Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| isoclinism | isoclinic class of S4 |
| automorphism | Aut(G) = C2×S4 (order 48) · Inn(G) = S4 · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = #72215293 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| general orthogonal | GO(3,𝔽3) |
| direct product | C2×S4 |
| unambiguous semidirect product | A4⋊C22 |
| unambiguous semidirect product | (C2×A4)⋊C2 |
| Fitting, socle | C23 ⋊ D3 |
| derived | A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | S4 |
| commutator, cosocle, hypocenter | A4 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C23 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 6 |
| transitive degree | μt(G) = 6 |
A minimal faithful permutation representation — 2 generators on 6 points:
(3,6,4,5);(1,4,6,2,3,5)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 3 | 6 | 6 | 8 | 1 | 3 | 6 | 6 | 8 |
| order | 1 | 2 | 2 | 4 | 3 | 2 | 2 | 2 | 4 | 6 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 |
| χ3 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 |
| χ4 | 3 | -1 | 1 | -1 | 0 | 3 | -1 | 1 | -1 | 0 |
| χ5 | 3 | -1 | -1 | 1 | 0 | 3 | -1 | -1 | 1 | 0 |
| χ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 |
| χ7 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 |
| χ8 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 1 |
| χ9 | 3 | -1 | 1 | -1 | 0 | -3 | 1 | -1 | 1 | 0 |
| χ10 | 3 | -1 | -1 | 1 | 0 | -3 | 1 | 1 | -1 | 0 |
| maximal subgroup of | C4×S4, C4⋊S4, GL(2,ℤ/4), … |
| maximal normal subgroup of | C4×S4, C4⋊S4, GL(2,ℤ/4), … |
| maximal quotient of | A4⋊Q8, C4×S4, C4⋊S4, … |
| automorphism group of | #41, C3×Q8, #226, … |
| outer automorphism group of | C2×Q8, #120, C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 48);