Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #6290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×Dic3⋊(C3⋊Dic3) |
| Sylow | C33 ⋊ C2×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×C3⋊D3 |
| commutator | C2×C33 | C22×C4 |
| Frattini | C22 | C2×D3×C3⋊D3 |
| Fitting, socle, prime radical | C23×C33 | C22 |
| cosocle | C22×C33 | C23 |
| hypocenter | C33 | C2×C4⋊C4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
768 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 272 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | #13154846, Dic3×Dic3⋊Dic3, #13156704, … |
| maximal normal subgroup of | #13154846, Dic3×Dic3⋊Dic3, #13156704, … |
| maximal quotient of | #13160262, C2×Dic3⋊(C3⋊(C3⋊C8)), #13160393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3574);