Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of C22⋊(C22⋊ΓL(1,𝔽8)) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
| direct product | C22×C22⋊(C22⋊ΓL(1,𝔽8)) |
| Fitting, socle, Hall | C26×C7 ⋊ C3 |
| derived | C24×C7 ⋊ C22×C3 |
| Sylow | C26 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C22×C22⋊A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C22⋊(C22⋊ΓL(1,𝔽8)) |
| commutator, cosocle, hypocenter | C24×C7 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C26×C7 | C3 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C22 | |
2000 conjugacy classes of subgroups.
| transitive degree | μt(G) = 84 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C22×ΓL(1,𝔽8)×C22⋊A4, C22×D7⋊(C22⋊A4), #72277085, … |
| maximal normal subgroup of | C22×ΓL(1,𝔽8)×C22⋊A4, C22×D7⋊(C22⋊A4), #72277208, … |
| maximal quotient of | #72230067, #72277370 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11714);