(193) D2946102 x C2885953 = D(2 * 3 * 19 * 43 * 601) x C(7^2 * 58897) D3361410 x C337 = D(2 * 3^2 * 5 * 13^3 * 17) x C(337) D4250070 x C17489 = D(2 * 3^6 * 5 * 11 * 53) x C(17489) D4269870 x C4993 = D(2 * 3^2 * 5 * 11 * 19 * 227) x C(4993) D4745250 x C3329 = D(2 * 3^3 * 5^3 * 19 * 37) x C(3329) D5627310 x C510313021 = D(2 * 3 * 5 * 13 * 47 * 307) x C(17^2 * 1765789) D6566670 x C18089 = D(2 * 3^4 * 5 * 11^2 * 67) x C(18089) D6958710 x C2129 = D(2 * 3^4 * 5 * 11^2 * 71) x C(2129) D8522514 x C8513 = D(2 * 3^2 * 7 * 11^2 * 13 * 43) x C(8513) D8786250 x C1249 = D(2 * 3^2 * 5^4 * 11 * 71) x C(1249) D9806850 x C3329 = D(2 * 3^2 * 5^2 * 19 * 31 * 37) x C(3329) D10484370 x C17921 = D(2 * 3^3 * 5 * 13 * 29 * 103) x C(17921) D23604750 x C2689 = D(2 * 3^3 * 5^3 * 13 * 269) x C(2689) D30365874 x C33125693 = D(2 * 3^3 * 7 * 11 * 67 * 109) x C(37^2 * 24197) D30958830 x C65729 = D(2 * 3^2 * 5 * 7 * 157 * 313) x C(65729) D48783150 x C2689 = D(2 * 3^2 * 5^2 * 13 * 31 * 269) x C(2689) D64124970 x C3053569 = D(2 * 3 * 5 * 7 * 13 * 83 * 283) x C(3053569) D76172850 x C26041 = D(2 * 3^2 * 5^2 * 13 * 29 * 449) x C(26041) D81617046 x C22517737 = D(2 * 3 * 7^2 * 19^2 * 769) x C(11^2 * 186097) D88387650 x C5209 = D(2 * 3^2 * 5^2 * 13 * 29 * 521) x C(5209) D89046426 x C2233297 = D(2 * 3 * 7^2 * 19^2 * 839) x C(11^2 * 18457) D93899610 x C2086657 = D(2 * 3^2 * 5 * 7 * 113 * 1319) x C(2086657) D94098510 x C21121 = D(2 * 3^4 * 5 * 11 * 59 * 179) x C(21121) D115579926 x C5081 = D(2 * 3^3 * 7^2 * 11^2 * 19^2) x C(5081) D127029870 x C28513 = D(2 * 3^4 * 5 * 11 * 53 * 269) x C(28513) D137425470 x C18433 = D(2 * 3 * 5 * 7 * 13 * 71 * 709) x C(18433) D148328334 x C19457 = D(2 * 3^4 * 7 * 11^2 * 23 * 47) x C(19457) D149696910 x C3169 = D(2 * 3^4 * 5 * 11 * 53 * 317) x C(3169) D170769606 x C389257 = D(2 * 3 * 7^2 * 19^2 * 1609) x C(11^2 * 3217) D176866998 x C12252733 = D(2 * 3 * 7 * 11 * 29 * 43 * 307) x C(17^2 * 42397) D211542450 x C974849 = D(2 * 3 * 5^2 * 7 * 31 * 67 * 97) x C(974849) D216386478 x C4481 = D(2 * 3^4 * 7 * 11^2 * 19 * 83) x C(4481) D232028874 x C1601 = D(2 * 3^4 * 7 * 11^2 * 19 * 89) x C(1601) D255218850 x C630169 = D(2 * 3^4 * 5^2 * 29 * 41 * 53) x C(630169) D278956062 x C641 = D(2 * 3^4 * 7 * 11^2 * 19 * 107) x C(641) D334658142 x C19457 = D(2 * 3^5 * 7^2 * 13 * 23 * 47) x C(19457) D414273150 x C59393 = D(2 * 3^3 * 5^2 * 19 * 31 * 521) x C(59393) D446667390 x C10753 = D(2 * 3^2 * 5 * 13 * 19 * 71 * 283) x C(10753) D488210814 x C4481 = D(2 * 3^5 * 7^2 * 13 * 19 * 83) x C(4481) D523503162 x C1601 = D(2 * 3^5 * 7^2 * 13 * 19 * 89) x C(1601) D556705710 x C23297 = D(2 * 3^5 * 5 * 11 * 59 * 353) x C(23297) D568361970 x C49921 = D(2 * 3^2 * 5 * 11 * 23 * 109 * 229) x C(49921) D617461650 x C1055489 = D(2 * 3^3 * 5^2 * 13 * 151 * 233) x C(1055489) D629380206 x C641 = D(2 * 3^5 * 7^2 * 13 * 19 * 107) x C(641) D734955606 x C130241 = D(2 * 3^4 * 7^2 * 11 * 19 * 443) x C(130241) D836777214 x C7681 = D(2 * 3^2 * 7^2 * 13 * 19 * 23 * 167) x C(7681) D889429086 x C3041 = D(2 * 3^5 * 7^2 * 13^3 * 17) x C(3041) D980514810 x C319489 = D(2 * 3^2 * 5 * 11 * 31 * 43 * 743) x C(319489) D1013127750 x C1500929 = D(2 * 3^7 * 5^3 * 17 * 109) x