Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×D3×D4 (order 96) · Inn(G) = C2×D3 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = #72219803 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D3 |
| commutator, socle | C2×C3 | C2×C4 |
| Fitting | C2×C3×C4 | C2 |
| prime radical | C22×C3 | C4 |
| cosocle | C3×C4 | C22 |
| hypercenter | C2×C4 | D3 |
| hypocenter | C3 | M16 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
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)(5,8)(9,11);(2,3)(4,5,6,9,7,8,10,11)Polycyclic presentation:
2^4,3|a^2=c,c^2=d,b^a=bd,e^a=e^218 conjugacy classes — view the full character table.
| maximal subgroup of | #805, #819, #822, … |
| maximal normal subgroup of | #805, #819, #822, … |
| maximal quotient of | #803, C4⋊(C3⋊C8), #830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 10);