Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2 |
| direct product | 3−1+2×#101 |
| unambiguous semidirect product | (C9×#101)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×C22⋊C4 |
| commutator | C22×C3 | C2×C32×C4 |
| Frattini, cosocle | C2×C3×C4 | C22×C32 |
| prime radical | C22×C32 | C2×C3×C4 |
| LCS | G ▷ C22×C3 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C4×3−1+2 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(10,11,13,17,14,16,22,24)(12,18,19,25,20,15,21,23);(2,5,6)(4,8,9)(10,12,14,20)(11,15,16,18)(13,21,22,19)(17,25,24,23)121 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 574);