Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of S6 |
| automorphism | Aut(G) = PΓL(2,𝔽9) (order 1440) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| symmetric | S6 |
| symplectic | Sp(4,𝔽2) |
| projective symplectic | PSp(4,𝔽2) |
| projective special semilinear | PΣL(2,𝔽9) |
| socle, derived | A6 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | A6 | C2 |
| LCS | G ▷ A6 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 6 |
| transitive degree | μt(G) = 6 |
A minimal faithful permutation representation — 2 generators on 6 points:
(1,2);(1,2,3,4,5,6)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 15 | 15 | 40 | 40 | 45 | 90 | 90 | 120 | 120 | 144 |
| order | 1 | 2 | 2 | 3 | 3 | 2 | 4 | 4 | 6 | 6 | 5 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 |
| χ3 | 5 | 1 | -3 | 2 | -1 | 1 | -1 | -1 | 0 | 1 | 0 |
| χ4 | 5 | 3 | -1 | -1 | 2 | 1 | -1 | 1 | -1 | 0 | 0 |
| χ5 | 5 | -3 | 1 | -1 | 2 | 1 | -1 | -1 | 1 | 0 | 0 |
| χ6 | 5 | -1 | 3 | 2 | -1 | 1 | -1 | 1 | 0 | -1 | 0 |
| χ7 | 9 | 3 | 3 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | -1 |
| χ8 | 9 | -3 | -3 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | -1 |
| χ9 | 10 | 2 | -2 | 1 | 1 | -2 | 0 | 0 | 1 | -1 | 0 |
| χ10 | 10 | -2 | 2 | 1 | 1 | -2 | 0 | 0 | -1 | 1 | 0 |
| χ11 | 16 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 |
| maximal subgroup of | PΓL(2,𝔽9), C2×S6, A8, … |
| maximal normal subgroup of | PΓL(2,𝔽9), C2×S6 |
| maximal quotient of | #13088273, ΣL(2,𝔽9), C2×S6, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 763);