Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9×C16 · Schur multiplier (H2) = C22 |
| direct product | C9×#1322 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C22⋊C4 |
| commutator | C22 | C2×C9×C16 |
| Frattini, cosocle | C22×C3×C8 | C22×C3 |
| socle | C22×C3 | C3×C22⋊C8 |
| prime radical | C23×C3 | C2×C3×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C22×C8×C9 ◁ G | |
162 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 576 |
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)(18,19,20,23,21,24,26,29,22,25,27,30,28,31,32,33);(10,12,13,17)(11,14,15,16)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 54);