Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #120⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#120⋊ΓL(1,𝔽8) |
| unambiguous semidirect product | (C2×#120)⋊ΓL(1,𝔽8) |
| Fitting, Hall | C2×C7×#120 ⋊ C3 |
| derived | C24×C7 ⋊ C22×C3 |
| Sylow | C2×#120 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C2×#120⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #120⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C24×C7 | C22×C3 |
| Frattini | C22 | C22×C22⋊ΓL(1,𝔽8) |
| Fitting | C2×C7×#120 | C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C2 | |
284 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽8)×#120⋊C3, C2×D7⋊(#120⋊C3), #72277208, … |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×#120⋊C3, C2×D7⋊(#120⋊C3), #72277208, … |
| maximal quotient of | #72277208 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10065);