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) = C4 |
| direct product | C9×#1922 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C2×D4 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C22×C3 | C3×C4×D4 |
| prime radical | C23×C3 | C3×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C22×C8×C9 ◁ G | |
279 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)(10,11)(12,15)(13,14)(16,17)(18,19,22,26,23,21,30,25)(20,24,28,32,29,27,33,31);(11,14)(15,17)(18,20,22,28,23,29,30,33)(19,24,26,32,21,27,25,31);(10,12,13,16)(11,15,14,17)(18,21)(19,23)(20,27)(22,25)(24,29)(26,30)(28,31)(32,33)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 654);