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Properties

Label 26T34
26T34 1 6 1->6 26 1->26 2 4 2->4 21 2->21 3 3->2 16 3->16 13 4->13 24 4->24 5 11 5->11 19 5->19 9 6->9 14 6->14 7 22 7->22 8 8->5 17 8->17 9->3 25 9->25 10 10->1 20 10->20 12 11->12 15 11->15 12->10 23 12->23 13->8 18 13->18 14->2 14->20 15->9 15->22 16->24 17->10 17->26 18->4 18->15 19->11 19->17 20->5 20->19 21->12 22->6 22->23 23->13 23->25 24->7 24->14 25->1 25->16 26->8 26->18
Degree $26$
Order $4056$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:(C_3\times Q_8)$

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Copy content magma:G := TransitiveGroup(26, 34);
 

Group invariants

Abstract group:  $C_{13}^2:(C_3\times Q_8)$
Copy content magma:IdentifyGroup(G);
 
Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
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 $n$:  $26$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $34$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
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:  $(1,26,8,17,10,20,5,19,11,15,9,25)(2,21,12,23,13,18,4,24,7,22,6,14)(3,16)$, $(1,6,9,3,2,4,13,8,5,11,12,10)(14,20,19,17,26,18,15,22,23,25,16,24)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $Q_8$
$12$:  $C_6\times C_2$
$24$:  24T4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T34 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{26}$ $1$ $1$ $0$ $()$
2A $2^{12},1^{2}$ $169$ $2$ $12$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,12)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$
3A1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 7,12)( 2,10, 8)( 3,13, 4)( 5, 6, 9)(14,16,22)(15,19,18)(17,25,23)(20,21,24)$
3A-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1,12, 7)( 2, 8,10)( 3, 4,13)( 5, 9, 6)(14,22,16)(15,18,19)(17,23,25)(20,24,21)$
4A $4^{6},2$ $338$ $4$ $19$ $( 1,25, 8,14)( 2,16, 7,23)( 3,20, 6,19)( 4,24, 5,15)( 9,18,13,21)(10,22,12,17)(11,26)$
4B $4^{6},2$ $338$ $4$ $19$ $( 1,16,12,15)( 2,23,11,21)( 3,17,10,14)( 4,24, 9,20)( 5,18, 8,26)( 6,25, 7,19)(13,22)$
4C $4^{6},1^{2}$ $338$ $4$ $18$ $( 1, 5,12, 8)( 2,10,11, 3)( 4, 7, 9, 6)(14,25,22,24)(15,20,21,16)(17,23,19,26)$
6A1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1,10, 7, 8,12, 2)( 3, 5,13, 6, 4, 9)(14,17,16,25,22,23)(15,21,19,24,18,20)$
6A-1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1, 2,12, 8, 7,10)( 3, 9, 4, 6,13, 5)(14,23,22,25,16,17)(15,20,18,24,19,21)$
12A1 $12^{2},2$ $338$ $12$ $23$ $( 1,16,10,25, 7,22, 8,23,12,14, 2,17)( 3,18, 5,20,13,15, 6,21, 4,19, 9,24)(11,26)$
12A-1 $12^{2},2$ $338$ $12$ $23$ $( 1,17, 2,14,12,23, 8,22, 7,25,10,16)( 3,24, 9,19, 4,21, 6,15,13,20, 5,18)(11,26)$
12B1 $12^{2},2$ $338$ $12$ $23$ $( 1,24,10,16, 9,14,12,20, 3,15, 4,17)( 2,26, 7,23, 5,19,11,18, 6,21, 8,25)(13,22)$
12B-1 $12^{2},2$ $338$ $12$ $23$ $( 1,17, 4,15, 3,20,12,14, 9,16,10,24)( 2,25, 8,21, 6,18,11,19, 5,23, 7,26)(13,22)$
12C1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1,11, 4, 5, 3, 7,12, 2, 9, 8,10, 6)(14,23,15,25,19,20,22,26,21,24,17,16)$
12C-1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 6,10, 8, 9, 2,12, 7, 3, 5, 4,11)(14,16,17,24,21,26,22,20,19,25,15,23)$
13A $13,1^{13}$ $24$ $13$ $12$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$
