Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#1500 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C22⋊C4 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C22×C3 | C2×C3×C22⋊C4 |
| prime radical | C3×#121 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C2×C42×C9 ◁ G | |
357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(11,14,15,16)(18,19)(20,26,28,23)(21,24)(22,25)(27,32,33,30)(29,31);(10,11)(12,16)(13,15)(14,17)(18,20)(19,23)(21,27)(22,28)(24,30)(25,26)(29,33)(31,32);(18,21,22,29)(19,24,25,31)(20,27,28,33)(23,30,26,32)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 232);