Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#1613 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #120 |
| commutator | C22×C4 | C23×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C22×C3 | C3×#120 |
| prime radical | C3×#534 | C2×C3 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C22×C4×C9 ◁ G | |
432 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 432 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) = 25 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,16)(12,14,17,15)(19,22)(20,24)(23,25);(11,14)(12,17)(15,16)(18,19)(20,25)(21,23)(22,24);(20,24)(22,25)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 345);