Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #4245 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C23×#4247 |
| Hall | C3×C7 ⋊ C23×D4 |
| Sylow | C7 ⋊ C23×D12 |
| Sylow | C3 ⋊ C23×C22⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | D3×D7 |
| commutator, cosocle | C2×C3×C7 | C25 |
| Frattini | C2 | C23×D3×D7 |
| Fitting, socle | C24×C3×C7 | C22 |
| hypocenter | C3×C7 | C23×D4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C24 | |
2724 conjugacy classes of subgroups; only the 728 normal subgroups are shown.
This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 728 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 672 |
216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11576);