Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C22×C7×GL(2,𝔽3) |
| Fitting | C22×C7×Q8 ⋊ D3 |
| derived | SL(2,𝔽3) ⋊ C23×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C7 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C7 |
| Frattini | C2 | C22×C7×S4 |
| Fitting | C22×C7×Q8 | D3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23×C7 | |
426 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11383);