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Magma
magma: G := TransitiveGroup(10, 35);
Group action invariants
Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $(A_6 : C_2) : C_2$ | ||
CHM label: | $L(10).2^{2}=P|L(2,9)$ | ||
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)(4,7)(5,8)(9,10), (1,2,10)(3,4,5)(6,7,8), (1,7,3,4,2,5,6,8), (3,6)(4,7)(5,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 5: None
Low degree siblings
12T220, 20T201, 20T204, 20T208, 24T2960, 30T264, 36T2341, 40T1198, 40T1199, 40T1201, 45T187Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6,10)( 7, 9)$ |
$ 5, 5 $ | $144$ | $5$ | $( 1, 7, 4,10, 5)( 2, 6, 8, 3, 9)$ |
$ 10 $ | $144$ | $10$ | $( 1, 6, 7, 8, 4, 3,10, 9, 5, 2)$ |
$ 2, 2, 2, 2, 1, 1 $ | $45$ | $2$ | $( 1, 2)( 4, 7)( 5, 8)( 9,10)$ |
$ 4, 4, 1, 1 $ | $90$ | $4$ | $( 1, 9, 2,10)( 4, 8, 7, 5)$ |
$ 8, 1, 1 $ | $180$ | $8$ | $( 1, 8,10, 4, 2, 5, 9, 7)$ |
$ 8, 2 $ | $180$ | $8$ | $( 1, 9, 2, 4, 7, 3, 8, 5)( 6,10)$ |
$ 2, 2, 2, 1, 1, 1, 1 $ | $30$ | $2$ | $( 1, 2)( 3, 6)( 9,10)$ |
$ 3, 3, 3, 1 $ | $80$ | $3$ | $( 2, 9,10)( 3, 7, 5)( 4, 8, 6)$ |
$ 6, 3, 1 $ | $240$ | $6$ | $( 2, 5, 9, 3,10, 7)( 4, 6, 8)$ |
$ 4, 4, 2 $ | $90$ | $4$ | $( 1,10, 2, 9)( 3, 6)( 4, 8, 7, 5)$ |
$ 4, 4, 1, 1 $ | $180$ | $4$ | $( 1, 4, 2, 7)( 5,10, 8, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $1440=2^{5} \cdot 3^{2} \cdot 5$ | 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: | 1440.5841 | magma: IdentifyGroup(G);
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Character table: |
2 5 4 1 1 5 4 3 3 3 1 1 3 4 3 2 1 2 1 . . . . . . . . . 5 1 . . . . . . . 1 1 1 . . 1a 2a 3a 6a 2b 4a 8a 8b 2c 5a 10a 4b 4c 2P 1a 1a 3a 3a 1a 2b 4a 4a 1a 5a 5a 2b 2b 3P 1a 2a 1a 2a 2b 4a 8a 8b 2c 5a 10a 4b 4c 5P 1a 2a 3a 6a 2b 4a 8a 8b 2c 1a 2c 4b 4c 7P 1a 2a 3a 6a 2b 4a 8a 8b 2c 5a 10a 4b 4c X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 X.3 1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 X.4 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 1 X.5 9 -3 . . 1 1 -1 -1 1 -1 1 1 1 X.6 9 -3 . . 1 1 1 1 -1 -1 -1 -1 1 X.7 9 3 . . 1 1 -1 1 1 -1 1 -1 -1 X.8 9 3 . . 1 1 1 -1 -1 -1 -1 1 -1 X.9 10 2 1 -1 2 -2 . . . . . . 2 X.10 10 -2 1 1 2 -2 . . . . . . -2 X.11 16 . -2 . . . . . -4 1 1 . . X.12 16 . -2 . . . . . 4 1 -1 . . X.13 20 . 2 . -4 . . . . . . . . |
magma: CharacterTable(G);