C(1500929) D1132795170 x C673 = D(2 * 3^2 * 5 * 13^3 * 17 * 337) x C(673) D1375605990 x C5660929 = D(2 * 3^5 * 5 * 11 * 53 * 971) x C(5660929) D1399942950 x C86017 = D(2 * 3 * 5^2 * 7 * 31 * 41 * 1049) x C(86017) D1640161950 x C12289 = D(2 * 3 * 5^2 * 7 * 31 * 41 * 1229) x C(12289) D1860832350 x C4649 = D(2 * 3^2 * 5^2 * 13 * 31^2 * 331) x C(4649) D1890460110 x C8183809 = D(2 * 3 * 5 * 7 * 11 * 503 * 1627) x C(8183809) D2026970010 x C2017 = D(2 * 3^2 * 5 * 13 * 17 * 101 * 1009) x C(2017) D2346457410 x C47881 = D(2 * 3^4 * 5 * 11^2 * 89 * 269) x C(47881) D2793152670 x C414721 = D(2 * 3 * 5 * 7 * 11 * 449 * 2693) x C(414721) D3456092250 x C1489 = D(2 * 3^2 * 5^3 * 13^2 * 61 * 149) x C(1489) D3463796490 x C380929 = D(2 * 3 * 5 * 7 * 11 * 433 * 3463) x C(380929) D3806356554 x C663937 = D(2 * 3^2 * 7^2 * 11 * 13 * 103 * 293) x C(663937) D4093252650 x C251969 = D(2 * 3^3 * 5^2 * 19^2 * 37 * 227) x C(251969) D4886775270 x C5953 = D(2 * 3^2 * 5 * 13^2 * 23 * 61 * 229) x C(5953) D5199328530 x C39937 = D(2 * 3^2 * 5 * 11 * 19 * 263 * 1051) x C(39937) D5380709866 x C218273725 = D(2 * 7 * 31 * 41 * 71 * 4259) x C(5^2 * 8730949) D5393204910 x C6307841 = D(2 * 3^3 * 5 * 19 * 23 * 43 * 1063) x C(6307841) D5407421166 x C26417 = D(2 * 3^5 * 7^2 * 17 * 19^2 * 37) x C(26417) D6142108050 x C10499329 = D(2 * 3^2 * 5^2 * 13 * 47 * 89 * 251) x C(10499329) D6472796022 x C40433 = D(2 * 3^4 * 7^2 * 11^2 * 23 * 293) x C(40433) D6643346490 x C85121 = D(2 * 3^3 * 5 * 11^2 * 43 * 4729) x C(85121) D7438860990 x C455393 = D(2 * 3^4 * 5 * 11^2 * 71 * 1069) x C(455393) D7910331750 x C1850369 = D(2 * 3^3 * 5^3 * 19 * 37 * 1667) x C(1850369) D7919822250 x C1111553 = D(2 * 3^3 * 5^3 * 19 * 37 * 1669) x C(1111553) D9246904758 x C25601 = D(2 * 3^5 * 7^3 * 13 * 17 * 251) x C(25601) D9798637554 x C204150168209 = D(2 * 3^4 * 19 * 37 * 97 * 887) x C(13^2 * 61^2 * 324641)
SmallGroup(6, 1) = (S3) x C(5) SmallGroup(6, 2) = (C6) x C(1) SmallGroup(12, 1) = (C3 : C4) x C(1) SmallGroup(20, 1) = (C5 : C4) x C(19) SmallGroup(24, 3) = (SL(2,3)) x C(5 * 7) SmallGroup(24, 12) = (S4) x C(7 * 41) SmallGroup(28, 1) = (C7 : C4) x C(13) SmallGroup(28, 2) = (C28) x C(1) SmallGroup(48, 28) = (C2 . S4 = SL(2,3) . C2) x C(7 * 83) SmallGroup(48, 29) = (GL(2,3)) x C(7 * 83) SmallGroup(56, 1) = (C7 : C8) x C(1) SmallGroup(56, 11) = ((C2 x C2 x C2) : C7) x C(3 * 5 * 13) SmallGroup(60, 5) = (A5) x C(2^3 * 31 * 61) SmallGroup(72, 3) = (Q8 : C9) x C(5 * 29) SmallGroup(72, 19) = ((C3 x C3) : C8) x C(23) SmallGroup(72, 39) = ((C3 x C3) : C8) x C(17) SmallGroup(80, 49) = ((C2 x C2 x C2 x C2) : C5) x C(3 * 7^2 * 19 * 97) SmallGroup(88, 1) = (C11 : C8) x C(43) SmallGroup(96, 3) = (((C4 x C2) : C4) : C3) x C(7 * 11 * 13) SmallGroup(108, 3) = ((C2 x C2) : C27) x C(5 * 29 * 173) SmallGroup(114, 5) = (D114) x C(7^2 * 97) SmallGroup(120, 5) = (SL(2,5)) x C(3^2 * 7^2 * 13 * 19 * 41) SmallGroup(120, 34) = (S5) x C(7^2 * 13 * 19 * 181) SmallGroup(120, 38) = ((C5 x A4) : C2) x C(19 * 37) SmallGroup(144, 3) = ((C4 x C4) : C9) x C(5 * 19) SmallGroup(144, 114) = ((C3 x C3) : C16) x C(47) SmallGroup(152, 1) = (C19 : C8) x C(37 * 73) SmallGroup(160, 199) = (((C2 x Q8) : C2) : C5) x C(3 * 