Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of C3⋊PSU(3,𝔽2) |
| automorphism | Aut(G) = D3×AΓL(1,𝔽9) (order 864) · Inn(G) = G · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊PSU(3,𝔽2) |
| Fitting, socle, Sylow | C33 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, prime radical, cosocle | C3×C3⋊D3 | C22 |
| Fitting, socle, hypocenter | C33 | Q8 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3)(4,10,9,11)(6,7,12,8);(2,3)(5,7,9,11)(6,10,8,12)Polycyclic presentation:
2^3,3^3|a^2=c,b^2=c,b^a=bc,d^a=d^2e,e^a=de,f^a=f^2,d^b=e^2,e^b=d,d^c=d^2,e^c=e^2| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 2 | 8 | 8 | 8 | 9 | 18 | 18 | 18 | 18 | 54 | 54 |
| order | 1 | 3 | 3 | 3 | 3 | 2 | 4 | 6 | 12 | 12 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 |
| χ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 |
| χ5 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 |
| χ6 | 2 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 |
| χ7 | 2 | -1 | 2 | -1 | -1 | 2 | -2 | -1 | 1 | 1 | 0 | 0 |
| χ8 | 2 | -1 | 2 | -1 | -1 | -2 | 0 | 1 | 0 | 0 | ||
| χ9 | 2 | -1 | 2 | -1 | -1 | -2 | 0 | 1 | 0 | 0 | ||
| χ10 | 8 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ11 | 8 | -4 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
| χ12 | 8 | -4 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | #89091, AGL(1,𝔽9)⋊D3, D3×PSU(3,𝔽2), … |
| maximal normal subgroup of | #89091, AGL(1,𝔽9)⋊D3, D3×PSU(3,𝔽2), … |
| maximal quotient of | #88945, #88946, C2×C3⋊PSU(3,𝔽2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 161);