Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C32×#366 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | D8 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(23,24,25);(3,4,5,6)(8,12)(9,13)(10,16)(14,22)(15,20)(17,21)(18,19)(26,27,28);(7,8,13,19,10,14,17,20,11,15,21,22,16,18,9,12)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3216);