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) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3 |
| commutator | C2×C3 | C2×C4 |
| Fitting, prime radical | C23×C3 | C2 |
| socle, cosocle | C22×C3 | C22 |
| hypercenter | C23 | D3 |
| hypocenter | C3 | C22⋊C4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C23 | |
34 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)(9,11);(2,3)(4,5,6,7)(8,9)(10,11)Polycyclic presentation:
2^4,3|a^2=c,b^a=bd,e^a=e^218 conjugacy classes — view the full character table.
| maximal subgroup of | #806, #834, #877, … |
| maximal normal subgroup of | #806, #834, #877, … |
| maximal quotient of | #830, C3⋊#95, #832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 19);