Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of S4 |
| automorphism | Aut(G) = C2×S4 (order 48) · Inn(G) = S4 · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = D3×S42 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×S4 |
| unambiguous semidirect product | A4⋊(C2×C3) |
| Fitting, socle | C22×C3 ⋊ D3 |
| derived | A4 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | S4 |
| commutator, cosocle, hypocenter | A4 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C3 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 7 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 7 points:
(1,2,4,3)(5,6,7);(2,3,4)15 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 42);