7 * 13) SmallGroup(168, 42) = (PSL(3,2)) x C(2^2 * 13^2 * 31 * 61) SmallGroup(180, 24) = ((C15 x C3) : C4) x C(29) SmallGroup(180, 25) = ((C15 x C3) : C4) x C(11) SmallGroup(192, 3) = ((C8 x C8) : C3) x C(5 * 7 * 139 * 277) SmallGroup(192, 1023) = ((((C4 x C4) : C2) : C2) : C3) x C(11 * 31) SmallGroup(192, 1025) = (((C2 x C2) . (C2 x C2 x C2 x C2)) : C3) x C(11 * 31) SmallGroup(216, 3) = (Q8 : C27) x C(5 * 359) SmallGroup(220, 5) = (C55 : C4) x C(109) SmallGroup(240, 89) = (C2 . S5 = SL(2,5) . C2) x C(7 * 11^2 * 19) SmallGroup(240, 90) = (SL(2,5) : C2) x C(7 * 11^2 * 19) SmallGroup(240, 191) = (((C2 x C2 x C2 x C2) : C5) : C3) x C(7 * 11) SmallGroup(252, 32) = ((C21 x C3) : C4) x C(17) SmallGroup(270, 3) = (D270) x C(17) SmallGroup(272, 1) = (C17 : C16) x C(271) SmallGroup(288, 3) = (((C4 x C2) : C4) : C9) x C(7 * 17) SmallGroup(288, 67) = (((C4 x C4) : C9) : C2) x C(11 * 43) SmallGroup(288, 836) = (((C2 x C2 x C2 x C2) : C9) : C2) x C(23 * 137) SmallGroup(304, 1) = (C19 : C16) x C(151) SmallGroup(320, 1581) = (C2 . (((C2 x C2 x C2 x C2) : C5) : C2) = (((C2 x Q8) : C2) : C5) . C2) x C(7 * 13 * 103) SmallGroup(320, 1582) = ((((C2 x Q8) : C2) : C5) : C2) x C(7 * 13 * 103) SmallGroup(336, 114) = (SL(2,7)) x C(3^2 * 5 * 13 * 19 * 113) SmallGroup(336, 118) = (C7 : (C2 . S4 = SL(2,3) . C2)) x C(11) SmallGroup(336, 119) = ((C7 x SL(2,3)) : C2) x C(11) SmallGroup(360, 41) = ((C5 x ((C2 x C2) : C9)) : C2) x C(29) SmallGroup(360, 57) = ((C15 x C3) : C8) x C(1) SmallGroup(360, 118) = (A6) x C(2^3 * 19^2 * 127) SmallGroup(360, 125) = ((C15 x C3) : C8) x C(59) SmallGroup(380, 3) = (C95 : C4) x C(37) SmallGroup(384, 3) = (((C8 x C2) : C8) : C3) x C(5^2 * 23 * 31) SmallGroup(384, 568) = (((C8 x C8) : C3) : C2) x C(7 * 83 * 331 * 661) SmallGroup(384, 569) = (C2 . ((((C4 x C2) : C4) : C3) : C2) = ((C4 . (C4 x C4)) : C3) . C2) x C(11 * 31 * 61) SmallGroup(384, 570) = (((C4 . (C4 x C4)) : C3) : C2) x C(11 * 31 * 61) SmallGroup(384, 5859) = ((C2 x C2 x ((C4 x C2) : C4)) : C3) x C(47 * 751) SmallGroup(384, 5868) = ((((C2 x C2 x Q8) : C2) : C2) : C3) x C(19^2 * 37 * 127) SmallGroup(384, 5870) = (((C4 x C4) : Q8) : C3) x C(19^2 * 37 * 127) SmallGroup(384, 5871) = (((C2 x C2 x C2) . (C2 x C2 x C2 x C2)) : C3) x C(19^2 * 37 * 127) SmallGroup(408, 37) = ((C17 x A4) : C2) x C(7) SmallGroup(432, 3) = ((C4 x C4) : C27) x C(5 * 47) SmallGroup(432, 233) = (((C3 x C3) : C3) : C16) x C(71 * 283) SmallGroup(432, 734) = ((((C3 x C3) : Q8) : C3) : C2) x C(11 * 17) SmallGroup(448, 1394) = ((C2 x C2 x C2 x C2 x C2 x C2) : C7) x C(3 * 5 * 23^2 * 79) SmallGroup(480, 217) = (A5 : C8) x C(19 * 227 * 907) SmallGroup(480, 219) = (SL(2,5) : C4) x C(19 * 23 * 37) SmallGroup(480, 1201) = ((C5 x ((C2 x C2 x C2 x C2) : C3)) : C2) x C(79 * 157) SmallGroup(496, 1) = (C31 : C16) x C(61) SmallGroup(496, 2) = (C496) x C(1) SmallGroup(504, 55) = ((C7 x ((C2 x C2) : C9)) : C2) x C(17) SmallGroup(504, 156) = (PSL(2,8)) x C(3^2 * 5 * 13 * 17 * 101) SmallGroup(540, 31) = (C9 x A5) x C(7^2 * 13^2 * 19 * 61^2 * 97) SmallGroup(560, 170) = (((C2 x C2 x C2) : C7) x D10) x C(13) SmallGroup(576, 3) = ((C8 x C8) : C9) x C(5 * 23 * 229) SmallGroup(576, 4) = ((C4 . (C4 x C4)) : C9) x C(5 * 239) SmallGroup(576, 183) = (C2 . (((C4 x C4) : C9) : C2) = (((C4 x C2) : C4) : C9) . C2) x C(11 * 263) SmallGroup(576, 184) = ((((C4 x C2) : C4) : C9) : C2) x C(11 * 263) SmallGroup(576, 4987) = ((C2 x C2) : (C2 . (((C2 x C2) : C9) : C2) = (Q8 : C9) . C2)) x C(31) SmallGroup(576, 4988) = (((C2 x C2 x Q8) : C9) : C2) x C(31) SmallGroup(624, 256) = ((C26 x C2 x C2 x C2) : C3) x C(11 * 571) SmallGroup(672, 1047) = (SL(2,7) : C2) x C(5^2 * 29 * 31) SmallGroup(672, 1201) = ((C2 x C14 x Q8) : C3) x C(41 * 163) SmallGroup(672, 1257) = ((C2 x C2 x ((C2 x C2 x C2) : C7)) : C3) x C(5^3 * 13 * 499 * 997) SmallGroup(672, 1265) = ((C7 x ((C2 x C2 x C2 x C2) : C3)) : C2) x C(23) SmallGroup(720, 105) = (C5 : (C2 . (((C2 x C2) : C9) : C2) = (Q8 : C9) . C2)) x C(59) SmallGroup(720, 106) = ((C5 x (Q8 : C9)) : C2) x C(59) SmallGroup(720, 766) = (C2 x A6) x C(7^2 * 13 * 19^3 * 181) SmallGroup(760, 6) = (C95 : C8) x C(1) SmallGroup(768, 1083477) = ((C16 x C16) : C3) x C(5^2 * 19 * 31 * 37 * 1109) SmallGroup(768, 1083478) = ((C2 . ((C2 x ((C4 x C2) : C2)) : C4) = (C2 x C2 x C2 x C2) . (C4 x C4)) : C3) x C(5 * 19 * 31 * 37) SmallGroup(768, 1083479) = (((((C4 x C2) : C4) : C2) : C4) : C3) x C(5 * 19 * 31 * 37) SmallGroup(792, 47) = ((C33 x C3) : C8) x C(1) SmallGroup(800, 202) = (((C2 x Q8) : C2) : C25) x C(3^3 * 17 * 101) SmallGroup(840, 134) = (C5 x PSL(3,2)) x C(3^2 * 13^3 * 17) SmallGroup(864, 73) = (((C4 x C4) : C27) : C2) x C(11 * 131 * 523) SmallGroup(864, 2666) = (((C2 x ((C3 x C3) : C4)) : C4) : C3) x C(5 * 17 * 19) SmallGroup(889, 1) = (C127 : C7) x C(3^4 * 11^2 * 19^2 * 113) SmallGroup(896, 564) = ((C2 . ((C2 x C2 x C2) . (C2 x C2 x C2)) = (C2 x C2 x C2 x C2) . (C2 x C2 x C2)) : C7) x C(3 * 5^2 * 31 * 149) SmallGroup(900, 120) = ((C15 x C15) : C4) x C(19) SmallGroup(900, 121) = ((C15 x C15) : C4) x C(1) SmallGroup(920, 6) = (C115 : C8) x C(229) SmallGroup(960, 641) = (C16 . A5 = SL(2,5) . C8) x C(23 * 229 * 1831) SmallGroup(960, 11357) = ((C2 x C2 x C2 x C2) : A5) x C(3^2 * 7^2 * 13 * 19 * 41 * 163 * 977) SmallGroup(960, 11358) = ((C2 x C2 x C2 x C2) : A5) x C(3^2 * 7^2 * 13 * 19 * 41 * 163 * 977) SmallGroup(992, 1) = (C31 : C32) x C(1) SmallGroup(992, 194) = ((C2 x C2 x C2 x C2 x C2) : C31) x C(3 * 5^2 * 7 * 41) SmallGroup(1008, 57) = ((C28 x C4) : C9) x C(5) SmallGroup(1080, 53) = ((C5 x ((C2 x C2) : C27)) : C2) x C(59 * 353) SmallGroup(1080, 260) = (C3 . A6) x C(2^3 * 17 * 271) SmallGroup(1080, 267) = ((C5 x ((C3 x C3) : C3)) : C8) x C(71) SmallGroup(1088, 1631) = (C17 : C64) x C(1087) SmallGroup(1152, 153314) = (((C8 x C2) : C8) : C9) x C(5 * 479) SmallGroup(1152, 153931) = (((C8 x C8) : C9) : C2) x C(11 * 47) SmallGroup(1152, 153932) = (C2 . ((((C4 x C2) : C4) : C9) : C2) = ((C4 . (C4 x C4)) : C9) . C2) x C(17 * 31) SmallGroup(1152, 153933) = (((C4 . (C4 x C4)) : C9) : C2) x C(17 * 31) SmallGroup(1152, 154457) = (((C4 x C4 x C2 x C2) : C9) : C2) x C(31 * 557) SmallGroup(1152, 154458) = ((C2 x C2 x C2) : (C2 . (((C2 x C2) : C9) : C2) = (Q8 : C9) . C2)) x C(47 * 563) SmallGroup(1152, 154459) = ((C2 x C2) . (((C2 x C2 x C2 x C2) : C9) : C2) = ((((C2 x Q8) : C2) : C2) : C9) . C2) x