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) = C25 |
| direct product | C2×#41×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C24×C32 |
| socle | C22×C3 | C23×C32 |
| prime radical | C2×C32×#41 | C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
656 conjugacy classes of subgroups; only the 490 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 490 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(11,12,14,13,16,18,17,15,19);(3,4)(5,8)(6,7)(9,10)(12,15,16)(14,18,19);(4,7)(8,10);(3,5,6,9)(4,8,7,10)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2659);