Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C27 |
| direct product | C4×C9×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C9 | C23 |
| commutator | C22 | C23×C4×C9 |
| Frattini, socle, cosocle | C23×C3 | C24×C3 |
| prime radical | C2×C3×#120 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C9 ◁ G | |
1836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21);(5,6)(7,9)(8,10)(11,12);(6,9)(10,12);(6,10)(9,12)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1031);