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Magma
magma: G := TransitiveGroup(14, 20);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_7 \wr C_2$ | ||
CHM label: | $[D(7)^{2}]2=D(7)wr2$ | ||
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), (2,12)(4,10)(6,8), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 7: None
Low degree siblings
14T20, 28T53 x 2, 28T54 x 2, 28T55 x 2, 28T57Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $14$ | $2$ | $( 4,14)( 6,12)( 8,10)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $49$ | $2$ | $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$ |
$ 7, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $7$ | $( 2, 4, 6, 8,10,12,14)$ |
$ 7, 2, 2, 2, 1 $ | $28$ | $14$ | $( 2, 4, 6, 8,10,12,14)( 3,13)( 5,11)( 7, 9)$ |
$ 7, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $7$ | $( 2, 6,10,14, 4, 8,12)$ |
$ 7, 2, 2, 2, 1 $ | $28$ | $14$ | $( 2, 6,10,14, 4, 8,12)( 3,13)( 5,11)( 7, 9)$ |
$ 7, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $7$ | $( 2, 8,14, 6,12, 4,10)$ |
$ 7, 2, 2, 2, 1 $ | $28$ | $14$ | $( 2, 8,14, 6,12, 4,10)( 3,13)( 5,11)( 7, 9)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $14$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$ |
$ 4, 4, 4, 2 $ | $98$ | $4$ | $( 1, 2)( 3, 4,13,14)( 5, 6,11,12)( 7, 8, 9,10)$ |
$ 14 $ | $28$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$ |
$ 14 $ | $28$ | $14$ | $( 1, 2, 5, 6, 9,10,13,14, 3, 4, 7, 8,11,12)$ |
$ 14 $ | $28$ | $14$ | $( 1, 2, 7, 8,13,14, 5, 6,11,12, 3, 4, 9,10)$ |
$ 7, 7 $ | $4$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ |
$ 7, 7 $ | $8$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$ |
$ 7, 7 $ | $8$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$ |
$ 7, 7 $ | $4$ | $7$ | $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$ |
$ 7, 7 $ | $8$ | $7$ | $( 1, 5, 9,13, 3, 7,11)( 2, 8,14, 6,12, 4,10)$ |
$ 7, 7 $ | $4$ | $7$ | $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $392=2^{3} \cdot 7^{2}$ | 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: | 392.37 | magma: IdentifyGroup(G);
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Character table: |
2 3 2 3 1 1 1 1 1 1 2 2 1 1 1 1 . . 1 . 1 7 2 1 . 2 1 2 1 2 1 1 . 1 1 1 2 2 2 2 2 2 1a 2a 2b 7a 14a 7b 14b 7c 14c 2c 4a 14d 14e 14f 7d 7e 7f 7g 7h 7i 2P 1a 1a 1a 7b 7b 7c 7c 7a 7a 1a 2b 7d 7g 7i 7g 7h 7e 7i 7f 7d 3P 1a 2a 2b 7c 14c 7a 14a 7b 14b 2c 4a 14f 14d 14e 7i 7f 7h 7d 7e 7g 5P 1a 2a 2b 7b 14b 7c 14c 7a 14a 2c 4a 14e 14f 14d 7g 7h 7e 7i 7f 7d 7P 1a 2a 2b 1a 2a 1a 2a 1a 2a 2c 4a 2c 2c 2c 1a 1a 1a 1a 1a 1a 11P 1a 2a 2b 7c 14c 7a 14a 7b 14b 2c 4a 14f 14d 14e 7i 7f 7h 7d 7e 7g 13P 1a 2a 2b 7a 14a 7b 14b 7c 14c 2c 4a 14d 14e 14f 7d 7e 7f 7g 7h 7i X.1 1 1 1 1 1 1 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 -1 1 1 1 1 1 1 X.3 1 -1 1 1 -1 1 -1 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 -1 1 1 1 1 1 1 X.5 2 . -2 2 . 2 . 2 . . . . . . 2 2 2 2 2 2 X.6 4 -2 . A J C L B K . . . . . D M N F O E X.7 4 -2 . B K A J C L . . . . . E N O D M F X.8 4 -2 . C L B K A J . . . . . F O M E N D X.9 4 . . D . F . E . -2 . J L K C N O B M A X.10 4 . . E . D . F . -2 . K J L A O M C N B X.11 4 . . F . E . D . -2 . L K J B M N A O C X.12 4 . . D . F . E . 2 . -J -L -K C N O B M A X.13 4 . . E . D . F . 2 . -K -J -L A O M C N B X.14 4 . . F . E . D . 2 . -L -K -J B M N A O C X.15 4 2 . A -J C -L B -K . . . . . D M N F O E X.16 4 2 . B -K A -J C -L . . . . . E N O D M F X.17 4 2 . C -L B -K A -J . . . . . F O M E N D X.18 8 . . G . I . H . . . . . . H P Q G R I X.19 8 . . H . G . I . . . . . . I Q R H P G X.20 8 . . I . H . G . . . . . . G R P I Q H A = -2*E(7)-2*E(7)^2-E(7)^3-E(7)^4-2*E(7)^5-2*E(7)^6 B = -2*E(7)-E(7)^2-2*E(7)^3-2*E(7)^4-E(7)^5-2*E(7)^6 C = -E(7)-2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5-E(7)^6 D = 2*E(7)^3+2*E(7)^4 E = 2*E(7)^2+2*E(7)^5 F = 2*E(7)+2*E(7)^6 G = 2*E(7)+2*E(7)^3+2*E(7)^4+2*E(7)^6 H = 2*E(7)^2+2*E(7)^3+2*E(7)^4+2*E(7)^5 I = 2*E(7)+2*E(7)^2+2*E(7)^5+2*E(7)^6 J = -E(7)^3-E(7)^4 K = -E(7)^2-E(7)^5 L = -E(7)-E(7)^6 M = E(7)+E(7)^3+E(7)^4+E(7)^6 N = E(7)^2+E(7)^3+E(7)^4+E(7)^5 O = E(7)+E(7)^2+E(7)^5+E(7)^6 P = -E(7)-2*E(7)^3-2*E(7)^4-E(7)^6 Q = -2*E(7)^2-E(7)^3-E(7)^4-2*E(7)^5 R = -2*E(7)-E(7)^2-E(7)^5-2*E(7)^6 |
magma: CharacterTable(G);