Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C23×D3 (order 48) · Inn(G) = D3 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = #72213893 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | D3×C8 |
| Fitting | C3×C8 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D3 |
| commutator, hypocenter | C3 | C2×C8 |
| Frattini | C4 | C2×D3 |
| Fitting | C3×C8 | C2 |
| socle | C2×C3 | C2×C4 |
| prime radical | C2×D3 | C4 |
| cosocle | C3×C4 | C22 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C8 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 11 points:
(2,3);(1,2,3)(4,5,6,8,7,9,10,11)24 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C16, #798, C2×D3×C8, … |
| maximal normal subgroup of | D3×C16, #798, C2×D3×C8, … |
| maximal quotient of | D3×C16, #798, C8×Dic3, … |
| outer automorphism group of | #33983 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 4);