Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C8 · Schur multiplier (H2) = C23×C32 |
| direct product | C22×C8×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C32 |
| commutator | C3 | C22×C32×C8 |
| Frattini, cosocle | C3×C4 | C23×C32 |
| socle | C23×C3 | C32×C4 |
| prime radical | C23×H(3,𝔽3) | C4 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(13,14,16)(15,18,20)(17,21,19);(3,4)(14,17,18)(16,20,21);(5,6,7,8,9,10,11,12)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1890);