Finite Group DatabasebetaFinite Group Databasebeta| order | 23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#22547870 |
| Fitting, socle, derived | C3×C31×C43 ⋊ C2×C3 |
| Hall | C3×#16860688 ⋊ C2 |
| Hall | C31×C43 ⋊ C3×D3 |
| Sylow | C43 ⋊ C3⋊(D31⋊C3) |
| Sylow | C31 ⋊ C3⋊(D43⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C31×C43 | C2×C3 |
| LCS | G ▷ C3×C31×C43 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3999 |
A minimal faithful permutation representation — 2 generators on 77 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,36)(12,26,22,20,16,13)(19,32,34,23,25,37)(21,38,39,30,29,33)(31,41,42,40,35,43)(45,49,50,46,47,51)(48,57,58,52,53,65)(54,68,69,60,61,73)(55,67,63,62,59,56)(64,66,74,70,71,72)(76,77);(1,2,5,11,19,22,7,6,12,21,31,39,15,26,14,25,35,42,28,37,38,27,18,29,34,36,43,41,32,10,16,24,33,40,30,20,4,8,13,23,17,9,3)(44,45,48,54,62,47,51,56,64,71,53,59,65,69,74,61,68,72,73,58,67,57,66,70,63,50,49,55,60,52,46)(75,76,77)Polycyclic presentation:
2,3^2,31,43|c^a=c^2,d^a=d^30,e^a=e^42,d^b=d^5,e^b=e^36675 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23994, 45);