Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)×AGL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C7 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×AGL(1,𝔽8) |
| unambiguous semidirect product | C7⋊(C3×AGL(1,𝔽8)) |
| unambiguous semidirect product | (C23×ΓL(1,𝔽8))⋊C7 |
| Fitting, socle, derived | C23×C7 ⋊ C3×C7 |
| Hall | C7×AGL(1,𝔽8) ⋊ C3 |
| Sylow | C23 ⋊ C7×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C23×C7 | C3×C7 |
| LCS | G ▷ C23×C7 | |
| UCS | C1 | |
28 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×AΓL(1,𝔽8), AGL(1,𝔽7)×AGL(1,𝔽8), AGL(1,𝔽8)×AΓL(1,𝔽8) |
| maximal normal subgroup of | ΓL(1,𝔽8)×AΓL(1,𝔽8), AGL(1,𝔽7)×AGL(1,𝔽8) |
| maximal quotient of | AGL(1,𝔽8)×AΓL(1,𝔽8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 222);