Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #19128628⋊D3 |
| Fitting, socle, derived | C3×C7×C127 ⋊ C22×C3 |
| Hall | C3×#14581888 ⋊ C22 |
| Hall | C7×C127 ⋊ C2×C3×D3 |
| Sylow | C127 ⋊ D3×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C2×C3⋊(D127⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C127 | C22×C3 |
| LCS | G ▷ C3×C7×C127 | |
| UCS | C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 2667 |
A minimal faithful permutation representation — 2 generators on 137 points:
(1,2,3)(5,8,9,6,7,10)(11,12,18)(13,22,34)(14,16,23)(15,26,17)(19,35,51)(20,36,33)(21,39,25)(24,48,30)(27,52,74)(28,53,47)(29,58,38)(31,49,32)(37,71,43)(40,75,102)(41,76,66)(42,81,57)(44,69,55)(45,50,72)(46,54,67)(56,99,62)(59,103,128)(60,104,88)(61,109,80)(63,93,78)(64,73,100)(65,91,90)(68,77,95)(70,97,89)(79,126,84)(83,129,108)(85,118,106)(86,101,127)(87,121,114)(92,105,124)(94,123,116)(96,115,120)(98,125,113)(107,112,110)(111,135,131)(117,130,132)(119,137,134)(122,133,136);(2,3)(4,5,7,10,9,8,6)(12,13)(14,16)(15,19)(17,18)(20,24)(21,27)(22,30)(23,32)(25,34)(26,36)(28,37)(29,40)(31,43)(33,45)(35,46)(38,51)(39,54)(41,56)(42,59)(44,62)(47,64)(48,55)(49,53)(50,68)(52,65)(57,74)(58,77)(60,79)(61,82)(63,84)(66,86)(67,89)(69,91)(70,92)(71,78)(72,76)(73,96)(75,87)(80,102)(81,105)(83,107)(85,110)(88,112)(90,113)(93,115)(94,117)(95,116)(97,121)(98,122)(99,106)(100,104)(101,119)(103,111)(108,128)(109,130)(114,132)(118,133)(120,136)(123,137)(124,134)(125,135)(126,131)(127,129)Polycyclic presentation:
2^2,3^2,7,127|d^a=d^2,f^a=f^126,e^b=e^6,e^c=e^2,f^c=f^107336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 201);