Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#24922583 |
| derived | C3×C7×C127 ⋊ C3×C4 |
| Hall | C3×#14581888 ⋊ 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⋊#19128628 |
| commutator, hypocenter | C3×C7×C127 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C7×C127 | C2×C3 |
| prime radical | C2×C3×#14581888 | 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)(52,67,83)(55,69,80)(58,72,77)(61,75,74)(64,78,71)(66,81,68)(70,86,99)(73,88,96)(76,91,93)(79,94,90)(82,97,87)(84,100,85)(89,103,113)(92,105,110)(95,108,107)(98,111,104)(101,114,102)(106,117,124)(109,119,121)(112,122,118)(115,125,116)(120,128,132)(123,130,129)(126,133,127)(131,136,137)(134,138,135)(139,141,140);(1,2,3)(8,9,11,14,13,12,10)(15,16,18,22,29,39,52,67,84,101,115,126,134,139,138,122,94,60,34,19,23,30,40,53,68,85,86,51,66,82,98,112,123,131,133,111,78,47,26,33,43,56,71,87,102,103,69,38,50,64,79,95,109,120,125,97,63,36,46,59,74,90,104,116,117,88,54,28,37,48,61,76,92,106,114,81,49,62,77,93,107,118,127,128,105,72,41,21,27,35,45,58,73,89,100,65,80,96,110,121,129,135,136,119,91,57,31,17,20,25,32,42,55,70,83,99,113,124,132,137,140,141,130,108,75,44,24)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^1071032 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 126);