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Properties

Label 14T13
14T13 1 8 1->8 13 1->13 2 4 2->4 9 2->9 12 2->12 3 10 3->10 11 3->11 6 4->6 4->10 4->11 5 5->9 5->12 6->8 6->8 6->13 7 14 7->14 8->10 10->12 12->14 14->2
Degree $14$
Order $196$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7^2$

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Copy content magma:G := TransitiveGroup(14, 13);
 

Group invariants

Abstract group:  $D_7^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $196=2^{2} \cdot 7^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $14$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $13$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[1/2.D(7)^{2}]2$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(2,4,6,8,10,12,14)$, $(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)$, $(1,13)(2,12)(3,11)(4,10)(5,9)(6,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$14$:  $D_{7}$ x 2
$28$:  $D_{14}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T13 x 2, 28T36 x 3

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{14}$ $1$ $1$ $0$ $()$
2A $2^{7}$ $7$ $2$ $7$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$
2B $2^{7}$ $7$ $2$ $7$ $( 1, 2)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$
2C $2^{6},1^{2}$ $49$ $2$ $6$ $( 1, 5)( 2,10)( 4, 8)( 7,13)( 9,11)(12,14)$
7A1 $7^{2}$ $2$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
7A2 $7^{2}$ $2$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$
7A3 $7^{2}$ $2$ $7$ $12$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$
7B1 $7^{2}$ $2$ $7$ $12$ $( 1,11, 7, 3,13, 9, 5)( 2, 6,10,14, 4, 8,12)$
7B2 $7^{2}$ $2$ $7$ $12$ $( 1, 7,13, 5,11, 3, 9)( 2,10, 4,12, 6,14, 8)$
7B3 $7^{2}$ $2$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2,14,12,10, 8, 6, 4)$
7C1 $7,1^{7}$ $4$ $7$ $6$ $( 2, 6,10,14, 4, 8,12)$
7C2 $7,1^{7}$ $4$ $7$ $6$ $( 2,10, 4,12, 6,14, 8)$
7C3 $7,1^{7}$ $4$ $7$ $6$ $( 2,14,12,10, 8, 6, 4)$
7D1 $7^{2}$ $4$ $7$ $12$ $( 1,13,11, 9, 7, 5, 3)( 2, 8,14, 6,12, 4,10)$
7D2 $7^{2}$ $4$ $7$ $12$ $( 1,13,11, 9, 7, 5, 3)( 2,12, 8, 4,14,10, 6)$
7D3 $7^{2}$ $4$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2,10, 4,12, 6,14, 8)$
7E1 $7^{2}$ $4$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
7E2 $7^{2}$ $4$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2,12, 8, 4,14,10, 6)$
7E3 $7^{2}$ $4$ $7$ $12$ $( 1,11, 7, 3,13, 9, 5)( 2,10, 4,12, 6,14, 8)$
14A1 $14$ $14$ $14$ $13$ $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
14A3 $14$ $14$ $14$ $13$ $( 1,12, 7, 4,13,10, 5, 2,11, 8, 3,14, 9, 6)$
14A5 $14$ $14$ $14$ $13$ $( 1,14,11,10, 7, 6, 3, 2,13,12, 9, 8, 5, 4)$
14B1 $14$ $14$ $14$ $13$ $( 1, 4,11, 8, 7,12, 3, 2,13, 6, 9,10, 5,14)$
14B3 $14$ $14$ $14$ $13$ $( 1, 8, 3, 6, 5, 4, 7, 2, 9,14,11,12,13,10)$
14B5 $14$ $14$ $14$ $13$ $( 1, 6, 7,14,13, 8, 5, 2,11,10, 3, 4, 9,12)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 7A1 7A2 7A3 7B1 7B2 7B3 7C1 7C2 7C3 7D1 7D2 7D3 7E1 7E2 7E3 14A1 14A3 14A5 14B1 14B3 14B5
Size 1 7 7 49 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 14 14 14 14 14 14
2 P 1A 1A 1A 1A 7A2 7A3 7A1 7B2 7B3 7B1 7C2 7C3 7C1 7D2 7D3 7D1 7E2 7E3 7E1 7A1 7A3 7A2 7B1 7B3 7B2
7 P 1A 2A 2B 2C 7A3 7A1 7A2 7B3 7B1 7B2 7C3 7C1 7C2 7D3 7D1 7D2 7E3 7E1 7E2 14A3 14A5 14A1 14B3 14B5 14B1
Type
