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Properties

Label 14T15
Degree $14$
Order $294$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(14, 15);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $15$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^2:S_3$
CHM label:   $[7^{2}:3_{3}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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);
 

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, 42T62

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 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);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  294.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 7A1 7A-1 7A2 7A-2 7A3 7A-3 7B1 7B-1 7C1 7C2 7C3 14A1 14A-1 14A3 14A-3 14A5 14A-5
Size 1 21 98 3 3 3 3 3 3 6 6 6 6 6 21 21 21 21 21 21
2 P 1A 1A 3A 7A-1 7A1 7A-2 7A3 7A-3 7A2 7B1 7B-1 7C2 7C3 7C1 7A3 7A1 7A-3 7A-2 7A-1 7A2
3 P 1A 2A 1A 7A2 7A-2 7A-3 7A1 7A-1 7A3 7B-1 7B1 7C3 7C1 7C2 14A-5 14A3 14A5 14A1 14A-3 14A-1
7 P 1A 2A 3A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A
Type
294.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
294.7.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
294.7.2a R 2 0 −1 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0
294.7.3a1 C 3 1 0 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−2+2ζ7 2ζ7−1+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3
294.7.3a2 C 3 1 0 2ζ72+ζ73 ζ7−3+2ζ7−2 2ζ7−3+ζ7−1 ζ7+2ζ73 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73
294.7.3a3 C 3 1 0 2ζ7−3+ζ7−1 ζ7+2ζ73 ζ7−2+2ζ7 2ζ7−1+ζ72 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1
294.7.3a4 C 3 1 0 ζ7+2ζ73 2ζ7−3+ζ7−1 2ζ7−1+ζ72 ζ7−2+2ζ7 2ζ72+ζ73 ζ7−3+2ζ7−2 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7
294.7.3a5 C 3 1 0 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+2ζ7−2 2ζ72+ζ73 2ζ7−3+ζ7−1 ζ7+2ζ73 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72
294.7.3a6 C 3 1 0 ζ7−2+2ζ7 2ζ7−1+ζ72 2ζ72+ζ73 ζ7−3+2ζ7−2 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2
294.7.3b1 C 3 −1 0 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−2+2ζ7 2ζ7−1+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+1+ζ73 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3
294.7.3b2 C 3 −1 0 2ζ72+ζ73 ζ7−3+2ζ7−2 2ζ7−3+ζ7−1 ζ7+2ζ73 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+1+ζ73 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73
294.7.3b3 C 3 −1 0 2ζ7−3+ζ7−1 ζ7+2ζ73 ζ7−2+2ζ7 2ζ7−1+ζ72 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1
294.7.3b4 C 3 −1 0 ζ7+2ζ73 2ζ7−3+ζ7−1 2ζ7−1+ζ72 ζ7−2+2ζ7 2ζ72+ζ73 ζ7−3+2ζ7−2 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7
294.7.3b5 C 3 −1 0 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+2ζ7−2 2ζ72+ζ73 2ζ7−3+ζ7−1 ζ7+2ζ73 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−2+1+ζ72 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72
294.7.3b6 C 3 −1 0 ζ7−2+2ζ7 2ζ7−1+ζ72 2ζ72+ζ73 ζ7−3+2ζ7−2 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−2+1+ζ72 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2
294.7.6a1 C 6 0 0 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 ζ7−3+3+ζ7+ζ72 −ζ7−3+2−ζ7−ζ72 −1 −1 −1 0 0 0 0 0 0
294.7.6a2 C 6 0 0 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −ζ7−3+2−ζ7−ζ72 ζ7−3+3+ζ7+ζ72 −1 −1 −1 0 0 0 0 0 0
294.7.6b1 R 6 0 0 2ζ7−3+2+2ζ73 2ζ7−3+2+2ζ73 2ζ7−1+2+2ζ7 2ζ7−1+2+2ζ7 2ζ7−2+2+2ζ72 2ζ7−2+2+2ζ72 −1 −1 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 0 0 0 0 0 0
294.7.6b2 R 6 0 0 2ζ7−2+2+2ζ72 2ζ7−2+2+2ζ72 2ζ7−3+2+2ζ73 2ζ7−3+2+2ζ73 2ζ7−1+2+2ζ7 2ζ7−1+2+2ζ7 −1 −1 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 0 0 0 0 0 0
294.7.6b3 R 6 0 0 2ζ7−1+2+2ζ7 2ζ7−1+2+2ζ7 2ζ7−2+2+2ζ72 2ζ7−2+2+2ζ72 2ζ7−3+2+2ζ73 2ζ7−3+2+2ζ73 −1 −1 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 0 0 0 0 0 0

magma: CharacterTable(G);