Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1343 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #99 |
| commutator | C22×C4 | C2×C32×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C22×C32 |
| prime radical | C32×#392 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C23×C32 ◁ C2×C32×C22⋊C4 ◁ G | |
666 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 666 conjugacy classes of subgroups — too many to draw. Only its 120 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 — 2 generators on 22 points:
(1,2,3)(7,8)(9,17,19,12)(10,15)(11,16)(13,18,21,22)(14,20);(4,5,6)(8,12)(9,18)(13,15)(14,21)(16,17)(19,22)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2403);