Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C33×#101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | C22⋊C4 |
| commutator | C22 | C2×C33×C4 |
| Frattini, cosocle | C2×C4 | C22×C33 |
| prime radical | C22×C33 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C2×C33×C4 ◁ G | |
532 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)(10,11,13,17,14,16,22,24)(12,18,19,25,20,15,21,23);(4,5,6)(10,12,14,20)(11,15,16,18)(13,21,22,19)(17,25,24,23);(7,8,9)297 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2181);