Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| isoclinism | isoclinic class of #204 |
| automorphism | Aut(G) = C2×D3×D4 (order 96) · Inn(G) = D12 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #72219759 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C3×C8 ⋊ C2 |
| Sylow | C3 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D12 |
| commutator, cosocle | C3×C4 | C22 |
| Frattini | C4 | C2×D3 |
| Fitting | C3×C8 | C2 |
| socle | C2×C3 | D4 |
| prime radical | D12 | C2 |
| hypercenter | C8 | D3 |
| hypocenter | C3 | SD16 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2,3)(4,5,6,8,7,9,10,11);(2,3)(5,8)(6,10)(9,11)Polycyclic presentation:
2^4,3|b^2=c,c^2=d,b^a=bc,c^a=cd,e^a=e^215 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#204, #904, #908, … |
| maximal normal subgroup of | C2×#204, #904, #908, … |
| maximal quotient of | #816, #817, #821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 6);