Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#10621538 |
| Fitting | C22×C7×Q8 ⋊ C2×C3 |
| Hall | C2×#6551 ⋊ C3 |
| Sylow | C7 ⋊ C2×2−1+4⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D7⋊A4 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Frattini | C2 | C22×D7⋊A4 |
| Fitting | C22×C7×Q8 | C2×C3 |
| socle | C22×C7 | C22×A4 |
| hypercenter | C23 | D7⋊A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C22 ◁ C23 | |
408 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
58 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11481);