Finite Group DatabasebetaFinite Group Databasebeta| order | 10668 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×#19128624 |
| Fitting, socle | C2×C7×C127 ⋊ C2×C3 |
| derived, Hall | C7×C127 ⋊ C22×C3 |
| Hall | C2×C7×D127 ⋊ C3 |
| Hall | #14581888 ⋊ C22 |
| Hall | C2×D127 ⋊ ΓL(1,𝔽8) |
| Sylow | C127 ⋊ C22×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C2×D127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #19128624 |
| commutator, cosocle, hypocenter | C7×C127 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C127 | C2×C3 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
314 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽8)×D127⋊C3, D3×#19128624, C2×C3×#19128624, … |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×D127⋊C3, C2×C3×#19128624 |
| maximal quotient of | C2×#30630663, D3×#19128624, C2×C3×#19128624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10668, 52);