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×#125 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C33 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C33 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ G | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(12,16)(14,17)(18,19,21,23)(20,22,24,25);(4,5,6)(10,11,13,15)(12,14,16,17)(18,19)(20,22)(21,23)(24,25);(7,8,9)(10,12)(11,14)(13,16)(15,17)(18,20,21,24)(19,22,23,25)378 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3973);