Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#2008 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23×C32 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32×C4 ◁ G | |
1140 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)(9,17)(10,15)(11,14)(12,16)(13,18)(19,22)(20,21);(4,5,6)(7,9)(8,13)(10,20)(11,18)(12,19)(14,22)(15,17)(16,21);(8,14)(10,11)(13,17)(15,16)(18,20)(21,22)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3068);