C(47 * 563) SmallGroup(1152, 154460) = ((C2 x C2) . (((C2 x C2 x C2 x C2) : C9) : C2) = ((((C2 x Q8) : C2) : C2) : C9) . C2) x C(47 * 563) SmallGroup(1152, 154461) = (((((C2 x Q8) : C2) : C2) : C9) : C2) x C(47 * 563) SmallGroup(1152, 154468) = (((((C4 x C4) : C2) : C2) : C9) : C2) x C(31 * 61) SmallGroup(1152, 154469) = (((((C4 x C4) : C2) : C2) : C9) : C2) x C(31 * 61) SmallGroup(1152, 157673) = (((C2 x ((C2 x Q8) : C2)) : C2) : C9) x C(5 * 7 * 139 * 1667) SmallGroup(1170, 11) = (D1170) x C(29) SmallGroup(1176, 182) = ((C7 x C7 x Q8) : C3) x C(293 * 1171) SmallGroup(1200, 99) = (C25 : (C2 . S4 = SL(2,3) . C2)) x C(23 * 229 * 457) SmallGroup(1200, 100) = ((C25 x SL(2,3)) : C2) x C(23 * 229 * 457) SmallGroup(1200, 680) = ((C5 x C5) : (C2 . S4 = SL(2,3) . C2)) x C(7 * 149) SmallGroup(1200, 681) = (((C5 x C5 x Q8) : C3) : C2) x C(7 * 149) SmallGroup(1224, 58) = ((C51 x C3) : C8) x C(67) SmallGroup(1260, 91) = ((C105 x C3) : C4) x C(41) SmallGroup(1280, 1116360) = (C4 . (((C2 x C2 x C2 x C2) : C5) : C4) = (((C2 x Q8) : C2) : C5) . C8) x C(79 * 157 * 313 * 2503) SmallGroup(1296, 3) = ((C4 x C4) : C81) x C(5 * 89) SmallGroup(1296, 3081) = ((C3 x C3 x C3) : (C2 . S4 = SL(2,3) . C2)) x C(11 * 17) SmallGroup(1296, 3082) = ((((C3 x C3 x C3) : Q8) : C3) : C2) x C(11 * 17) SmallGroup(1296, 3084) = ((C3 x C3 x C3) : (C2 . S4 = SL(2,3) . C2)) x C(11 * 17) SmallGroup(1296, 3085) = ((((C3 x C3 x C3) : Q8) : C3) : C2) x C(11 * 17) SmallGroup(1296, 3098) = ((C3 x C3 x C3 x C3) : C16) x C(107) SmallGroup(1320, 13) = (SL(2,11)) x C(3^4 * 7^2 * 11 * 19) SmallGroup(1344, 814) = ((C2 x C2 x C2) . PSL(3,2)) x C(3^2 * 7 * 11 * 13 * 41) SmallGroup(1344, 827) = (C7 : (C2 . (((C4 x C4) : C3) : C2) = (((C4 x C2) : C4) : C3) . C2)) x C(13 * 311) SmallGroup(1344, 828) = ((C7 x (((C4 x C2) : C4) : C3)) : C2) x C(13 * 311) SmallGroup(1344, 6311) = (PSL(3,2) : C8) x C(19^2 * 127) SmallGroup(1344, 6312) = (C4 . (PSL(3,2) : C2) = SL(2,7) . C4) x C(13^2 * 31 * 61 * 337) SmallGroup(1344, 6316) = (C4 x SL(2,7)) x C(13 * 31 * 61) SmallGroup(1344, 10093) = ((C7 x (((C2 x Q8) : C2) : C2)) : C3) x C(83 * 331) SmallGroup(1344, 11686) = ((C2 x C2 x C2) : PSL(3,2)) x C(3^2 * 7 * 11 * 13 * 41) SmallGroup(1368, 82) = ((C19 x Q8) : C9) x C(5 * 37 * 443) SmallGroup(1404, 14) = ((C26 x C2) : C27) x C(5 * 89) SmallGroup(1404, 160) = (((C3 x C3 x C3) : C13) : C4) x C(17 * 883) SmallGroup(1440, 1542) = (C9 x (((C2 x Q8) : C2) : C5)) x C(13^2 * 31 * 61) SmallGroup(1440, 4591) = (SL(2,9) : C2) x C(7^2 * 13 * 19 * 103) SmallGroup(1440, 4593) = (SL(2,9) : C2) x C(7^2 * 13 * 19 * 103) SmallGroup(1440, 4594) = (C2 . (A6 : C2) = SL(2,9) . C2) x C(7^2 * 13 * 19 * 103) SmallGroup(1440, 4597) = (SL(2,9) : C2) x C(7 * 11^2 * 19 * 197) SmallGroup(1440, 4598) = (C2 x SL(2,9)) x C(7^2 * 13 * 19 * 167) SmallGroup(1440, 4615) = (((SL(2,5) : C2) : C2) : C3) x C(5 * 7 * 83) SmallGroup(1440, 5847) = ((A4 x A5) : C2) x C(7 * 59 * 2477) SmallGroup(1540, 21) = (C385 : C4) x C(769) SmallGroup(1600, 1015) = (((C2 x C2) . (C2 x C2 x C2 x C2)) : C25) x C(3 * 7 * 41) SmallGroup(1600, 9876) = (D10 x (((C2 x Q8) : C2) : C5)) x C(79 * 157 * 313) SmallGroup(1638, 23) = (D1638) x