Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26×C3 |
| direct product | C23×(C3⋊Dic3)⋊C3 |
| derived | C32 ⋊ C23×C3×C4 |
| Sylow | H(3,𝔽3) ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C23×C3×C4 |
| Frattini | C2×C3 | C23×C3×D3 |
| Fitting, prime radical | C24×H(3,𝔽3) | C2 |
| socle | C24×C3 | C3×D3 |
| cosocle | C2×C32 | C24×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C24 | |
960 conjugacy classes of subgroups; only the 437 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 437 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11720334, #13141147, #13141150, … |
| maximal normal subgroup of | C2×#11720334, #13141147, #13141150, … |
| maximal quotient of | C23×(C3⋊(C3⋊C8))⋊C3, C22×#3748⋊C3, C22×C4×(C3⋊Dic3)⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2614);