Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×#239 |
| Fitting | C3×C7×Q8 ⋊ C2×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ #41 |
| Sylow | C7 ⋊ C3×#239 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C2×C3×C7 | C23×C3 |
| Fitting | C3×C7×Q8 | C2×C3 |
| hypercenter | Q8 | D3×ΓL(1,𝔽8) |
| hypocenter | C3×C7 | C3×#41 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
172 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
75 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#41×ΓL(1,𝔽8), ΓL(1,𝔽8)×#1009, AGL(1,𝔽7)×#239, … |
| maximal normal subgroup of | D3×#41×ΓL(1,𝔽8), ΓL(1,𝔽8)×#1009, AGL(1,𝔽7)×#239, … |
| maximal quotient of | ΓL(1,𝔽8)×#3799, ΓL(1,𝔽8)×#3796 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 654);