Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of (C3⋊#857)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | (C3⋊#857)⋊C3 |
| Fitting | C32×C42 ⋊ D3 |
| derived | C3×C42⋊C3 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C42⋊C3 | C2×C3 |
| Frattini, socle | C22×C3 | C3×S4 |
| Fitting | C32×C42 | D3 |
| LCS | G ▷ C3×C42⋊C3 | |
| UCS | C1 | |
100 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3^3,2^4|e^2=g,f^2=gh,c^a=c^2,d^a=d^2,e^a=fh,f^a=eh,g^a=gh,c^b=cd,e^c=fg,f^c=ef,g^c=gh,h^c=g46 conjugacy classes — view the full character table.
| maximal subgroup of | #13146253, #13146254, C2×(C3⋊#857)⋊C3, … |
| maximal normal subgroup of | #13146253, #13146254, C2×(C3⋊#857)⋊C3, … |
| maximal quotient of | (C3⋊#4621)⋊C3, (C3⋊#4622)⋊C3, (C3⋊#4623)⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 686);