Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of S4×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | S4×#157 |
| Fitting, socle | C22×C32 ⋊ C4×D3 |
| derived | C32×A4 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C32×A4 | C2×C4 |
| Fitting, socle | C22×C32 | C4×D3 |
| prime radical | C3⋊D3×S4 | C2 |
| cosocle | A4×C3⋊D3 | C22 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 | |
194 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
30 conjugacy classes — view the full character table.
| maximal subgroup of | S4×AGL(1,𝔽9), S4×GO+(4,𝔽2), S4×PSU(3,𝔽2), … |
| maximal normal subgroup of | S4×AGL(1,𝔽9), S4×GO+(4,𝔽2), S4×PSU(3,𝔽2), … |
| maximal quotient of | #13180179, #13180180, #13180181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4668);