Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C24×C4 |
| direct product | C9×#2472 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C23 | C24×C9 |
| Frattini, socle, cosocle | C23×C3 | C24×C3 |
| prime radical | C2×C3×#120 | C2×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
1179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(11,12)(18,19)(20,21)(22,23)(24,25);(10,11,13,12)(15,17)(19,23)(20,24);(10,11,13,12)(14,15)(16,17)(18,20,22,24)(19,21,23,25);(10,12,13,11)(18,21)(19,24)(20,23)(22,25)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1204);