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×#1518 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C22×C3 | C2×C3×C22⋊C4 |
| prime radical | C23×C3 | C2×C3×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C42×C9 ◁ G | |
330 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)(11,15)(12,13)(16,17)(18,19)(20,26,28,23)(21,24)(22,25)(27,32,29,31)(30,33);(10,11,14,17)(12,15,13,16)(18,20,22,28)(19,23,25,26)(21,29,30,27)(24,32,33,31);(10,12,14,13)(11,16,17,15)(18,21,22,30)(19,24,25,33)(20,27,28,29)(23,31,26,32)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 250);