Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C7 · Schur multiplier (H2) = C23 |
| direct product | C22×C7×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C22×C3×C7) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C23×C7) |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊(C22×C7) |
| Fitting, socle | C22×C72 ⋊ C2×C3 |
| derived | C7 ⋊ C23×C3×C7 |
| Hall | C22×C7×D7 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C23 |
| Sylow | C72 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C7 | AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7 | C23×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C22×C72 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C7 | |
133 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 244);