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Magma
magma: G := TransitiveGroup(28, 35);
Group action invariants
Degree $n$: | $28$ | 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: | $C_7^2:C_4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $14$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9)(2,10)(3,11)(4,12)(5,6)(7,8)(13,25)(14,26)(15,27)(16,28)(17,22)(18,21)(19,24)(20,23), (1,28,21,8)(2,27,22,7)(3,25,4,26)(5,24,18,11)(6,23,17,12)(9,20,14,15)(10,19,13,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 7: None
Degree 14: 14T12
Low degree siblings
14T12 x 4, 28T35 x 3Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $7$ | $( 3, 8,12,16,20,24,27)( 4, 7,11,15,19,23,28)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $7$ | $( 3,12,20,27, 8,16,24)( 4,11,19,28, 7,15,23)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $7$ | $( 3,16,27,12,24, 8,20)( 4,15,28,11,23, 7,19)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $49$ | $2$ | $( 1, 2)( 3, 4)( 5,25)( 6,26)( 7,27)( 8,28)( 9,22)(10,21)(11,24)(12,23)(13,18) (14,17)(15,20)(16,19)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $49$ | $4$ | $( 1, 3, 2, 4)( 5, 7,25,27)( 6, 8,26,28)( 9,11,22,24)(10,12,21,23)(13,15,18,20) (14,16,17,19)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $49$ | $4$ | $( 1, 4, 2, 3)( 5,27,25, 7)( 6,28,26, 8)( 9,24,22,11)(10,23,21,12)(13,20,18,15) (14,19,17,16)$ |
$ 7, 7, 7, 7 $ | $4$ | $7$ | $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3, 8,12,16,20,24,27) ( 4, 7,11,15,19,23,28)$ |
$ 7, 7, 7, 7 $ | $4$ | $7$ | $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,12,20,27, 8,16,24) ( 4,11,19,28, 7,15,23)$ |
$ 7, 7, 7, 7 $ | $4$ | $7$ | $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$ |
$ 7, 7, 7, 7 $ | $4$ | $7$ | $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,20, 8,24,12,27,16) ( 4,19, 7,23,11,28,15)$ |
$ 7, 7, 7, 7 $ | $4$ | $7$ | $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,24,16, 8,27,20,12) ( 4,23,15, 7,28,19,11)$ |
$ 7, 7, 7, 7 $ | $4$ | $7$ | $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,12,20,27, 8,16,24) ( 4,11,19,28, 7,15,23)$ |
$ 7, 7, 7, 7 $ | $4$ | $7$ | $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$ |
$ 7, 7, 7, 7 $ | $4$ | $7$ | $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,20, 8,24,12,27,16) ( 4,19, 7,23,11,28,15)$ |
$ 7, 7, 7, 7 $ | $4$ | $7$ | $( 1,14,25,10,22, 6,18)( 2,13,26, 9,21, 5,17)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $196=2^{2} \cdot 7^{2}$ | 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: | 196.8 | magma: IdentifyGroup(G);
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Character table: |
2 2 . . . 2 2 2 . . . . . . . . . 7 2 2 2 2 . . . 2 2 2 2 2 2 2 2 2 1a 7a 7b 7c 2a 4a 4b 7d 7e 7f 7g 7h 7i 7j 7k 7l 2P 1a 7b 7c 7a 1a 2a 2a 7i 7k 7e 7h 7j 7l 7g 7f 7d 3P 1a 7c 7a 7b 2a 4b 4a 7l 7f 7k 7j 7g 7d 7h 7e 7i 5P 1a 7b 7c 7a 2a 4a 4b 7i 7k 7e 7h 7j 7l 7g 7f 7d 7P 1a 1a 1a 1a 2a 4b 4a 1a 1a 1a 1a 1a 1a 1a 1a 1a X.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 X.3 1 1 1 1 -1 J -J 1 1 1 1 1 1 1 1 1 X.4 1 1 1 1 -1 -J J 1 1 1 1 1 1 1 1 1 X.5 4 A C B . . . D G I I G E H H F X.6 4 B A C . . . F I H H I D G G E X.7 4 C B A . . . E H G G H F I I D X.8 4 D E F . . . C I H H I B G G A X.9 4 E F D . . . B G I I G A H H C X.10 4 F D E . . . A H G G H C I I B X.11 4 G H I . . . I C A E F G D B H X.12 4 H I G . . . G B C F D H E A I X.13 4 I G H . . . H A B D E I F C G X.14 4 G H I . . . I F E A C G B D H X.15 4 H I G . . . G D F C B H A E I X.16 4 I G H . . . H E D B A I C F G A = -2*E(7)-E(7)^2-2*E(7)^3-2*E(7)^4-E(7)^5-2*E(7)^6 B = -E(7)-2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5-E(7)^6 C = -2*E(7)-2*E(7)^2-E(7)^3-E(7)^4-2*E(7)^5-2*E(7)^6 D = 2*E(7)^2+2*E(7)^5 E = 2*E(7)^3+2*E(7)^4 F = 2*E(7)+2*E(7)^6 G = E(7)^2+E(7)^3+E(7)^4+E(7)^5 H = E(7)+E(7)^3+E(7)^4+E(7)^6 I = E(7)+E(7)^2+E(7)^5+E(7)^6 J = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);