Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#1496 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C2×D4 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C3×C42 | C23×C3 |
| socle | C22×C3 | C2×C3×C22⋊C4 |
| prime radical | C3×#532 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C2×C42×C9 ◁ G | |
369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,16)(13,15)(14,17)(18,19,21,25,22,24,28,31)(20,23,27,29,26,30,32,33);(11,15)(12,13)(16,17)(18,20)(19,23)(21,27)(22,26)(24,30)(25,29)(28,32)(31,33);(10,12,14,13)(11,16,17,15)(19,24)(20,26)(25,31)(27,32)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 228);