Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1827 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #117 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C2×C32 | #381 |
| prime radical | C2×C32×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C32×#117 ◁ G | |
786 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)(9,16,18,12)(10,15)(11,14)(13,17,20,22)(19,21);(4,5,6)(7,9)(8,12)(10,17)(11,18)(13,15)(14,16)(19,22)(20,21);(7,9)(8,13,14,20)(10,17)(11,18)(12,15,16,21)(19,22)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2887);