Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#2197 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C32 |
| prime radical | C32×B(4,𝔽2) | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32 ◁ C2×C32×D4 ◁ G | |
774 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 774 conjugacy classes of subgroups — too many to draw. Only its 168 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)(9,15)(10,13)(11,14)(12,16)(17,21)(18,20)(19,22);(4,5,6)(7,9,11,17)(8,12,14,19)(10,16,18,22)(13,15,20,21);(10,18)(16,22)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3257);