Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C82×C9 · Schur multiplier (H2) = C4 |
| direct product | C9×#1311 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C9 | C22 |
| commutator | C2 | C82×C9 |
| Frattini, cosocle | C3×C4×C8 | C22×C3 |
| socle, prime radical | C22×C3 | C3×C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C9 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(18,19,20,24,21,25,26,31,22,23,27,29,28,30,32,33);(10,11,12,14,13,15,16,17)(19,23)(24,29)(25,30)(31,33)720 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 43);