Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#5950 |
| Fitting, Hall | C7×#585 ⋊ C3 |
| Sylow | #585 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C3×C7 |
| Frattini | C2 | C2×C7×C22⋊A4 |
| Fitting | C7×#585 | C3 |
| socle | C2×C7 | C2×C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C4×C7 | |
330 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9,12,15)(10,14,22,20,11,19,17,21,16,13,18,23);(10,17)(11,18)(13,20)(14,21);(8,16)(9,19)(10,11)(12,22)(13,14)(15,23)(17,18)(20,21)154 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#5950, C7×#87547, D7×#5950, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#5950, C7×#87547, D7×#5950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11611);