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×#1826 |
| 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×C22⋊C4 |
| prime radical | C24×C32 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C23×C32×C4 ◁ G | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(7,8)(9,12)(10,11)(13,14)(15,16,18,21,19,17,22,20);(4,5,6)(8,11)(12,14);(7,9,10,13)(8,12,11,14)(15,17)(16,19)(18,20)(21,22)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2886);