Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1884 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | #120 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#386 |
| prime radical | C3×#532 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C22×C4×C9 ◁ G | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,19,20,23)(13,16,21,22)(14,17)(15,18)(24,25);(10,12,14,22,15,20,24,16)(11,13,17,23,18,21,25,19);(10,13)(11,16)(12,17)(14,23)(15,21)(18,22)(19,24)(20,25)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 616);