Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#1354 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #99 |
| commutator | C22×C4 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C4⋊C4 | C22×C3 |
| socle | C22×C3 | C3×#99 |
| prime radical | C23×C3 | C3×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C23×C9 ◁ C2×C9×C4⋊C4 ◁ G | |
210 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(11,14,15,16)(18,19,21,25,22,24,20,23);(10,11)(12,16)(13,15)(14,17)(18,20,22,21)(19,23)(24,25)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 86);