C(17) SmallGroup(1680, 572) = (C35 : (C2 . S4 = SL(2,3) . C2)) x C(1) SmallGroup(1680, 573) = ((C35 x SL(2,3)) : C2) x C(1) SmallGroup(1728, 20789) = (((C3 x C3) : (C4 . (C4 x C4))) : C3) x C(5 * 11 * 59 * 1297) SmallGroup(1728, 46097) = (((C2 x C6 x A4) : C3) : C4) x C(11 * 43 * 773) SmallGroup(1768, 3) = (C221 : C8) x C(883) SmallGroup(1776, 134) = (C37 : (C2 . S4 = SL(2,3) . C2)) x C(7 * 1553) SmallGroup(1776, 135) = ((C37 x SL(2,3)) : C2) x C(7 * 1553) SmallGroup(1776, 256) = ((C74 x C2 x C2 x C2) : C3) x C(11 * 43) SmallGroup(1800, 602) = ((C15 x C15) : C8) x C(29 * 173) SmallGroup(1824, 1192) = ((C2 x C38 x Q8) : C3) x C(23 * 37) SmallGroup(1836, 42) = ((C17 x ((C3 x C3) : C3)) : C4) x C(11) SmallGroup(1872, 619) = ((C39 x C3) : C16) x C(311) SmallGroup(1920, 237225) = (C5 : (C2 . ((((C4 x C2) : C4) : C3) : C2) = ((C4 . (C4 x C4)) : C3) . C2)) x C(19 * 607) SmallGroup(1920, 237226) = ((C5 x ((C4 . (C4 x C4)) : C3)) : C2) x C(19 * 607) SmallGroup(1920, 239654) = ((C5 x ((Q8 x Q8) : C3)) : C2) x C(79 * 631) SmallGroup(1920, 239656) = (C5 : ((C2 x C2) . (((C2 x C2 x C2 x C2) : C3) : C2) = ((((C4 x C4) : C2) : C2) : C3) . C2)) x C(127) SmallGroup(1920, 239657) = ((C5 x ((((C4 x C4) : C2) : C2) : C3)) : C2) x C(79 * 631) SmallGroup(1920, 239660) = ((C5 x ((((C4 x C4) : C2) : C2) : C3)) : C2) x C(127) SmallGroup(1920, 240998) = ((C2 x C2 x C2 x C2 x C2) : A5) x C(3^2 * 7 * 13 * 17 * 23) SmallGroup(1920, 240999) = (C2 . ((C2 x C2 x C2 x C2) : A5)) x C(3^2 * 7 * 13 * 17 * 23) SmallGroup(1920, 241000) = ((C2 x C2 x C2 x C2) : SL(2,5)) x C(3^2 * 7 * 13 * 19 * 37 * 73) SmallGroup(1920, 241001) = ((C2 x C2 x C2 x C2 x C2) . A5) x C(3^2 * 7 * 11 * 13 * 179) SmallGroup(1920, 241002) = ((C2 x C2 x C2 x C2) : SL(2,5)) x C(3^2 * 7 * 13 * 19 * 37 * 73) SmallGroup(1920, 241003) = (C2 . ((C2 x C2 x C2 x C2) : A5)) x C(3^2 * 7 * 13 * 17 * 23) SmallGroup(1920, 241004) = (((C2 x Q8) : C2) : A5) x C(3^2 * 7 * 13 * 17 * 23)
(alt 5) (lie a 1 4) (lie a 1 5) x C(2^3 * 31 * 61) (alt 6) (lie a 1 9) x C(2^3 * 19^2 * 127) (alt 7) x C(2 * 7^2 * 13 * 19 * 97 * 2521) (alt 10) x C(2^2 * 19 * 23 * 29 * 173 * 78887) (lie 2a 2 9) (tits g 2 2) x C(2^2 * 11 * 23 * 263) (lie 2a 5 4) x C(2 * 7 * 13 * 23 * 43 * 71 * 2897 * 11587 * 23173 * 30529) (lie a 1 7) (lie a 2 2) x C(2^2 * 13^2 * 31 * 61) (lie a 1 8) (tits 2g 2 3) x C(3^2 * 5 * 13 * 17 * 101) (lie a 2 2) x C(2^2 * 13^2 * 31 * 61) (lie a 5 2) x C(3 * 5^2 * 13 * 19 * 37 * 47 * 73 * 389 * 1021 * 12689 * 10079354881 * 20158709761) (lie b 2 3) (lie 2a 3 4) x C(2 * 7^4 * 13 * 23^2 * 79 * 467 * 2801) (lie c 2 4) x C(3^2 * 7^2 * 13^3 * 19 * 23 * 3041 * 979201) (lie c 3 2) x C(3 * 5 * 7^2 * 19 * 59 * 293 * 2477 * 1451521) (lie g 2 4) x C(3^2 * 5 * 11 * 23 * 137 * 307 * 499 * 997 * 1836473) (spor-m22) x C(2 * 5 * 13 * 79 * 109 * 157 * 313)
