Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C28×C3 |
| direct product | C32×#3483 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| socle | C22×C32 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
3606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2,3)(8,12)(13,14)(15,16)(17,20)(18,22)(19,21);(4,5,6)(7,8,11,12)(9,13,10,14)(15,17,20,16)(18,22,19,21);(7,9,11,10)(8,13,12,14)(15,18,20,19)(16,21,17,22);(7,9,11,10)(8,14,12,13)(15,19,20,18)(16,21,17,22);(7,10,11,9)(8,14,12,13)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4543);