Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of (C3⋊D3)⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×(C3⋊D3)⋊A4 |
| Fitting | C24×C32 ⋊ C2×C3 |
| derived | C22×C32 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | (C3⋊D3)⋊A4 |
| commutator, cosocle, hypocenter | C22×C32 | C23×C3 |
| Frattini | C3 | C22×D3×A4 |
| Fitting | C24×C32 | C2×C3 |
| socle | C24×C3 | C3×D3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C22 | |
928 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 928 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | #13146399, #3752⋊A4, C2×((C3⋊D3)⋊A4)⋊C4, … |
| maximal normal subgroup of | #13146399, #3752⋊A4, C2×((C3⋊D3)⋊A4)⋊C4, … |
| maximal quotient of | C2×#634⋊A4, C2×C4×(C3⋊D3)⋊A4, C2×C4⋊((C3⋊D3)⋊A4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4026);