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) = C24×C3 |
| direct product | C32×#1873 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| prime radical | C24×C32 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C23×C32×C4 ◁ G | |
1080 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)(8,10)(11,12)(13,14)(15,16,19,23,20,24,18,22)(17,25,26,21,28,29,27,30);(4,5,6)(7,8,9,10)(15,17)(16,21)(18,26)(19,27)(20,28)(22,29)(23,25)(24,30);(12,14)(15,18,20,19)(16,22,24,23)(17,26,28,27)(21,29,30,25)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2933);