Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3291 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ G | |
1380 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)(7,8,13,18)(9,19,20,21)(10,15,14,22)(11,17,12,16);(4,5,6)(7,9)(8,14)(10,18)(11,22)(12,15)(13,20)(16,21)(17,19);(7,10)(8,9)(11,21)(12,19)(13,14)(15,16)(17,22)(18,20);(7,10)(8,15)(9,16)(11,19)(12,21)(13,14)(17,20)(18,22)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4351);