Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×#226 |
| unambiguous semidirect product | (C7×#226)⋊C3 |
| Sylow | C7 ⋊ C3×#226 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽8)×S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C2×C3 |
| Fitting | C7×Q8 | C3×D3 |
| socle | C2×C7 | C3×S4 |
| prime radical | ΓL(1,𝔽8)×SL(2,𝔽3) | C2 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#985, #72218844, #72221415, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#985, #72218844, ΓL(1,𝔽8)×#983, … |
| maximal quotient of | ΓL(1,𝔽8)×#859 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 518);