Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #89091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C33 ⋊ #103 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #89091 |
| commutator | C3×#157 | C2×C4 |
| Fitting, socle | C2×C33 | SD16 |
| prime radical | C2×C3×C3⋊D3 | D4 |
| cosocle | C2×C3×#157 | C22 |
| hypocenter | C33 | #103 |
| LCS | G ▷ C3×#157 ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
106 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^5,3^3|a^2=c,b^2=d,c^2=e,b^a=bcde,f^a=fg,g^a=f,h^a=h^2,c^b=ce,f^b=fg^2,g^b=g^2,f^c=f^2g,g^c=fg,f^e=f^2,g^e=g^236 conjugacy classes — view the full character table.
| maximal subgroup of | D3×PSU(3,𝔽2)⋊C4, (AGL(1,𝔽9)⋊D3)⋊C4, #13180140, … |
| maximal normal subgroup of | D3×PSU(3,𝔽2)⋊C4, (AGL(1,𝔽9)⋊D3)⋊C4, #13180140, … |
| maximal quotient of | #13165089, #13165092, #13165093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4066);