ZM(3, 2, alpha=2) x C(5) ZM(3, 2^2, alpha=2) x C(1) ZM(5, 2^2, alpha=2) x C(19) ZM(7, 2^2, alpha=2) x C(13) ZM(7, 2^3, alpha=2) x C(1) ZM(11, 2^3, alpha=2) x C(43) ZM(17, 2^4, alpha=2) x C(271) ZM(17, 2^6, alpha=8) x C(1087) ZM(19, 2^3, alpha=2) x C(37 * 73) ZM(19, 2^4, alpha=2) x C(151) ZM(31, 2^4, alpha=2) x C(61) ZM(31, 2^5, alpha=2) x C(1) ZM(67, 2^6, alpha=2) x C(2143) ZM(79, 2^5, alpha=2) x C(157 * 313) ZM(79, 2^6, alpha=2) x C(631) ZM(127, 7, alpha=7) x C(3^4 * 11^2 * 19^2 * 113) ZM(127, 2^7, alpha=2) x C(1) ZM(139, 2^7, alpha=2) x C(4447 * 8893) ZM(257, 2^10, alpha=8) x C(263167) ZM(257, 2^10, alpha=32) x C(4111 * 8221 * 263071) ZM(263, 2^8, alpha=2) x C(16831) ZM(271, 2^7, alpha=2) x C(541 * 4327) ZM(307, 2^3, alpha=2) x C(17^2 * 577 * 1153) ZM(383, 2^8, alpha=2) x C(1531) ZM(1279, 2^9, alpha=2) x C(2557 * 5113) ZM(2063, 2^11, alpha=2) x C(528127) ZM(4111, 2^11, alpha=2) x C(8221 * 1052287) ZM(4157, 2^12, alpha=2) x C(1064191 * 2128381 * 4256761 * 8513521 * 17027041) ZM(6143, 2^12, alpha=2) x C(24571) ZM(8191, 2^12, alpha=2) x C(16381) ZM(8191, 2^13, alpha=2) x C(1) ZM(8719, 2^13, alpha=2) x C(279007 * 8928223) ZM(32831, 2^15, alpha=2) x C(33618943) ZM(33791, 2^15, alpha=2) x C(2162623) ZM(34819, 2^15, alpha=2) x C(1114207 * 570473983) ZM(65537, 2^16, alpha=2) x C(4295032831) ZM(65537, 2^18, alpha=8) x C(17180131327) ZM(65537, 2^26, alpha=2048) x C(4398113619967) ZM(65537, 2^30, alpha=32768) x C(70369817919487) ZM(65551, 2^16, alpha=2) x C(536993791) ZM(81919, 2^16, alpha=2) x C(655351) ZM(131071, 2^17, alpha=2) x C(1) ZM(262147, 2^18, alpha=2) x C(34360131583) ZM(524287, 2^18, alpha=2) x C(1048573) ZM(524287, 2^19, alpha=2) x C(1) ZM(1048583, 2^20, alpha=2) x C(274879741951) ZM(1048703, 2^20, alpha=2) x C(17181949951) ZM(1114111, 2^19, alpha=2) x C(2228221 * 17825767) ZM(1310719, 2^20, alpha=2) x C(10485751) ZM(2622511, 2^20, alpha=2) x C(5245021 * 10490041 * 5370900991 * 171868831711 * 343737663421) ZM(16785407, 2^24, alpha=2) x C(68753027071)
ZM(3, 2^2, -1) x C(1) ZM(5, 2^2, -1) x C(19) ZM(7, 2^2, -1) x C(13) ZM(7, 2^3, -1) x C(1) ZM(11, 2^3, -1) x C(43) ZM(17, 2^4, -1) x C(271) ZM(19, 2^3, -1) x C(37 * 73) ZM(19, 2^4, -1) x C(151) ZM(31, 2^4, -1) x C(61) ZM(31, 2^5, -1) x C(1) ZM(5 * 11, 2^2, -1) x C(109) ZM(67, 2^6, -1) x C(2143) ZM(79, 2^5, -1) x C(157 * 313) ZM(79, 2^6, -1) x C(631) ZM(5 * 19, 2^2, -1) x C(37) ZM(127, 2^7, -1) x C(1) ZM(139, 2^7, -1) x C(4447 * 8893) ZM(13 * 17, 2^3, -1) x C(883) ZM(263, 2^8, -1) x C(16831) ZM(271, 2^7, -1) x C(541 * 4327) ZM(307, 2^3, -1) x C(17^2 * 577 * 1153) ZM(383, 2^8, -1) x C(1531) ZM(23 * 47, 2^4, -1) x C(2161) ZM(1279, 2^9, -1) x C(2557 * 5113) ZM(19 * 79, 2^4, -1) x C(3001) ZM(2063, 2^11, -1) x C(528127) ZM(5 * 11 * 59, 2^2, -1) x C(1297) ZM(5 * 19 * 37, 2^2, -1) x C(73) ZM(4111, 2^11, -1) x C(8221 * 1052287) ZM(4157, 2^12, -1) x C(1064191 * 2128381 * 4256761 * 8513521 * 17027041) ZM(17 * 271, 2^4, -1) x C(541) ZM(41 * 131, 2^5, -1) x C(42967) ZM(6143, 2^12, -1) x C(24571) ZM(41 * 163, 2^5, -1) x C(1303) ZM(8191, 2^12, -1) x C(16381) ZM(8191, 2^13, -1) x C(1) ZM(8719, 2^13, -1) x C(279007 * 8928223) ZM(7^2 * 13 * 19, 2^2, -1) x C(181) ZM(7 * 11^2 * 19, 2^2, -1) x C(241) ZM(109 * 151, 2^6, -1) x C(131671) ZM(13 * 23 * 59, 2^3, -1) x C(35281) ZM(5^2 * 19 * 47, 2^2, -1) x C(8929) ZM(83 * 331, 2^6, -1) x C(2647) ZM(32831, 2^15, -1) x C(33618943) ZM(33791, 2^15, -1) x C(2162623) ZM(34819, 2^15, -1) x C(1114207 * 570473983) ZM(19 * 2053, 2^4, -1) x C(197^2 * 787) ZM(65537, 2^16, -1) x C(4295032831) ZM(65551, 