Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| isoclinism | isoclinic class of C7⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×C7⋊S4 |
| unambiguous semidirect product | C7⋊(C7×S4) |
| unambiguous semidirect product | A4⋊(C7×D7) |
| Fitting, socle | C22×C72 ⋊ D3 |
| derived | C7×A4 ⋊ C2×C7 |
| Sylow | C72 ⋊ S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | C7⋊S4 |
| commutator, cosocle, hypocenter | C7×A4 | C2×C7 |
| Frattini | C1 | G |
| Fitting, socle | C22×C72 | D3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 ◁ C7 | |
59 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
119 conjugacy classes — view the full character table.
| maximal subgroup of | C7×C7⋊#1020, #72214846, C7×D7×S4, … |
| maximal normal subgroup of | #72214846, C7×D7×S4, D7×C7⋊S4, … |
| maximal quotient of | C7×C7⋊#1020, #72275546 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 241);