Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#1584 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C42 | C23×C32 |
| socle, prime radical | C22×C32 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C32×C4×C8 ◁ G | |
456 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)(7,8,10,9,11,12,14,13)(15,16,19,22)(17,23,25,30)(18,26,28,21)(20,27,29,24);(4,5,6)(15,17,19,25)(16,20,22,29)(18,27,28,24)(21,30,26,23);(7,9,10,12,11,13,14,8)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2644);