Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#2197 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C9 |
| Frattini, cosocle | C2×C3×D4 | C23×C3 |
| socle | C2×C3 | C3×B(4,𝔽2) |
| prime radical | C3×B(4,𝔽2) | C2×C3 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C9 ◁ C2×D4×C9 ◁ G | |
387 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 84 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)(12,18)(13,16)(14,17)(15,19)(20,24)(21,23)(22,25);(10,12,14,20)(11,15,17,22)(13,19,21,25)(16,18,23,24);(13,21)(19,25)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 929);