Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C16 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×Q8×C16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C16 | C22 |
| commutator | C2 | C22×C32×C16 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle, prime radical | C22×C32 | C22×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C16 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(7,8,9,10)(11,14,13,12);(4,5,6)(7,11,9,13)(8,12,10,14);(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)720 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3242);