Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#2165 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #117 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C2×C32 | #432 |
| prime radical | C23×C32 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C32×#97 ◁ G | |
384 conjugacy classes of subgroups; only the 228 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 228 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(7,8,9,14,10,15,17,22,11,12,16,19,18,13,20,21);(4,5,6)(8,12)(9,16)(13,15)(17,20);(8,13,12,15)(9,16)(14,22,19,21)(17,20)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3225);