Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #3688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×#3688 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | (C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C22×C3 |
| Frattini | C2×C3 | C2×C3×S4 |
| Fitting | C2×C32×Q8 | D3 |
| socle, prime radical | C22×C32 | S4 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C22×C3 | |
148 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#3688, #13146302, #13146306, … |
| maximal normal subgroup of | Dic3×#3688, #13146302, #13146306, … |
| maximal quotient of | C3×C4×#3688, C3×C4⋊#3688, C3×#10589493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2310);