Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1279 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | D4 |
| commutator | C4 | C32×C4×C8 |
| Frattini, cosocle | C2×C42 | C22×C32 |
| socle, prime radical | C23×C32 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ C2×C32×C42 ◁ G | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(7,8)(9,10)(12,16)(13,14)(17,18)(19,20,22,25,23,26,21,24);(4,5,6)(8,10)(11,12,14,17,15,18,13,16)(20,24,26,25)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2339);