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Magma
magma: G := TransitiveGroup(14, 22);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_7^2:C_3:C_4$ | ||
CHM label: | $[1/6_-.F_{42}(7)^{2}]2_{2}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | 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: | (2,4,6,8,10,12,14), (1,6,13,8)(2,9,12,5)(3,4,11,10)(7,14), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8), (1,11,9)(2,4,8)(3,5,13)(6,12,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $6$: $S_3$ $12$: $C_3 : C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 7: None
Low degree siblings
28T75, 42T119, 42T125Siblings 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, 1, 1 $ | $1$ | $1$ | $()$ |
$ 7, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $7$ | $( 2, 4, 6, 8,10,12,14)$ |
$ 7, 7 $ | $12$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ |
$ 7, 7 $ | $12$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$ |
$ 7, 7 $ | $12$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $49$ | $2$ | $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$ |
$ 3, 3, 3, 3, 1, 1 $ | $98$ | $3$ | $( 3, 5, 9)( 4,10, 6)( 7,13,11)( 8,12,14)$ |
$ 6, 6, 1, 1 $ | $98$ | $6$ | $( 3,11, 9,13, 5, 7)( 4, 8, 6,14,10,12)$ |
$ 4, 4, 4, 2 $ | $147$ | $4$ | $( 1, 6,13, 8)( 2, 9,12, 5)( 3, 4,11,10)( 7,14)$ |
$ 4, 4, 4, 2 $ | $147$ | $4$ | $( 1,12,11, 8)( 2, 7, 4, 5)( 3,14, 9, 6)(10,13)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $588=2^{2} \cdot 3 \cdot 7^{2}$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 588.33 | magma: IdentifyGroup(G);
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Character table: |
2 2 . . . . 2 1 1 2 2 3 1 . . . . 1 1 1 . . 7 2 2 2 2 2 . . . . . 1a 7a 7b 7c 7d 2a 3a 6a 4a 4b 2P 1a 7a 7c 7d 7b 1a 3a 3a 2a 2a 3P 1a 7a 7d 7b 7c 2a 1a 2a 4b 4a 5P 1a 7a 7c 7d 7b 2a 3a 6a 4a 4b 7P 1a 1a 1a 1a 1a 2a 3a 6a 4b 4a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 1 1 -1 -1 X.3 1 1 1 1 1 -1 1 -1 D -D X.4 1 1 1 1 1 -1 1 -1 -D D X.5 2 2 2 2 2 -2 -1 1 . . X.6 2 2 2 2 2 2 -1 -1 . . X.7 12 5 -2 -2 -2 . . . . . X.8 12 -2 A C B . . . . . X.9 12 -2 B A C . . . . . X.10 12 -2 C B A . . . . . A = E(7)^2-2*E(7)^3-2*E(7)^4+E(7)^5 B = E(7)-2*E(7)^2-2*E(7)^5+E(7)^6 C = -2*E(7)+E(7)^3+E(7)^4-2*E(7)^6 D = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);