Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C32×#1974 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C22×C32 | C4×D4 |
| prime radical | C2×C32×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C22×C32×C8 ◁ G | |
900 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)(8,10)(11,12,15,19,16,20,14,17)(13,21,23,26,24,25,22,18);(4,5,6)(7,8,9,10)(12,17,20,19)(18,25,26,21);(11,13,14,22,16,24,15,23)(12,18,19,21,20,26,17,25)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3034);