Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#24922584 |
| derived | C3×C7×C127 ⋊ C3×C4 |
| Hall | C3×#14581887 ⋊ C4 |
| Hall | C7×C127 ⋊ C3×Dic3 |
| Sylow | C127 ⋊ C3⋊(ΓL(1,𝔽8)⋊C4) |
| Sylow | C7 ⋊ Dic3×C127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#19128629 |
| commutator, hypocenter | C3×C7×C127 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C7×C127 | C2×C3 |
| prime radical | C2×C3×#14581887 | C2 |
| LCS | G ▷ C3×C7×C127 | |
| UCS | C1 ◁ C2 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 10668 |
A minimal faithful permutation representation — 2 generators on 141 points:
(2,3)(4,5,6,7)(9,12,13,10,11,14)(16,17,19)(18,21,26)(20,24,23)(22,28,36)(25,31,33)(27,34,30)(29,38,49)(32,41,46)(35,44,43)(37,47,40)(39,51,65)(42,54,62)(45,57,59)(48,60,56)(50,63,53)(52,67,68)(55,70,81)(58,73,78)(61,76,75)(64,79,72)(66,82,69)(71,87,85)(74,90,98)(77,93,95)(80,96,92)(83,99,89)(84,88,86)(91,104,102)(94,107,112)(97,110,109)(100,113,106)(101,105,103)(108,118,116)(111,121,123)(114,124,120)(115,119,117)(122,129,127)(125,132,131)(126,130,128)(133,137,135)(134,138,136)(139,141,140);(1,2,3)(8,9,11,14,13,12,10)(15,16,18,22,29,39,52,68,85,102,116,127,135,140,137,121,93,57,31,17,20,25,32,42,55,71,88,65,81,98,112,123,131,136,129,107,73,41,21,27,35,45,58,74,91,105,82,49,62,78,95,109,120,128,118,90,54,28,37,48,61,77,94,108,119,99,63,36,46,59,75,92,106,117,104,70,38,50,64,80,97,111,122,130,113,79,47,26,33,43,56,72,89,103,87,51,66,83,100,114,125,133,138,124,96,60,34,19,23,30,40,53,69,86,67,84,101,115,126,134,139,141,132,110,76,44,24)Polycyclic presentation:
2,3^2,7,127,2|a^2=f,c^a=c^2,d^a=d^6,d^b=d^2,e^b=e^191032 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 127);