Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of D7×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C22×D7×S4 |
| unambiguous semidirect product | (C7×S4)⋊C23 |
| unambiguous semidirect product | (C7⋊S4)⋊C23 |
| unambiguous semidirect product | (A4×D7)⋊C23 |
| Fitting, socle | C24×C7 ⋊ C2×D3 |
| derived | C7×A4 ⋊ C24 |
| Sylow | C7 ⋊ C23×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7×S4 |
| commutator, cosocle, hypocenter | C7×A4 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C24×C7 | C2×D3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 ◁ C22 | |
1402 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | A4×D7×S4, C22×S4×AGL(1,𝔽7), S4×D7⋊A4, … |
| maximal normal subgroup of | A4×D7×S4, C22×S4×AGL(1,𝔽7), S4×D7⋊A4, … |
| maximal quotient of | D4×D7×S4, C2×D7×GL(2,ℤ/4), C22×D7×#1020 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11695);