Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #10624296 |
| automorphism | Aut(G) = PΓL(2,𝔽9) (order 1440) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, prime radical, cosocle, hypocenter | A6 | C2 |
| LCS | G ▷ A6 | |
| UCS | C1 | |
25 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 10 |
A minimal faithful permutation representation — 2 generators on 10 points:
(2,5,6)(3,10,7)(4,9,8);(1,8,9,7,2,6,3,5)(4,10)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
|---|---|---|---|---|---|---|---|---|
| size | 1 | 45 | 80 | 90 | 90 | 90 | 144 | 180 |
| order | 1 | 2 | 3 | 4 | 8 | 8 | 5 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 |
| χ3 | 9 | 1 | 0 | 1 | 1 | 1 | -1 | -1 |
| χ4 | 9 | 1 | 0 | 1 | -1 | -1 | -1 | 1 |
| χ5 | 10 | 2 | 1 | -2 | 0 | 0 | 0 | 0 |
| χ6 | 10 | -2 | 1 | 0 | 0 | 0 | ||
| χ7 | 10 | -2 | 1 | 0 | 0 | 0 | ||
| χ8 | 16 | 0 | -2 | 0 | 0 | 0 | 1 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | PΓL(2,𝔽9), C2×#10624296, M11 |
| maximal normal subgroup of | PΓL(2,𝔽9), C2×#10624296 |
| maximal quotient of | #13088278, C2×#10624296 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 765);