13B $13^{2}$ $24$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13C $13^{2}$ $24$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13D1 $13^{2}$ $24$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13D2 $13^{2}$ $24$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13E1 $13^{2}$ $24$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13E2 $13^{2}$ $24$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,26,25,24,23,22,21,20,19,18,17,16,15)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 4A 4B 4C 6A1 6A-1 12A1 12A-1 12B1 12B-1 12C1 12C-1 13A 13B 13C 13D1 13D2 13E1 13E2
Size 1 169 169 169 338 338 338 169 169 338 338 338 338 338 338 24 24 24 24 24 24 24
2 P 1A 1A 3A-1 3A1 2A 2A 2A 3A1 3A-1 6A1 6A-1 6A-1 6A1 6A1 6A-1 13A 13B 13C 13D2 13D1 13E2 13E1
3 P 1A 2A 1A 1A 4A 4B 4C 2A 2A 4A 4A 4B 4B 4C 4C 13A 13B 13C 13D1 13D2 13E1 13E2
13 P 1A 2A 3A1 3A-1 4A 4B 4C 6A1 6A-1 12A1 12A-1 12B1 12B-1 12C1 12C-1 1A 1A 1A 1A 1A 1A 1A
Type
4056.bf.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4056.bf.1b R 1 1 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1
4056.bf.1c R 1 1 1 1 −1 1 −1 1 1 −1 1 −1 −1 −1 1 1 1 1 1 1 1 1
4056.bf.1d R 1 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 1 −1 1 1 1 1 1 1 1
4056.bf.1e1 C 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 1
4056.bf.1e2 C 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 1
4056.bf.1f1 C 1 1 ζ3−1 ζ3 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 1 1 1 1 1 1 1
4056.bf.1f2 C 1 1 ζ3 ζ3−1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 1 1 1 1 1 1 1
4056.bf.1g1 C 1 1 ζ3−1 ζ3 −1 1 −1 ζ3 ζ3−1 −ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 ζ3−1 1 1 1 1 1 1 1
4056.bf.1g2 C 1 1 ζ3 ζ3−1 −1 1 −1 ζ3−1 ζ3 −ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 ζ3 1 1 1 1 1 1 1
4056.bf.1h1 C 1 1 ζ3−1 ζ3 1 −1 −1 ζ3 ζ3−1 ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 −ζ3−1 1 1 1 1 1 1 1
4056.bf.1h2 C 1 1 ζ3 ζ3−1 1 −1 −1 ζ3−1 ζ3 ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 −ζ3 1 1 1 1 1 1 1
4056.bf.2a S 2 −2 2 2 0 0 0 −2 −2 0 0 0 0 0 0 2 2 2 2 2 2 2
4056.bf.2b1 C 2 −2 2ζ3−1 2ζ3 0 0 0 −2ζ3 −2ζ3−1 0 0 0 0 0 0 2 2 2 2 2 2 2
4056.bf.2b2 C 2 −2 2ζ3 2ζ3−1 0 0 0 −2ζ3−1 −2ζ3 0 0 0 0 0 0 2 2 2 2 2 2 2
4056.bf.24a R 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 −2 −2 −2 −2 11 −2
4056.bf.24b R 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 −2 −2 11 −2 −2 −2
4056.bf.24c R 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 11 −2 −2 −2 −2 −2
4056.bf.24d1 R 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2ζ13−6+2ζ13−5+2ζ13−2−1+2ζ132+2ζ135+2ζ136 −2 −2ζ13−6−2ζ13−5−2ζ13−2−3−2ζ132−2ζ135−2ζ136 −2 3ζ13−6+3ζ13−5+3ζ13−2+6+3ζ132+3ζ135+3ζ136 −2 −3ζ13−6−3ζ13−5−3ζ13−2+3−3ζ132−3ζ135−3ζ136
4056.bf.24d2 R 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2ζ13−6−2ζ13−5−2ζ13−2−3−2ζ132−2ζ135−2ζ136 −2 2ζ13−6+2ζ13−5+2ζ13−2−1+2ζ132+2ζ135+2ζ136 −2 −3ζ13−6−3ζ13−5−3ζ13−2+3−3ζ132−3ζ135−3ζ136 −2 3ζ13−6+3ζ13−5+3ζ13−2+6+3ζ132+3ζ135+3ζ136
4056.bf.24e1 R 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3ζ13−6+3ζ13−5+3ζ13−2+6+3ζ132+3ζ135+3ζ136 −2 −3ζ13−6−3ζ13−5−3ζ13−2+3−3ζ132−3ζ135−3ζ136 −2 −2ζ13−6−2ζ13−5−2ζ13−2−3−2ζ132−2ζ135−2ζ136 −2 2ζ13−6+2ζ13−5+2ζ13−2−1+2ζ132+2ζ135+2ζ136
4056.bf.24e2 R 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3ζ13−6−3ζ13−5−3ζ13−2+3−3ζ132−3ζ135−3ζ136 −2 3ζ13−6+3ζ13−5+3ζ13−2+6+3ζ132+3ζ135+3ζ136 −2 2ζ13−6+2ζ13−5+2ζ13−2−1+2ζ132+2ζ135+2ζ136 −2 −2ζ13−6−2ζ13−5−2ζ13−2−3−2ζ132−2ζ135−2ζ136

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed