Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C3⋊D3)⋊(D127⋊C7) |
| unambiguous semidirect product | ((C3⋊D3)⋊D127)⋊C7 |
| unambiguous semidirect product | (C3⋊(D127⋊C7))⋊D3 |
| Fitting, socle, derived, Hall | C32×C127 ⋊ C22×C7 |
| Hall | C32×C127⋊C7 ⋊ C22 |
| Hall | (C3⋊D3)⋊D127 ⋊ C7 |
| Hall | C127⋊C7 ⋊ D32 |
| Sylow | C127 ⋊ D32×C7 |
| Sylow | C32 ⋊ C2×D127⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C127 | C22×C7 |
| LCS | G ▷ C32×C127 | |
| UCS | C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 762 |
A minimal faithful permutation representation — 2 generators on 133 points:
(2,3)(4,5)(7,8,14,38,97,113,42)(9,19,54,66,86,53,29)(10,21,58,100,108,25,69)(11,27,76,67,131,68,26)(12,30,34,61,125,90,73)(13,32,41,107,77,99,72)(15,43,105,23,64,85,101)(16,44,24,70,133,40,103)(17,49,119,128,75,18,20)(22,63,82,95,118,62,55)(28,80,89,132,59,120,130)(31,37,51,50,87,45,110)(33,84,81,92,93,116,57)(35,91,129,121,39,102,46)(36,48,56,79,126,123,111)(47,114,112,106,98,78,74)(52,60,109,83,122,104,71)(65,115,117,124,96,94,88);(1,2,3)(5,6)(8,9)(10,12)(11,15)(13,18)(14,22)(16,25)(17,19)(20,29)(21,33)(23,36)(24,39)(26,42)(27,45)(28,31)(30,50)(32,55)(34,57)(35,59)(37,62)(38,65)(40,68)(41,43)(44,73)(46,75)(47,77)(48,53)(49,81)(51,84)(52,83)(54,79)(56,87)(58,88)(60,90)(61,63)(64,94)(66,96)(67,98)(69,101)(70,104)(71,72)(74,108)(76,111)(78,113)(80,116)(82,118)(85,120)(86,115)(89,122)(91,123)(92,93)(95,126)(97,127)(99,129)(100,102)(103,121)(105,132)(106,133)(107,130)(109,110)(112,131)(114,128)(117,125)(119,124)Polycyclic presentation:
2^2,7,3^2,127|e^a=e^2,f^a=f^126,d^b=d^2,e^b=e^2,f^c=f^4117 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 169);