Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#24922580 |
| derived | C3×C7×C127 ⋊ C3×C4 |
| Hall | C3×#14581887 ⋊ C4 |
| Hall | C7×C127 ⋊ C3×Dic3 |
| Sylow | C127 ⋊ Dic3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3⋊((C127⋊C3)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#19128625 |
| 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)(9,10,11)(12,14,13)(16,20,21,17,18,22)(19,28,29,23,24,38)(25,41,42,31,32,55)(26,40,36,34,30,27)(33,60,61,44,45,78)(35,58,59,47,43,52)(37,39,57,49,50,53)(46,83,84,63,64,107)(48,81,82,66,62,75)(51,56,80,68,69,76)(54,96,94,72,73,71)(65,112,88,86,87,114)(67,110,111,89,85,104)(70,79,109,91,92,105)(74,122,123,97,98,100)(77,127,120,101,102,95)(90,135,119,116,113,133)(93,108,115,117,118,134)(99,139,138,124,125,131)(103,141,140,128,129,126)(106,136,137,130,132,121);(1,2,3)(8,9,10,12,11,13,14)(15,16,19,25,33,46,65,88,115,136,141,139,135,112,116,89,66,47,34,18,22,27,37,51,70,93,119,84,109,127,122,110,83,108,124,97,72,50,24,30,38,52,71,95,121,138,111,61,80,96,81,60,79,106,128,101,69,32,43,53,55,75,100,126,140,123,82,42,57,58,41,56,77,103,130,92,45,62,73,76,78,104,131,137,120,94,59,29,40,28,39,54,74,99,118,64,85,98,102,105,107,133,117,91,68,49,36,21,20,26,35,48,67,90,87,113,125,129,132,134,114,86,63,44,31,23,17)Polycyclic presentation:
2,3^2,7,127,2|a^2=f,c^a=c^2,e^a=e^126,d^b=d^2,e^b=e^19912 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 131);