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) = C22×C3 |
| direct product | C32×#1485 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C42 | C23×C32 |
| socle | C22×C32 | C2×C22⋊C4 |
| prime radical | C23×C32 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C2×C32×C42 ◁ G | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(7,8,11,14,12,9,10,13)(15,16)(17,24)(18,22)(19,28)(20,23)(21,26)(25,29)(27,30);(4,5,6)(7,9,12,8)(10,14,11,13)(15,17,20,27)(16,21,23,29)(18,26,19,25)(22,30,28,24);(7,10,12,11)(8,13,9,14)(15,18,20,19)(16,22,23,28)(17,25,27,26)(21,24,29,30)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2545);