Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×PSU(3,𝔽2) |
| commutator, prime radical, cosocle | C2×C3×C3⋊D3 | C23 |
| Fitting, socle | C2×C33 | C2×Q8 |
| hypocenter | C33 | #125 |
| LCS | G ▷ C2×C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
134 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^5,3^3|a^2=e,b^2=d,c^2=d,c^a=ce,h^a=h^2,c^b=cde,f^b=g,g^b=f^2,f^c=fg^2,g^c=f^2g^2,f^d=f^2,g^d=g^230 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#64243, #13180616, #13180618, … |
| maximal normal subgroup of | D3×#64243, #13180616, #13180618, … |
| maximal quotient of | #13165798, #13165802, #13165803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4119);