Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C32×#335 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | D4 |
| commutator | C4 | C22×C32×C8 |
| Frattini, cosocle | C42 | C23×C32 |
| socle, prime radical | C23×C32 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ C2×C32×C42 ◁ G | |
600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(19,20,21);(4,7)(5,8)(9,10)(11,12,13,15,14,16,17,18)(22,23,24);(3,4,8,10,6,9,5,7)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2623);