Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#10621660 |
| Fitting | C22×C3×C7×Q8 ⋊ C2 |
| Hall | C3×C7 ⋊ C2×2−1+4 |
| Sylow | C7 ⋊ C2×#1007 |
| Sylow | C3 ⋊ C2×#6551 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D21 |
| commutator, cosocle | C2×C3×C7 | C25 |
| Frattini | C2 | C24×D21 |
| Fitting | C22×C3×C7×Q8 | C2 |
| socle | C22×C3×C7 | C24 |
| hypercenter | C22×Q8 | D21 |
| hypocenter | C3×C7 | C2×2−1+4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
1588 conjugacy classes of subgroups; only the 587 normal subgroups are shown.
This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 587 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 672 |
234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11666);