Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of (C3⋊D3)⋊(C42⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3×C4 |
| unambiguous semidirect product | (C3⋊D3)⋊(C42⋊C3) |
| Fitting, derived | C32×C42 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C32×C42 | C2×C3 |
| Frattini, socle | C22×C3 | D3×A4 |
| LCS | G ▷ C32×C42 | |
| UCS | C1 | |
144 conjugacy classes of subgroups.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3,2^2,3^2,2^2|c^2=g,d^2=h,e^a=e^2,f^a=f^2,c^b=dg,d^b=cd,e^b=ef,g^b=h,h^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #13146253, C2×(C3⋊D3)⋊(C42⋊C3), #13147226, … |
| maximal normal subgroup of | #13146253, C2×(C3⋊D3)⋊(C42⋊C3), #13147226, … |
| maximal quotient of | #13136828, #13136831, (C3⋊Dic3)⋊(C42⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 726);