Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1855 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C22×C32 | C2×C4⋊C4 |
| prime radical | C24×C32 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C23×C32×C4 ◁ G | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(7,8)(9,10)(11,12,15,18,16,19,14,23)(13,20,21,17,22,25,24,26);(4,5,6)(11,13,15,21,16,22,14,24)(12,17,18,25,19,26,23,20);(8,10)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2915);