Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#1553 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C2×C22⋊C4 |
| 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 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C42×C9 ◁ G | |
363 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)(12,14)(13,15)(16,17)(18,19,22,25,23,26,21,30)(20,27,28,32,29,24,31,33);(11,14)(12,17)(15,16)(18,20)(19,24)(21,28)(22,31)(23,29)(25,32)(26,27)(30,33);(11,15)(14,16)(18,21,23,22)(19,25,26,30)(20,28,29,31)(24,32,27,33)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 285);