Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#1272 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C22⋊C4 |
| commutator | C2×C4 | C2×C8×C9 |
| Frattini, cosocle | C2×C3×C42 | C22×C3 |
| socle, prime radical | C23×C3 | C3×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C2×C42×C9 ◁ G | |
249 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(10,11)(12,16,17,14)(13,15)(19,22,23,24)(26,27,28,30,29,31,32,33);(10,12)(11,14)(13,17)(15,16)(18,19,21,23)(20,24,25,22)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4);