Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#1280 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C22⋊C4 |
| commutator | C22 | C4×C8×C9 |
| Frattini, cosocle | C23×C3×C4 | C22×C3 |
| socle | C23×C3 | C3×C22⋊C4 |
| prime radical | C24×C3 | C2×C3×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C23×C4×C9 ◁ G | |
396 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(14,15)(16,20,21,18)(17,19)(22,23,24,26,25,27,28,29);(10,11,12,13)(14,16)(15,18)(17,21)(19,20)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 12);