Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C9 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×C3×C9×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C9 | C22 |
| commutator | C2 | C24×C3×C9 |
| Frattini, cosocle | C2×C3 | C24×C32 |
| socle | C22×C32 | C23×C3 |
| prime radical | C2×C32×#41 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C9 ◁ G | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2)(11,12,13);(3,4)(5,8)(6,7)(9,10)(14,15,16,17,18,19,20,21,22);(4,7)(8,10);(3,5,6,9)(4,8,7,10)540 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2657);