Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C33 · Schur multiplier (H2) = C23×C33 |
| direct product | C2×C33×D8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C33 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C33 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ C2×C33×C4 ◁ G | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(11,12,13);(3,4,5,6,7,8,9,10)(14,15,16);(4,10)(5,9)(6,8)(17,18,19)378 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3980);