Show commands:
Magma
magma: G := TransitiveGroup(8, 49);
Group action invariants
| Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $49$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $A_8$ | ||
| CHM label: | $A8$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2)(3,4,5,6,7,8), (1,2,3) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Low degree siblings
15T72 x 2, 28T433, 35T36Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{8}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{4}$ | $105$ | $2$ | $4$ | $(1,2)(3,4)(5,6)(7,8)$ |
| 2B | $2^{2},1^{4}$ | $210$ | $2$ | $2$ | $(1,2)(3,4)$ |
| 3A | $3,1^{5}$ | $112$ | $3$ | $2$ | $(1,2,3)$ |
| 3B | $3^{2},1^{2}$ | $1120$ | $3$ | $4$ | $(1,2,3)(4,5,6)$ |
| 4A | $4^{2}$ | $1260$ | $4$ | $6$ | $(1,2,3,4)(5,6,7,8)$ |
| 4B | $4,2,1^{2}$ | $2520$ | $4$ | $4$ | $(1,2,3,4)(5,6)$ |
| 5A | $5,1^{3}$ | $1344$ | $5$ | $4$ | $(1,2,3,4,5)$ |
| 6A | $3,2^{2},1$ | $1680$ | $6$ | $4$ | $(1,2,3)(4,5)(6,7)$ |
| 6B | $6,2$ | $3360$ | $6$ | $6$ | $(1,2,3,4,5,6)(7,8)$ |
| 7A1 | $7,1$ | $2880$ | $7$ | $6$ | $(1,2,3,4,5,6,7)$ |
| 7A-1 | $7,1$ | $2880$ | $7$ | $6$ | $(1,3,4,5,6,7,2)$ |
| 15A1 | $5,3$ | $1344$ | $15$ | $6$ | $(1,2,3,4,5)(6,7,8)$ |
| 15A-1 | $5,3$ | $1344$ | $15$ | $6$ | $(1,3,4,5,2)(6,7,8)$ |
Malle's constant $a(G)$: $1/2$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 20160.a | magma: IdentifyGroup(G);
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| Character table: |
| 1A | 2A | 2B | 3A | 3B | 4A | 4B | 5A | 6A | 6B | 7A1 | 7A-1 | 15A1 | 15A-1 | ||
| Size | 1 | 105 | 210 | 112 | 1120 | 1260 | 2520 | 1344 | 1680 | 3360 | 2880 | 2880 | 1344 | 1344 | |
| 2 P | 1A | 1A | 1A | 3A | 3B | 2A | 2B | 5A | 3A | 3B | 7A1 | 7A-1 | 15A1 | 15A-1 | |
| 3 P | 1A | 2A | 2B | 1A | 1A | 4A | 4B | 5A | 2B | 2A | 7A-1 | 7A1 | 5A | 5A | |
| 5 P | 1A | 2A | 2B | 3A | 3B | 4A | 4B | 1A | 6A | 6B | 7A-1 | 7A1 | 3A | 3A | |
| 7 P | 1A | 2A | 2B | 3A | 3B | 4A | 4B | 5A | 6A | 6B | 1A | 1A | 15A-1 | 15A1 | |
| Type |
magma: CharacterTable(G);