Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C22×C32×SD32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | D8 |
| commutator, Frattini, cosocle | C8 | C24×C32 |
| prime radical | C22×C32×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
1200 conjugacy classes of subgroups; only the 600 normal subgroups are shown.
This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 600 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 — 4 generators on 26 points:
(1,2)(21,22,23);(3,4)(24,25,26);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,12)(7,19)(8,10)(9,17)(11,15)(14,20)(16,18)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4469);