Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#3310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C32×D42 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C4×D4 ◁ G | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(7,8,10,12)(9,13,14,11)(15,16)(17,20)(18,21)(19,22);(4,5,6)(7,9)(8,11)(10,14)(12,13)(15,17,19,18)(16,20,22,21);(16,21)(17,18)(20,22);(9,14)(11,13)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4370);