Finite Group DatabasebetaFinite Group Databasebeta| order | 23226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C72×C79 ⋊ C2×C3 |
| Hall | #26042478 ⋊ C2 |
| Hall | C7⋊D553 ⋊ C3 |
| Sylow | C79 ⋊ #64424 |
| Sylow | C72 ⋊ D79⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C79 | C2×C3 |
| LCS | G ▷ C72×C79 | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3871 |
A minimal faithful permutation representation — 3 generators on 93 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,63)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,67,68,49,50,78)(34,66,51,52,48,60)(37,42,65,54,55,61)(40,74,73,58,59,57)(53,79,71,70,69,76)(56,64,75,72,62,77)(81,84,85,82,83,86)(88,91,92,89,90,93);(1,2,5,11,19,32,51,28,43,44,27,42,62,31,48,59,61,63,76,70,52,33,20,4,8,13,23,37,56,73,45,15,26,14,25,40,50,69,68,75,77,78,79,67,58,36,10,16,24,38,57,72,54,35,22,7,6,12,21,34,53,71,47,65,74,66,46,64,55,18,29,39,41,60,49,30,17,9,3)(87,88,90,93,92,91,89);(80,81,83,86,85,84,82)Polycyclic presentation:
2,3,7^2,79|c^a=c^6,d^a=d^6,e^a=e^78,c^b=c^2,d^b=d^2,e^b=e^23651 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23226, 56);