Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C32 |
| direct product | C22×D4×H(3,𝔽3) |
| unambiguous semidirect product | C4⋊(C23×H(3,𝔽3)) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C24×C32 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(9,10,12)(11,14,16)(13,17,15);(3,4)(10,13,14)(12,16,17);(5,6,7,8);(6,8)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2652);