196.9.1a R 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
196.9.1b R 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
196.9.1c R 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
196.9.1d R 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
196.9.2a1 R 2 0 2 0 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 ζ7−2+ζ72 0 ζ7−3+ζ73 ζ7−1+ζ7
196.9.2a2 R 2 0 2 0 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 ζ7−1+ζ7 0 ζ7−2+ζ72 ζ7−3+ζ73
196.9.2a3 R 2 0 2 0 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 ζ7−3+ζ73 0 ζ7−1+ζ7 ζ7−2+ζ72
196.9.2b1 R 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 0 ζ7−3+ζ73 0 0
196.9.2b2 R 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 0 ζ7−2+ζ72 0 0
196.9.2b3 R 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 0 ζ7−1+ζ7 0 0
196.9.2c1 R 2 −2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 0 −ζ7−3−ζ73 0 0
196.9.2c2 R 2 −2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 0 −ζ7−2−ζ72 0 0
196.9.2c3 R 2 −2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 0 −ζ7−1−ζ7 0 0
196.9.2d1 R 2 0 −2 0 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 −ζ7−2−ζ72 0 −ζ7−3−ζ73 −ζ7−1−ζ7
196.9.2d2 R 2 0 −2 0 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 −ζ7−1−ζ7 0 −ζ7−2−ζ72 −ζ7−3−ζ73
196.9.2d3 R 2 0 −2 0 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 −ζ7−3−ζ73 0 −ζ7−1−ζ7 −ζ7−2−ζ72
196.9.4a1 R 4 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 −ζ7−2−1−ζ72 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 ζ7−3+2+ζ73 ζ7−1+2+ζ7 −ζ7−1−1−ζ7 −ζ7−1−1−ζ7 ζ7−2+2+ζ72 −ζ7−3−1−ζ73 0 0 0 0 0 0
196.9.4a2 R 4 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 −ζ7−1−1−ζ7 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 ζ7−2+2+ζ72 ζ7−3+2+ζ73 −ζ7−3−1−ζ73 −ζ7−3−1−ζ73 ζ7−1+2+ζ7 −ζ7−2−1−ζ72 0 0 0 0 0 0
196.9.4a3 R 4 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 −ζ7−3−1−ζ73 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 ζ7−1+2+ζ7 ζ7−2+2+ζ72 −ζ7−2−1−ζ72 −ζ7−2−1−ζ72 ζ7−3+2+ζ73 −ζ7−1−1−ζ7 0 0 0 0 0 0
196.9.4b1 R 4 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 −ζ7−1−1−ζ7 ζ7−3+2+ζ73 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−3−1−ζ73 ζ7−2+2+ζ72 −ζ7−2−1−ζ72 ζ7−1+2+ζ7 0 0 0 0 0 0
196.9.4b2 R 4 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 −ζ7−3−1−ζ73 ζ7−2+2+ζ72 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−2−1−ζ72 ζ7−1+2+ζ7 −ζ7−1−1−ζ7 ζ7−3+2+ζ73 0 0 0 0 0 0
196.9.4b3 R 4 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 −ζ7−2−1−ζ72 ζ7−1+2+ζ7 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−1−1−ζ7 ζ7−3+2+ζ73 −ζ7−3−1−ζ73 ζ7−2+2+ζ72 0 0 0 0 0 0
196.9.4c1 R 4 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 ζ7−2+2+ζ72 −ζ7−3−1−ζ73 ζ7−3+2+ζ73 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 ζ7−1+2+ζ7 −ζ7−2−1−ζ72 −ζ7−1−1−ζ7 −ζ7−1−1−ζ7 0 0 0 0 0 0
196.9.4c2 R 4 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 ζ7−1+2+ζ7 −ζ7−2−1−ζ72 ζ7−2+2+ζ72 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 ζ7−3+2+ζ73 −ζ7−1−1−ζ7 −ζ7−3−1−ζ73 −ζ7−3−1−ζ73 0 0 0 0 0 0
196.9.4c3 R 4 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 ζ7−3+2+ζ73 −ζ7−1−1−ζ7 ζ7−1+2+ζ7 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 ζ7−2+2+ζ72 −ζ7−3−1−ζ73 −ζ7−2−1−ζ72 −ζ7−2−1−ζ72 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed