Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = D3×B(4,𝔽2) (order 384) · Inn(G) = C2×D3 · Out(G) = #120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×D12 |
| Fitting | C2×C3×C4 ⋊ C2 |
| Sylow | C3 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3 |
| commutator, cosocle | C2×C3 | C23 |
| Frattini | C2 | C22×D3 |
| Fitting | C2×C3×C4 | C2 |
| socle | C22×C3 | C22 |
| hypercenter | C2×C4 | D3 |
| hypocenter | C3 | C2×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 9 points:
(3,4,5)(6,7,8,9);(1,2)(4,5)(7,9);(1,2)18 conjugacy classes — view the full character table.
| maximal subgroup of | #809, #821, #823, … |
| maximal normal subgroup of | #809, #821, #823, … |
| maximal quotient of | #869, C4×D12, #874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 36);