Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = D3×#120 (order 192) · Inn(G) = C2×D3 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#67 |
| Fitting | C23×C3 ⋊ C2 |
| Sylow | C3 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3 |
| commutator, cosocle | C2×C3 | C23 |
| Frattini | C2 | C22×D3 |
| Fitting | C23×C3 | C2 |
| socle | C22×C3 | C22 |
| hypercenter | C23 | D3 |
| hypocenter | C3 | C2×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C23 | |
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)(7,9);(4,5)(6,7)(8,9);(1,2)18 conjugacy classes — view the full character table.
| maximal subgroup of | #806, #67⋊C4, #882, … |
| maximal normal subgroup of | #806, #67⋊C4, #882, … |
| maximal quotient of | C2×Dic3⋊C4, #924, C2×(C2×D3)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 43);