Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C27 · Schur multiplier (H2) = C22 |
| direct product | C27×C4⋊Q8 |
| unambiguous semidirect product | C4⋊(Q8×C27) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C27 | C23 |
| commutator | C22 | C23×C27 |
| Frattini, cosocle | C22×C9 | C23×C3 |
| socle, prime radical | C22×C3 | C23×C9 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C27 ◁ G | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(29,31)(32,33,36,39)(34,38,35,37);(32,34,36,35)(33,37,39,38);(28,29,30,31)(32,35,36,34)(33,38,39,37)378 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 184);