Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C26 |
| direct product | C9×#2901 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | C22×D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C2×C3 | C22×C3×C22⋊C4 |
| prime radical | C3×#585 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C2×C9×#41 ◁ G | |
1107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,14,18,15,12,19,17)(13,16,22,24,21,20,25,23);(10,12)(11,15)(13,20)(14,17)(16,21)(18,19)(22,23)(24,25);(10,13)(11,16)(12,20)(14,22)(15,21)(17,23)(18,24)(19,25);(13,21)(16,20)(22,25)(23,24)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1633);