Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×A8 (order 40320) · Inn(G) = C1 · Out(G) = C2×A8 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C26 |
| abelian | C24×C3 |
| multiplicative | (ℤ/168)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 11 points:
(1,2)(9,10,11);(3,4);(5,6);(7,8)48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 52);