Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×#1931 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #121 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| prime radical | C22×C32×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C22×C32×C4 ◁ G | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(7,8,10,9)(12,16)(13,20)(14,17)(18,25)(19,23)(22,26);(4,5,6)(7,9,10,8)(11,12,17,23,15,19,14,16)(13,18,24,25,20,26,21,22);(11,13,14,21,15,20,17,24)(12,18,16,22,19,26,23,25)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2991);