Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#1408 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #353 |
| commutator | C42 | C2×C4×C9 |
| Frattini, cosocle | C3×#127 | C22×C3 |
| socle | C2×C3 | C3×#353 |
| prime radical | C3×#493 | C2×C3 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×#127 ◁ G | |
180 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,19,20,15)(13,17,23,16)(14,18)(21,24,22,25);(10,12)(11,15)(13,22)(14,21)(16,24)(17,19)(18,25)(20,23)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 140);