# Properties

 Label 8T49 Degree $8$ Order $20160$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $A_8$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(8, 49);

## Group action invariants

 Degree $n$: $8$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $49$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $A_8$ CHM label: $A8$ Parity: $1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2)(3,4,5,6,7,8), (1,2,3) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 4: None

## Low degree siblings

15T72 x 2, 28T433, 35T36

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2$ $105$ $2$ $(1,3)(2,6)(4,5)(7,8)$ $3, 3, 1, 1$ $1120$ $3$ $(1,6,5)(2,4,3)$ $6, 2$ $3360$ $6$ $(1,4,6,3,5,2)(7,8)$ $2, 2, 1, 1, 1, 1$ $210$ $2$ $(4,6)(5,7)$ $4, 2, 1, 1$ $2520$ $4$ $(1,2)(4,5,6,7)$ $4, 4$ $1260$ $4$ $(1,3,2,8)(4,7,6,5)$ $7, 1$ $2880$ $7$ $(1,7,4,5,6,8,2)$ $7, 1$ $2880$ $7$ $(1,2,8,6,5,4,7)$ $3, 1, 1, 1, 1, 1$ $112$ $3$ $(1,7,2)$ $3, 2, 2, 1$ $1680$ $6$ $(1,2,7)(3,5)(4,6)$ $5, 1, 1, 1$ $1344$ $5$ $(1,5,8,2,4)$ $5, 3$ $1344$ $15$ $(1,8,4,5,2)(3,7,6)$ $5, 3$ $1344$ $15$ $(1,8,4,5,2)(3,6,7)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 20160.a magma: IdentifyGroup(G);
 Character table: 2 6 2 . . . . . 6 1 1 5 2 3 4 3 2 2 1 1 1 . . 1 2 1 1 1 . . 5 1 1 1 1 1 . . . . . . . . . 7 1 . . . . 1 1 . . . . . . . 1a 3a 5a 15a 15b 7a 7b 2a 3b 6a 2b 6b 4a 4b 2P 1a 3a 5a 15a 15b 7a 7b 1a 3b 3b 1a 3a 2b 2a 3P 1a 1a 5a 5a 5a 7b 7a 2a 1a 2a 2b 2b 4a 4b 5P 1a 3a 1a 3a 3a 7b 7a 2a 3b 6a 2b 6b 4a 4b 7P 1a 3a 5a 15b 15a 1a 1a 2a 3b 6a 2b 6b 4a 4b 11P 1a 3a 5a 15b 15a 7a 7b 2a 3b 6a 2b 6b 4a 4b 13P 1a 3a 5a 15b 15a 7b 7a 2a 3b 6a 2b 6b 4a 4b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 7 4 2 -1 -1 . . -1 1 -1 3 . 1 -1 X.3 14 -1 -1 -1 -1 . . 6 2 . 2 -1 . 2 X.4 20 5 . . . -1 -1 4 -1 1 4 1 . . X.5 21 6 1 1 1 . . -3 . . 1 -2 -1 1 X.6 21 -3 1 A /A . . -3 . . 1 1 -1 1 X.7 21 -3 1 /A A . . -3 . . 1 1 -1 1 X.8 28 1 -2 1 1 . . -4 1 -1 4 1 . . X.9 35 5 . . . . . 3 2 . -5 1 -1 -1 X.10 45 . . . . B /B -3 . . -3 . 1 1 X.11 45 . . . . /B B -3 . . -3 . 1 1 X.12 56 -4 1 1 1 . . 8 -1 -1 . . . . X.13 64 4 -1 -1 -1 1 1 . -2 . . . . . X.14 70 -5 . . . . . -2 1 1 2 -1 . -2 A = -E(15)-E(15)^2-E(15)^4-E(15)^8 = (-1-Sqrt(-15))/2 = -1-b15 B = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7

magma: CharacterTable(G);