Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C8 · Schur multiplier (H2) = C22×C33 |
| direct product | C33×C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C33×C4 | C22 |
| commutator | C2 | C2×C33×C8 |
| Frattini, cosocle | C2×C4 | C22×C33 |
| socle | C22×C33 | C2×C4 |
| prime radical | C23×C33 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33×C4 ◁ G | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3)(10,11)(12,13)(14,15,16,18,17,19,20,21);(4,5,6)(11,13);(7,8,9)540 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2178);