Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C26×C32 |
| direct product | C23×C4×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C32 |
| commutator | C3 | C23×C32×C4 |
| Frattini, cosocle | C2×C3 | C24×C32 |
| socle | C24×C3 | C2×C32 |
| prime radical | C24×H(3,𝔽3) | C2 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(11,12,14)(13,16,18)(15,19,17);(3,4)(12,15,16)(14,18,19);(5,6);(7,8,9,10)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2649);