Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#5912 |
| Fitting | C23×C4×C7 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×S4 |
| commutator, cosocle | C2×A4 | C23×C7 |
| Frattini | C2 | C22×C7×S4 |
| Fitting | C23×C4×C7 | D3 |
| socle | C23×C7 | C22×D3 |
| hypercenter | C2×C4×C7 | S4 |
| hypocenter | A4 | C7×#41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 | |
342 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,4,5,6,7)(8,9,11)(13,17)(16,19);(12,14,15,18)(13,16,17,19);(9,10)(12,13)(14,16)(15,17)(18,19)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11377);