Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)×C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3×C4 |
| direct product | ΓL(1,𝔽8)×C42⋊C3 |
| unambiguous semidirect product | (C42×ΓL(1,𝔽8))⋊C3 |
| unambiguous semidirect product | (C7×C42⋊C3)⋊C3 |
| unambiguous semidirect product | (C7⋊(C42⋊C3))⋊C3 |
| Fitting, derived, Hall | C42×C7 ⋊ C32 |
| Sylow | C42 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×C42⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C42×C7 | C32 |
| Frattini | C22 | A4×ΓL(1,𝔽8) |
| socle | C22×C7 | C3×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 84 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#5464, ΓL(1,𝔽8)×#127⋊C3, ΓL(1,𝔽8)×#126⋊C3, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#127⋊C3, ΓL(1,𝔽8)×#126⋊C3, AGL(1,𝔽7)×C42⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 242);