Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1280 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C22⋊C4 |
| commutator | C22 | C32×C4×C8 |
| Frattini, cosocle | C23×C4 | C22×C32 |
| prime radical | C24×C32 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C23×C32×C4 ◁ G | |
792 conjugacy classes of subgroups; only the 312 normal subgroups are shown.
This group has 792 conjugacy classes of subgroups — too many to draw. Only its 312 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(11,12)(13,17,18,15)(14,16)(19,20,21,23,22,24,25,26);(4,5,6)(7,8,9,10)(11,13)(12,15)(14,18)(16,17)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2340);