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Properties

Label 26T21
26T21 1 4 1->4 15 1->15 2 8 2->8 21 2->21 3 12 3->12 14 3->14 4->3 20 4->20 5 7 5->7 26 5->26 6 11 6->11 19 6->19 7->2 25 7->25 8->6 18 8->18 9 10 9->10 24 9->24 10->1 17 10->17 11->5 23 11->23 12->9 16 12->16 13 22 13->22 14->13 14->26 15->2 15->17 16->4 16->21 17->6 17->25 18->16 19->10 19->20 20->12 20->24 21->1 21->15 22->3 22->19 23->5 24->7 24->14 25->9 25->18 26->11 26->22
Degree $26$
Order $2028$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}:F_{13}$

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Copy content magma:G := TransitiveGroup(26, 21);
 

Group invariants

Abstract group:  $C_{13}:F_{13}$
Copy content magma:IdentifyGroup(G);
 
Order:  $2028=2^{2} \cdot 3 \cdot 13^{2}$
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 $t$:  $21$
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:  $(1,15,2,21)(3,14,13,22)(4,20,12,16)(5,26,11,23)(6,19,10,17)(7,25,9,24)(8,18)$, $(1,4,3,12,9,10)(2,8,6,11,5,7)(14,26,22,19,20,24)(15,17,25,18,16,21)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$156$:  $F_{13}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T21 x 5

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 $2^{12},1^{2}$ $169$ $2$ $12$ $( 1, 9)( 2, 8)( 3, 7)( 4, 6)(10,13)(11,12)(14,19)(15,18)(16,17)(20,26)(21,25)(22,24)$
3A1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 6, 8)( 2, 9, 4)( 3,12,13)( 7,11,10)(14,22,20)(15,25,16)(17,18,21)(19,24,26)$
3A-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 8, 6)( 2, 4, 9)( 3,13,12)( 7,10,11)(14,20,22)(15,16,25)(17,21,18)(19,26,24)$
4A1 $4^{6},2$ $169$ $4$ $19$ $( 1,25, 9,21)( 2,18, 8,15)( 3,24, 7,22)( 4,17, 6,16)( 5,23)(10,14,13,19)(11,20,12,26)$
4A-1 $4^{6},2$ $169$ $4$ $19$ $( 1,21, 9,25)( 2,15, 8,18)( 3,22, 7,24)( 4,16, 6,17)( 5,23)(10,19,13,14)(11,26,12,20)$
6A1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1, 2, 6, 9, 8, 4)( 3,10,12, 7,13,11)(14,26,22,19,20,24)(15,17,25,18,16,21)$
6A-1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1, 4, 8, 9, 6, 2)( 3,11,13, 7,12,10)(14,24,20,19,22,26)(15,21,16,18,25,17)$
12A1 $12^{2},2$ $169$ $12$ $23$ $( 1,17, 2,25, 6,18, 9,16, 8,21, 4,15)( 3,20,10,24,12,14, 7,26,13,22,11,19)( 5,23)$
12A-1 $12^{2},2$ $169$ $12$ $23$ $( 1,15, 4,21, 8,16, 9,18, 6,25, 2,17)( 3,19,11,22,13,26, 7,14,12,24,10,20)( 5,23)$
12A5 $12^{2},2$ $169$ $12$ $23$ $( 1,18, 4,25, 8,17, 9,15, 6,21, 2,16)( 3,14,11,24,13,20, 7,19,12,22,10,26)( 5,23)$
12A-5 $12^{2},2$ $169$ $12$ $23$ $( 1,16, 2,21, 6,15, 9,17, 8,25, 4,18)( 3,26,10,22,12,19, 7,20,13,24,11,14)( 5,23)$
13A $13^{2}$ $12$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13B $13^{2}$ $12$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13C1 $13^{2}$ $12$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13C2 $13^{2}$ $12$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13D1 $13^{2}$ $12$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13D2 $13^{2}$ $12$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13E1 $13^{2}$ $12$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13E2 $13^{2}$ $12$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13F1 $13^{2}$ $12$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13F2 $13^{2}$ $12$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13G1 $13^{2}$ $12$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13G2 $13^{2}$ $12$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13H1 $13,1^{13}$ $12$ $13$ $12$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$
13H2 $13,1^{13}$ $12$ $13$ $12$ $(14,19,24,16,21,26,18,23,15,20,25,17,22)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5 13A 13B 13C1 13C2 13D1 13D2 13E1 13E2 13F1 13F2 13G1 13G2 13H1 13H2
Size 1 169 169 169 169 169 169 169 169 169 169 169 12 12 12 12 12 12 12 12 12 12 12 12 12 12
2 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 6A1 6A-1 6A-1 6A1 13A 13B 13C2 13C1 13D2 13D1 13E2 13E1 13F2 13F1 13G2 13G1 13H2 13H1
3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 4A1 4A-1 4A1 4A-1 13A 13B 13C1 13C2 13D1 13D2 13E1 13E2 13F1 13F2 13G1 13G2 13H1 13H2
13 P 1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A
Type
2028.q.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 1
2028.q.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 1
2028.q.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.1d1 C 1 −1 1 1 −i i −1 −1 i −i i −i 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.1d2 C 1 −1 1 1 i −i −1 −1 −i i −i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.1e1 C 1 1 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.1e2 C 1 1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2028.q.12a R 12 0 0 0 0 0 0 0 0 0 0 0 −1 −1 12 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
2028.q.12b R 12 0 0 0 0 0 0 0 0 0 0 0 −1 12 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
2028.q.12c1 R 12 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136
2028.q.12c2 R 12 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136
2028.q.12d1 R 12 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136
2028.q.12d2 R 12 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136
2028.q.12e1 R 12 0 0 0 0 0 0 0 0 0 0 0 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 −1 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 −1
2028.q.12e2 R 12 0 0 0 0 0 0 0 0 0 0 0 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 −1 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −1
2028.q.12f1 R 12 0 0 0 0 0 0 0 0 0 0 0 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 −1 −1 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136
2028.q.12f2 R 12 0 0 0 0 0 0 0 0 0 0 0 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 −1 −1 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136
2028.q.12g1 R 12 0 0 0 0 0 0 0 0 0 0 0 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 −1 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136
2028.q.12g2 R 12 0 0 0 0 0 0 0 0 0 0 0 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 −1 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 −1 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 −1 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136
2028.q.12h1 R 12 0 0 0 0 0 0 0 0 0 0 0 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 −1 −1 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −1 −1 −1 −1
2028.q.12h2 R 12 0 0 0 0 0 0 0 0 0 0 0 −ζ13−6−ζ13−5−ζ13−2+5−ζ132−ζ135−ζ136 −1 −1 ζ13−6+ζ13−5+ζ13−2+6+ζ132+ζ135+ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 −2ζ13−6−2ζ13−5−2ζ13−2−2−2ζ132−2ζ135−2ζ136 2ζ13−6+2ζ13−5+2ζ13−2+2ζ132+2ζ135+2ζ136 −1 −1 −1 −1

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Regular extensions

Data not computed