Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C33×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C33 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C33 |
| prime radical | C23×C33 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ G | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(11,15)(12,19)(17,24)(18,22)(20,23)(21,25);(4,5,6)(10,11,16,22)(12,17,23,24)(13,15,14,18)(19,25,20,21);(7,8,9)(10,12,13,20)(11,17,15,21)(14,19,16,23)(18,25,22,24)378 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3974);