Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1873 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C2×D4 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C23×C3 | C2×C3×D4 |
| prime radical | C24×C3 | C3×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C23×C4×C9 ◁ G | |
540 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)(11,13)(14,15)(16,17)(18,19,22,26,23,27,21,25)(20,28,29,24,31,32,30,33);(10,11,12,13)(18,20)(19,24)(21,29)(22,30)(23,31)(25,32)(26,28)(27,33);(15,17)(18,21,23,22)(19,25,27,26)(20,29,31,30)(24,32,33,28)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 605);