Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#2962 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C24 |
| commutator | C22 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C23×C3×C4 |
| prime radical | C23×C3 | C22×C3×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,17)(13,15)(14,16)(18,19,21,24,22,25,28,31)(20,23,27,30,26,29,32,33);(10,12,14,13)(11,15,16,17);(10,13,14,12)(11,15,16,17)(18,20,21,27,22,26,28,32)(19,23,24,30,25,29,31,33);(20,26)(23,29)(27,32)(30,33)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1694);