Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#2104 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | #401 |
| commutator, Frattini, cosocle | C24 | C23×C32 |
| prime radical | C24×C32 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C24×C32 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(7,8)(9,15,17,12)(10,13,19,22)(11,14)(16,21)(18,20);(4,5,6)(7,9)(8,12)(10,18,19,16)(11,17)(13,21,22,20)(14,15);(7,10)(8,13,14,22)(9,16)(11,19)(12,20,15,21)(17,18)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3164);