Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of S4 |
| automorphism | Aut(G) = C22×C3×S4 (order 288) · Inn(G) = S4 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C22 |
| direct product | C2×C7×S4 |
| unambiguous semidirect product | A4⋊(C22×C7) |
| unambiguous semidirect product | (C2×A4)⋊(C2×C7) |
| Fitting, socle | C23×C7 ⋊ D3 |
| derived | A4 ⋊ C22×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | S4 |
| commutator, cosocle, hypocenter | A4 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C23×C7 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C7 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 2 generators on 13 points:
(3,4,6,5)(7,8,9,10,11,12,13);(1,2)(4,5,6)70 conjugacy classes — view the full character table.
| maximal subgroup of | S4×Dic7, #10621480, #10621482, … |
| maximal normal subgroup of | S4×Dic7, #10621480, #10621482, … |
| maximal quotient of | C7×A4⋊Q8, C4×C7×S4, C7×C4⋊S4, … |
| outer automorphism group of | C22×D29 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 214);