Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#24922587 |
| derived | C3×C7×C127 ⋊ C3×C4 |
| Hall | C3×#14581888 ⋊ C4 |
| Hall | C7×C127 ⋊ C3×Dic3 |
| Sylow | C127 ⋊ C3⋊(ΓL(1,𝔽8)⋊C4) |
| Sylow | C7 ⋊ C3⋊((C127⋊C3)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#19128632 |
| 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)(16,20,21,17,18,22)(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)(46,83,84,63,64,106)(48,81,82,66,62,75)(51,56,80,68,69,76)(54,95,93,72,73,71)(65,111,112,86,87,132)(67,109,110,88,85,103)(70,79,108,90,91,104)(74,119,120,96,97,99)(77,125,117,100,101,94)(89,134,135,114,113,130)(92,107,133,116,115,131)(98,137,138,121,122,128)(102,141,140,126,127,124)(105,139,136,123,129,118);(1,2,3)(8,9,11,14,13,12,10)(15,16,19,25,33,46,65,87,113,122,127,129,131,132,116,90,68,49,36,21,20,26,35,48,67,89,115,64,85,97,101,104,106,130,136,117,93,59,29,40,28,39,54,74,98,123,91,45,62,73,76,78,103,128,140,120,82,42,57,58,41,56,77,102,126,100,69,32,43,53,55,75,99,124,138,110,61,80,95,81,60,79,105,121,96,72,50,24,30,38,52,71,94,118,135,84,108,125,119,109,83,107,114,88,66,47,34,18,22,27,37,51,70,92,112,133,139,141,137,134,111,86,63,44,31,23,17)Polycyclic presentation:
2,3^2,7,127,2|a^2=f,c^a=c^2,d^a=d^6,e^a=e^126,d^b=d^2,e^b=e^107906 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 124);