Finite Group DatabasebetaFinite Group Databasebeta| order | 23226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C79 · Schur multiplier (H2) = C1 |
| direct product | C79×ΓL(1,𝔽8)⋊D7 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(D7×C79) |
| Fitting, socle, Hall | C72×C79 ⋊ C2×C3 |
| derived, Sylow | C72 ⋊ C2×C3×C79 |
| Hall | C7×ΓL(1,𝔽8)×C79 ⋊ C2 |
| Hall | C79×C7⋊D7 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×C79 |
| Hall | C7⋊D7 ⋊ C3×C79 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C79 | ΓL(1,𝔽8)⋊D7 |
| commutator, cosocle, hypocenter | C72 | C2×C3×C79 |
| Frattini | C1 | G |
| Fitting, socle | C72×C79 | C2×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C79 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1659 |
1738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23226, 38);