Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#1819 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C2×C22⋊C4 |
| commutator | C23 | C22×C32×C4 |
| Frattini, cosocle | C24 | C23×C32 |
| prime radical | C25×C32 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C25×C32 ◁ G | |
1512 conjugacy classes of subgroups; only the 336 normal subgroups are shown.
This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 336 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)(9,15,17,13)(10,12)(11,14)(16,21,22,19)(18,20);(4,5,6)(8,12)(13,19)(14,20)(15,21);(7,9,10,16)(8,13,12,19)(11,17,18,22)(14,15,20,21)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2879);