Finite Group DatabasebetaFinite Group Databasebeta| order | 2520 |
| isoclinism | isoclinic class of A7 |
| automorphism | Aut(G) = S7 (order 5040) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2×C3 |
| alternating | A7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 7 |
| transitive degree | μt(G) = 7 |
A minimal faithful permutation representation — 2 generators on 7 points:
(1,2,3,4,5,6,7);(1,2,3)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 70 | 105 | 210 | 280 | 360 | 360 | 504 | 630 |
| order | 1 | 3 | 2 | 6 | 3 | 7 | 7 | 5 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 6 | 3 | 2 | -1 | 0 | -1 | -1 | 1 | 0 |
| χ3 | 10 | 1 | -2 | 1 | 1 | 0 | 0 | ||
| χ4 | 10 | 1 | -2 | 1 | 1 | 0 | 0 | ||
| χ5 | 14 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | 0 |
| χ6 | 14 | -1 | 2 | -1 | 2 | 0 | 0 | -1 | 0 |
| χ7 | 15 | 3 | -1 | -1 | 0 | 1 | 1 | 0 | -1 |
| χ8 | 21 | -3 | 1 | 1 | 0 | 0 | 0 | 1 | -1 |
| χ9 | 35 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | S7, C2×A7, #72227517 |
| maximal normal subgroup of | S7, C2×A7 |
| maximal quotient of | #72209611, #72209622, C2×A7, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4,5,6,7), (1,2,3));