Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#2096 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C32 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C2×C32×C42 ◁ G | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(11,12,14,16)(13,17,18,15)(19,20,24,26,23,21,22,25);(4,5,6)(7,8)(9,10)(11,13,14,18)(12,15,16,17)(19,21,23,20)(22,25,24,26);(8,10)(19,20,23,21)(22,26,24,25)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3156);