Finite Group DatabasebetaFinite Group Databasebeta| order | 32046 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C109⋊C3)⋊D49 |
| Fitting, derived, Hall | C49×C109 ⋊ C2×C3 |
| Hall | C49×C109⋊C3 ⋊ C2 |
| Hall | D5341 ⋊ C3 |
| Hall | C109⋊C3 ⋊ D49 |
| Sylow | C109 ⋊ C3×D49 |
| Sylow | C49 ⋊ D109⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C49×C109 | C2×C3 |
| Frattini | C7 | (C109⋊C3)⋊D7 |
| socle | C7×C109 | C3×D7 |
| LCS | G ▷ C49×C109 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 158 |
| transitive degree | μt(G) = 5341 |
A minimal faithful permutation representation — 2 generators on 158 points:
(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,20)(12,23)(14,25)(17,29)(18,28)(19,27)(21,22)(24,34)(26,36)(30,37)(31,39)(32,38)(33,42)(35,44)(40,45)(41,46)(43,48)(47,49)(51,55,56,52,53,57)(54,63,64,58,59,73)(60,76,77,66,67,90)(61,75,71,69,65,62)(68,95,96,79,80,113)(70,93,94,82,78,87)(72,74,92,84,85,88)(81,118,119,98,99,140)(83,116,117,101,97,110)(86,91,115,103,104,111)(89,129,127,107,108,106)(100,145,146,121,122,158)(102,143,144,123,120,137)(105,114,142,124,125,138)(109,150,151,130,131,133)(112,154,148,134,135,128)(126,141,152,147,132,157)(136,156,149,139,155,153);(1,15,39,49,41,19,4,7,28,45,43,32,10,13,16,37,35,33,20,23,25,29,26,24,22,21,34,36,17,14,12,11,42,44,30,8,6,5,38,48,40,18,3,2,27,46,47,31,9)(50,51,54,60,68,81,100,119,142,154,150,143,118,141,130,107,85,59,65,73,87,106,128,149,151,117,77,92,93,76,91,112,136,125,80,97,108,111,113,137,147,124,103,84,71,56,55,61,70,83,102,122,146,152,155,156,157,158,145,123,101,82,69,53,57,62,72,86,105,126,144,96,115,129,116,95,114,139,134,104,67,78,88,90,110,133,153,148,127,94,64,75,63,74,89,109,132,99,120,131,135,138,140,121,98,79,66,58,52)Polycyclic presentation:
2,3,7,109,7|c^7=e^6,c^a=c^6e,d^a=d^108,e^a=e^6,d^b=d^63960 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32046, 21);