Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2 |
| Fitting, socle | C2×C3×C127 ⋊ C2×C3×C7 |
| derived | C3×C127 ⋊ C22×C3×C7 |
| Hall | C3×(C127⋊C3)⋊C7 ⋊ C22 |
| Hall | C2×C3⋊(D127⋊C3) ⋊ C7 |
| Hall | C3×C127⋊C3 ⋊ C22×C7 |
| Hall | C127⋊C7 ⋊ C2×C3×D3 |
| Sylow | C127 ⋊ C2×C3×D3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊((D127⋊C3)⋊C7) |
| commutator, cosocle, hypocenter | C3×C127 | C22×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C127 | C2×C3×C7 |
| LCS | G ▷ C3×C127 | |
| UCS | C1 ◁ C2 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 762 |
A minimal faithful permutation representation — 2 generators on 132 points:
(4,5)(7,23,55,51,20,120,42,12,27,25,57,75,21,18,14,13,28,100,11,76,69,8,15,56,32,37,78,70,9,30,17,84,39,38,34,16,10,46,99,19,40,35)(22,97,95,58,73,63,116,53,115,59,104,87,74,96,85,54,60,132,36,117,88,26,86,130,61,77,106,89,44,123,29,102,119,107,52,90,45,103,131,72,71,47)(24,118,122,108,62,66,113,79,109,128,126,125,67,94,64,92,124,81,114,127,68,48,80,110,82,101,33,121,41,65,91,129,83,111,50,31,49,98,43,93,105,112);(1,2)(3,4,5)(6,7,11,22,10,18,36,73,21,43,84,124,45,87,86,96,17,33,64,14,12,24,50,95,74,78,123,35,71,118,102,89,42,76,122,132,83,113,39,79,37,67,114,92,126,128,88,15,29,62,55,30,63,108,100,65,60,85,127,112,53,28,58,51,32,61,46,44,68,105,90,103,109,99,82,106,27,56,101,59,23,47,91,129,49,93,111,20,41,75,121,125,131,110,40,70,116,104,80,117,69,115,120,107,130,94,48,9,16,31,66,25,52,97,119,54,98,57,81,38,77,72,34,13,26,19,8)144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 159);