Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = D3×#120 (order 192) · Inn(G) = D3 · Out(G) = #120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
| direct product | C4×Dic3 |
| unambiguous semidirect product | C3⋊C42 |
| derived, Sylow | C3 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D3 |
| commutator, hypocenter | C3 | C42 |
| Frattini | C22 | C2×D3 |
| Fitting | C2×C3×C4 | C2 |
| socle, prime radical, cosocle | C22×C3 | C22 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C4 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 11 points:
(2,3)(8,9,10,11);(1,2,3)(4,5,6,7)24 conjugacy classes — view the full character table.
| maximal subgroup of | C8×Dic3, Dic3⋊C8, #815, … |
| maximal normal subgroup of | C8×Dic3, Dic3⋊C8, #815, … |
| maximal quotient of | C4×C3⋊C8, #803, C8×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 11);