Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×C4×D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | D8 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C22×C32 | C2×D8 |
| prime radical | C2×C32×D16 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C32×C42 ◁ C32×C4×C8 ◁ G | |
522 conjugacy classes of subgroups; only the 252 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 252 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 — 3 generators on 26 points:
(1,2,3)(7,8,9,10);(4,5,6)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3232);