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Magma
magma: G := TransitiveGroup(14, 15);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_7^2:S_3$ | ||
CHM label: | $[7^{2}:3_{3}]2$ | ||
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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (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) $2$: $C_2$ $6$: $S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 7: None
Low degree siblings
21T17, 21T18, 42T56, 42T57, 42T62Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 1, 1 $ | $98$ | $3$ | $( 3, 5, 9)( 4,10, 6)( 7,13,11)( 8,12,14)$ |
$ 7, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $7$ | $( 2, 4, 6, 8,10,12,14)$ |
$ 7, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $7$ | $( 2, 8,14, 6,12, 4,10)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $21$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$ |
$ 14 $ | $21$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$ |
$ 14 $ | $21$ | $14$ | $( 1, 2, 3, 6, 5,10, 7,14, 9, 4,11, 8,13,12)$ |
$ 14 $ | $21$ | $14$ | $( 1, 2, 3,10, 5, 4, 7,12, 9, 6,11,14,13, 8)$ |
$ 14 $ | $21$ | $14$ | $( 1, 2, 7, 8,13,14, 5, 6,11,12, 3, 4, 9,10)$ |
$ 14 $ | $21$ | $14$ | $( 1, 2, 7,14,13,12, 5,10,11, 8, 3, 6, 9, 4)$ |
$ 14 $ | $21$ | $14$ | $( 1, 2, 7,12,13, 8, 5, 4,11,14, 3,10, 9, 6)$ |
$ 7, 7 $ | $3$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ |
$ 7, 7 $ | $3$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$ |
$ 7, 7 $ | $6$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$ |
$ 7, 7 $ | $3$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2,10, 4,12, 6,14, 8)$ |
$ 7, 7 $ | $6$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2,12, 8, 4,14,10, 6)$ |
$ 7, 7 $ | $6$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2,14,12,10, 8, 6, 4)$ |
$ 7, 7 $ | $3$ | $7$ | $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$ |
$ 7, 7 $ | $3$ | $7$ | $( 1, 7,13, 5,11, 3, 9)( 2,12, 8, 4,14,10, 6)$ |
$ 7, 7 $ | $3$ | $7$ | $( 1, 7,13, 5,11, 3, 9)( 2,14,12,10, 8, 6, 4)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $294=2 \cdot 3 \cdot 7^{2}$ | 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: | 294.7 | magma: IdentifyGroup(G);
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Character table: |
2 1 . . . 1 1 1 1 1 1 1 1 1 . 1 . . 1 1 1 3 1 1 . . . . . . . . . . . . . . . . . . 7 2 . 2 2 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1a 3a 7a 7b 2a 14a 14b 14c 14d 14e 14f 7c 7d 7e 7f 7g 7h 7i 7j 7k 2P 1a 3a 7a 7b 1a 7c 7d 7f 7i 7k 7j 7f 7c 7g 7d 7h 7e 7j 7k 7i 3P 1a 1a 7b 7a 2a 14d 14e 14f 14c 14a 14b 7i 7k 7h 7j 7e 7g 7f 7d 7c 5P 1a 3a 7b 7a 2a 14e 14f 14d 14a 14b 14c 7k 7j 7g 7i 7h 7e 7c 7f 7d 7P 1a 3a 1a 1a 2a 2a 2a 2a 2a 2a 2a 1a 1a 1a 1a 1a 1a 1a 1a 1a 11P 1a 3a 7a 7b 2a 14b 14c 14a 14e 14f 14d 7d 7f 7h 7c 7e 7g 7k 7i 7j 13P 1a 3a 7b 7a 2a 14f 14d 14e 14b 14c 14a 7j 7i 7e 7k 7g 7h 7d 7c 7f 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 2 -1 2 2 . . . . . . . 2 2 2 2 2 2 2 2 2 X.4 3 . A /A -1 C D E /D /E /C F G M H O N /G /F /H X.5 3 . A /A -1 D E C /E /C /D G H N F M O /H /G /F X.6 3 . A /A -1 E C D /C /D /E H F O G N M /F /H /G X.7 3 . /A A -1 /E /C /D C D E /H /F O /G N M F H G X.8 3 . /A A -1 /D /E /C E C D /G /H N /F M O H G F X.9 3 . /A A -1 /C /D /E D E C /F /G M /H O N G F H X.10 3 . A /A 1 -E -C -D -/C -/D -/E H F O G N M /F /H /G X.11 3 . A /A 1 -D -E -C -/E -/C -/D G H N F M O /H /G /F X.12 3 . A /A 1 -C -D -E -/D -/E -/C F G M H O N /G /F /H X.13 3 . /A A 1 -/C -/D -/E -D -E -C /F /G M /H O N G F H X.14 3 . /A A 1 -/D -/E -/C -E -C -D /G /H N /F M O H G F X.15 3 . /A A 1 -/E -/C -/D -C -D -E /H /F O /G N M F H G X.16 6 . -1 -1 . . . . . . . I K P J Q R K I J X.17 6 . -1 -1 . . . . . . . J I Q K R P I J K X.18 6 . -1 -1 . . . . . . . K J R I P Q J K I X.19 6 . B /B . . . . . . . L L -1 L -1 -1 /L /L /L X.20 6 . /B B . . . . . . . /L /L -1 /L -1 -1 L L L A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 B = -3*E(7)-3*E(7)^2-2*E(7)^3-3*E(7)^4-2*E(7)^5-2*E(7)^6 = (5-Sqrt(-7))/2 = 2-b7 C = -E(7)^3 D = -E(7)^5 E = -E(7)^6 F = 2*E(7)^4+E(7)^6 G = 2*E(7)^2+E(7)^3 H = 2*E(7)+E(7)^5 I = -2*E(7)-2*E(7)^2-2*E(7)^5-2*E(7)^6 J = -2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5 K = -2*E(7)-2*E(7)^3-2*E(7)^4-2*E(7)^6 L = 2*E(7)^3+2*E(7)^5+2*E(7)^6 = -1-Sqrt(-7) = -1-i7 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)+E(7)^2+E(7)^5+2*E(7)^6 R = 2*E(7)^2+E(7)^3+E(7)^4+2*E(7)^5 |
magma: CharacterTable(G);