Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #102 |
| commutator | C8 | C32×C42 |
| Frattini, cosocle | C22×C8 | C22×C32 |
| prime radical | C23×C32 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(7,8,9,10)(11,12,15,19)(13,14,21,23)(16,18,24,26)(17,25,22,20);(4,5,6)(11,13,16,20,17,18,23,19,15,21,24,25,22,26,14,12)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2446);