Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| isoclinism | isoclinic class of AGL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C72 · Schur multiplier (H2) = C7 |
| direct product | C3×C7×AGL(1,𝔽8) |
| unambiguous semidirect product | (C23×C3)⋊C72 |
| unambiguous semidirect product | (C23×C3×C7)⋊C7 |
| Fitting, socle | C23×C3×C7 ⋊ C7 |
| derived, Sylow | C23 ⋊ C3×C72 |
| Hall | C23×C3 ⋊ C72 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C7 | AGL(1,𝔽8) |
| commutator, cosocle, hypocenter | C23 | C3×C72 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C7 | C7 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C3×C7 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
168 conjugacy classes — view the full character table.
| derived subgroup of | D3×#11924306 |
| maximal subgroup of | C3×AGL(1,𝔽8)2, C3×#11924306, D3×C7×AGL(1,𝔽8) |
| maximal normal subgroup of | C3×#11924306, D3×C7×AGL(1,𝔽8) |
| maximal quotient of | C3×AGL(1,𝔽8)2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 238);