Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#2069 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #396 |
| commutator | C22×C4 | C23×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#396 |
| prime radical | C22×C3×D4 | C22×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
324 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,15,17,14,18,13,22)(12,19,23,24,20,25,21,16);(10,12,13,21,14,20,15,23)(11,16,17,19,18,24,22,25);(12,20)(13,15)(17,22)(19,25)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 801);