Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C26×C3 |
| direct product | C32×#2901 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C22×D4 |
| commutator | C4 | C23×C32×C4 |
| Frattini, cosocle | C2×C4 | C24×C32 |
| socle | C2×C32 | C22×C22⋊C4 |
| prime radical | C32×#585 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C2×C32×#41 ◁ G | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(7,8,11,15,12,9,16,14)(10,13,19,21,18,17,22,20);(4,5,6)(7,9)(8,12)(10,17)(11,14)(13,18)(15,16)(19,20)(21,22);(7,10)(8,13)(9,17)(11,19)(12,18)(14,20)(15,21)(16,22);(10,18)(13,17)(19,22)(20,21)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3961);