Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C23×C4 |
| direct product | C9×#3137 |
| 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 | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C9×#117 ◁ G | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,15)(13,14)(16,17)(18,19,23,25)(20,26,27,33)(21,22,29,30)(24,31,32,28);(11,14)(15,17)(18,20,22,28,23,27,30,31)(19,24,21,26,25,32,29,33);(10,12,13,16)(11,15,14,17)(18,21,22,25,23,29,30,19)(20,26,28,32,27,33,31,24);(20,27)(24,32)(26,33)(28,31)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1869);