Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #857⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#857⋊C2 |
| unambiguous semidirect product | #857⋊(C2×C7) |
| Fitting | C7×#127 ⋊ D3 |
| derived | C42⋊C3 ⋊ C22×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | #857⋊C2 |
| commutator, cosocle, hypocenter | C42⋊C3 | C22×C7 |
| Frattini | C22 | C2×C7×S4 |
| Fitting | C7×#127 | D3 |
| socle | C22×C7 | C2×S4 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C7 | |
166 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 20 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) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,4,5,6,7)(8,9,19,11,13,16)(10,18,12)(14,17,15);(9,12)(13,17)(14,16)(18,19)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6406);