2^16, -1) x C(536993791) ZM(11 * 23 * 263, 2^3, -1) x C(12097) ZM(199 * 353, 2^7, -1) x C(1123951) ZM(81919, 2^16, -1) x C(655351) ZM(17^2 * 307, 2^4, -1) x C(2311) ZM(37 * 47 * 59, 2^4, -1) x C(205201) ZM(13 * 17 * 467, 2^3, -1) x C(15877) ZM(67 * 1607, 2^6, -1) x C(6427) ZM(37 * 41 * 73, 2^4, -1) x C(11971) ZM(131071, 2^17, -1) x C(1) ZM(5^3 * 19 * 67, 2^2, -1) x C(63649) ZM(5 * 17 * 31 * 61, 2^2, -1) x C(10369) ZM(7 * 79 * 307, 2^2, -1) x C(17^2 * 157) ZM(23 * 59 * 193, 2^4, -1) x C(523801) ZM(479 * 547, 2^8, -1) x C(1048051) ZM(262147, 2^18, -1) x C(34360131583) ZM(359 * 883, 2^8, -1) x C(5071951) ZM(13^2 * 31 * 61, 2^3, -1) x C(337) ZM(524287, 2^18, -1) x C(1048573) ZM(524287, 2^19, -1) x C(1) ZM(439 * 1297, 2^8, -1) x C(2593 * 20743 * 18212353) ZM(383 * 1531, 2^8, -1) x C(3061) ZM(5 * 11 * 79 * 173, 2^2, -1) x C(300673) ZM(281 * 2767, 2^8, -1) x C(12440431) ZM(131 * 6287, 2^7, -1) x C(25147) ZM(277 * 3323, 2^8, -1) x C(212671) ZM(1048583, 2^20, -1) x C(274879741951) ZM(1048703, 2^20, -1) x C(17181949951) ZM(1114111, 2^19, -1) x C(2228221 * 17825767) ZM(7^2 * 13 * 19 * 97, 2^2, -1) x C(2521) ZM(271 * 4339, 2^8, -1) x C(9406951) ZM(23 * 47 * 1103, 2^4, -1) x C(103681) ZM(1310719, 2^20, -1) x C(10485751) ZM(271 * 4877, 2^8, -1) x C(78031) ZM(5 * 11 * 61 * 487, 2^2, -1) x C(53569) ZM(5 * 11 * 59 * 659, 2^2, -1) x C(77761) ZM(19 * 23 * 29 * 173, 2^3, -1) x C(151201) ZM(263 * 8419, 2^8, -1) x C(35427151) ZM(263 * 8447, 2^8, -1) x C(4443121) ZM(17 * 19 * 61 * 113, 2^3, -1) x C(234361) ZM(263 * 9467, 2^8, -1) x C(151471) ZM(2622511, 2^20, -1) x C(5245021 * 10490041 * 5370900991 * 171868831711 * 343737663421) ZM(47 * 109 * 653, 2^5, -1) x C(61381) ZM(5^2 * 29 * 31 * 149, 2^2, -1) x C(8641) ZM(17 * 149 * 1447, 2^4, -1) x C(7330501) ZM(11 * 41 * 67 * 131, 2^3, -1) x C(719713) ZM(5 * 11 * 59 * 1297, 2^2, -1) x C(2593) ZM(13 * 23 * 79 * 233, 2^3, -1) x C(846721) ZM(23^2 * 79 * 137, 2^4, -1) x C(6301) ZM(41 * 163 * 977, 2^5, -1) x C(3907) ZM(7 * 11 * 17^2 * 307, 2^2, -1) x C(577) ZM(257 * 32911, 2^8, -1) x C(135330031) ZM(7 * 11 * 29 * 41 * 109, 2^2, -1) x C(259201) ZM(1151 * 9209, 2^10, -1) x C(84796471) ZM(2311 * 4621, 2^10, -1) x C(9241 * 18481 * 591391) ZM(17 * 139 * 4759, 2^4, -1) x C(1323001) ZM(17 * 139 * 5003, 2^4, -1) x C(170101) ZM(37 * 577 * 619, 2^5, -1) x C(1153 * 52814317) ZM(521 * 27091, 2^9, -1) x C(3467647) ZM(16785407, 2^24, -1) x C(68753027071) ZM(19^2 * 127 * 379, 2^4, -1) x C(14401) ZM(3061 * 6229, 2^10, -1) x C(6121 * 12457 * 783487) ZM(2687 * 8599, 2^11, -1) x C(92422051) ZM(5 * 11^2 * 139 * 277, 2^2, -1) x C(67033) ZM(37 * 53 * 73 * 179, 2^4, -1) x C(1385101) ZM(7^2 * 13^2 * 19 * 181, 2^2, -1) x C(32941) ZM(7^2 * 13 * 19^2 * 127, 2^2, -1) x C(3457) ZM(307 * 353 * 367, 2^3, -1) x C(17^2 * 4404733) ZM(5 * 17 * 31 * 101 * 151, 2^2, -1) x C(4727809) ZM(1049 * 41959, 2^10, -1) x C(5370751) ZM(5^2 * 17 * 67 * 1741, 2^2, -1) x C(87049) ZM(6991 * 9887, 2^12, -1) x C(1105920271) ZM(11 * 23 * 379 * 757, 2^3, -1) x C(383041) ZM(6151 * 12301, 2^12, -1) x C(6298111) ZM(2207 * 35311, 2^11, -1) x C(282487) ZM(6271 * 12541, 2^12, -1) x C(401311) ZM(71 * 1021 * 1277, 2^6, -1) x C(5215267)