Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of ((C3⋊S4)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×((C3⋊S4)⋊C3)⋊C2 |
| unambiguous semidirect product | ((C3⋊S4)⋊C3)⋊C22 |
| unambiguous semidirect product | #6258⋊C22 |
| unambiguous semidirect product | ((C3⋊D3)⋊A4)⋊C22 |
| Fitting | C23×C32 ⋊ C2×D3 |
| derived | C22⋊H(3,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ((C3⋊S4)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | C22⋊H(3,𝔽3) | C23 |
| Frattini | C3 | C2×D3×S4 |
| Fitting | C23×C32 | C2×D3 |
| socle | C23×C3 | D32 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
531 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 531 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 18 |
40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4000);