Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of C3⋊PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23×C3 |
| unambiguous semidirect product | C3⋊#63747 |
| Sylow | C33 ⋊ #95 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C3⋊PSU(3,𝔽2) |
| commutator | C3×C3⋊D3 | C42 |
| Fitting, socle | C22×C33 | Q8 |
| prime radical, cosocle | C22×C3×C3⋊D3 | C22 |
| hypocenter | C33 | #95 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C22 | |
194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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=c,b^2=cd,b^a=bce,f^a=g,g^a=f^2,h^a=h^2,f^b=fg^2,g^b=f^2g^2,f^c=f^2,g^c=g^248 conjugacy classes — view the full character table.
| maximal subgroup of | #13165099, #13165135, D3×#63747, … |
| maximal normal subgroup of | #13165099, #13165135, D3×#63747, … |
| maximal quotient of | #13155160, #13155161, #13155162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2776);