Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9×C16 · Schur multiplier (H2) = C2 |
| direct product | C9×#1367 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C9 | D4 |
| commutator | C4 | C2×C9×C16 |
| Frattini, cosocle | C3×C4×C8 | C22×C3 |
| socle, prime radical | C22×C3 | C3×C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C9 ◁ C4×C8×C9 ◁ G | |
114 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)(11,14)(12,16)(15,17)(18,19,20,23,21,24,26,29,22,25,27,30,28,31,32,33);(10,11,12,14,13,15,16,17)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 99);