Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #19128629⋊D3 |
| Fitting, socle, derived | C3×C7×C127 ⋊ C22×C3 |
| Hall | C3×#14581887 ⋊ 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)(40,75,103)(41,76,66)(42,81,57)(44,69,55)(45,50,72)(46,54,67)(56,100,62)(59,104,110)(60,105,89)(61,109,80)(63,94,78)(64,73,101)(65,92,91)(68,77,96)(70,98,90)(79,129,85)(82,112,108)(83,111,115)(86,121,107)(87,102,130)(88,123,117)(93,106,127)(95,125,119)(97,118,122)(99,128,116)(113,120,131)(114,132,134)(124,135,133)(126,137,136);(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)(63,85)(66,87)(67,90)(69,92)(70,93)(71,78)(72,76)(73,97)(75,88)(80,103)(81,106)(83,108)(84,110)(86,112)(89,113)(91,116)(94,118)(95,120)(96,119)(98,123)(99,124)(100,107)(101,105)(102,126)(104,114)(109,131)(111,130)(115,129)(117,133)(121,135)(122,136)(125,137)(127,134)(128,132)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^19336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 202);