Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#3200 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C3×#535 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C9×#41 ◁ G | |
615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,16)(13,15)(14,17)(18,19,22,26)(20,23,27,31)(21,24,29,25)(28,30,33,32);(10,12)(11,14)(13,17)(15,16)(18,20,22,27)(19,23,26,31)(21,28,29,33)(24,30,25,32);(12,17)(14,16)(19,24)(21,29)(23,30)(25,26)(28,33)(31,32);(11,15)(14,16)(20,27)(23,31)(28,33)(30,32)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1932);