Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of ((C22⋊C9)⋊C2)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×#1094⋊C4 |
| unambiguous semidirect product | (C2×#1094)⋊C4 |
| unambiguous semidirect product | (C2×(C22⋊C9)⋊C4)⋊C3 |
| derived | C22⋊C9 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ((C22⋊C9)⋊C2)⋊C3 |
| commutator, hypocenter | C22⋊C9 | C2×C3×C4 |
| Frattini | C2×C3 | C2×C3×S4 |
| Fitting, prime radical | C24×C32 | D3 |
| socle | C24×C3 | C3×D3 |
| cosocle | C2×C22⋊C9 | C22×C3 |
| LCS | G ▷ C22⋊C9 | |
| UCS | C1 ◁ C22 | |
249 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1094⋊C4, (#1094⋊C4)⋊C4, C4⋊(#1094⋊C4), … |
| maximal normal subgroup of | C4×#1094⋊C4, (#1094⋊C4)⋊C4, C4⋊(#1094⋊C4), … |
| maximal quotient of | C2×#1094⋊C8, #10589511⋊C3, C4×#1094⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2327);