Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A4×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×A4×#157 |
| unambiguous semidirect product | (C23×#157)⋊C3 |
| Fitting, socle | C23×C32 ⋊ C3×C4 |
| derived | C22×C32 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×#157 |
| commutator, hypocenter | C22×C32 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | C3×C4 |
| prime radical | C2×A4×C3⋊D3 | C2 |
| cosocle | C22×C3⋊D3 | C22×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C2 | |
226 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | #13180177, #13180178, #157×SL(2,ℤ/4), … |
| maximal normal subgroup of | #13180177, #13180178, #157×SL(2,ℤ/4), … |
| maximal quotient of | A4×#3787, A4×#3788, C4×A4×#157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4689);