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Properties

Label 26T16
26T16 1 19 1->19 23 1->23 2 21 2->21 25 2->25 3 14 3->14 3->23 4 16 4->16 4->25 5 5->14 18 5->18 6 6->16 20 6->20 7 7->18 22 7->22 8 8->20 24 8->24 9 9->22 26 9->26 10 15 10->15 10->24 11 17 11->17 11->26 12 12->15 12->19 13 13->17 13->21 14->6 14->10 15->3 15->13 16->7 16->9 17->1 17->2 18->8 18->8 19->2 20->7 20->9 21->3 21->13 22->6 22->10 23->4 23->12 24->5 24->11 25->5 25->11 26->4 26->12
Degree $26$
Order $1352$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}\wr C_2$

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Copy content magma:G := TransitiveGroup(26, 16);
 

Group invariants

Abstract group:  $D_{13}\wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $1352=2^{3} \cdot 13^{2}$
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 $t$:  $16$
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:  $(1,19)(2,21,13,17)(3,23,12,15)(4,25,11,26)(5,14,10,24)(6,16,9,22)(7,18,8,20)$, $(1,23,4,16,7,22,10,15,13,21,3,14,6,20,9,26,12,19,2,25,5,18,8,24,11,17)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T16, 26T19

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 $2^{13}$ $26$ $2$ $13$ $( 1,19)( 2,17)( 3,15)( 4,26)( 5,24)( 6,22)( 7,20)( 8,18)( 9,16)(10,14)(11,25)(12,23)(13,21)$
2B $2^{6},1^{14}$ $26$ $2$ $6$ $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)$
2C $2^{12},1^{2}$ $169$ $2$ $12$ $( 1,13)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(14,17)(15,16)(18,26)(19,25)(20,24)(21,23)$
4A $4^{6},2$ $338$ $4$ $19$ $( 1,21,13,23)( 2,19,12,25)( 3,17,11,14)( 4,15,10,16)( 5,26, 9,18)( 6,24, 8,20)( 7,22)$
13A1 $13^{2}$ $4$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13A2 $13^{2}$ $4$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13A3 $13^{2}$ $4$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13A4 $13^{2}$ $4$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13A5 $13^{2}$ $4$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13A6 $13^{2}$ $4$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13B1 $13,1^{13}$ $4$ $13$ $12$ $(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13B2 $13,1^{13}$ $4$ $13$ $12$ $(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13B3 $13,1^{13}$ $4$ $13$ $12$ $(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13B4 $13,1^{13}$ $4$ $13$ $12$ $(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13B5 $13,1^{13}$ $4$ $13$ $12$ $(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13B6 $13,1^{13}$ $4$ $13$ $12$ $(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13C1 $13^{2}$ $8$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13C2 $13^{2}$ $8$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13C4 $13^{2}$ $8$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13D1 $13^{2}$ $8$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13D2 $13^{2}$ $8$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13D3 $13^{2}$ $8$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13D4 $13^{2}$ $8$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13D5 $13^{2}$ $8$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13D6 $13^{2}$ $8$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13E1 $13^{2}$ $8$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13E2 $13^{2}$ $8$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13E3 $13^{2}$ $8$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13E4 $13^{2}$ $8$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13E5 $13^{2}$ $8$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13E6 $13^{2}$ $8$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
26A1 $26$ $52$ $26$ $25$ $( 1,22,11,15, 8,21, 5,14, 2,20,12,26, 9,19, 6,25, 3,18,13,24,10,17, 7,23, 4,16)$
26A3 $26$ $52$ $26$ $25$ $( 1,15, 5,20, 9,25,13,17, 4,22, 8,14,12,19, 3,24, 7,16,11,21, 2,26, 6,18,10,23)$
26A5 $26$ $52$ $26$ $25$ $( 1,21,12,25,10,16, 8,20, 6,24, 4,15, 2,19,13,23,11,14, 9,18, 7,22, 5,26, 3,17)$
26A7 $26$ $52$ $26$ $25$ $( 1,14, 6,17,11,20, 3,23, 8,26,13,16, 5,19,10,22, 2,25, 7,15,12,18, 4,21, 9,24)$
26A9 $26$ $52$ $26$ $25$ $( 1,20,13,22,12,24,11,26,10,15, 9,17, 8,19, 7,21, 6,23, 5,25, 4,14, 3,16, 2,18)$
26A11 $26$ $52$ $26$ $25$ $( 1,26, 7,14,13,15, 6,16,12,17, 5,18,11,19, 4,20,10,21, 3,22, 9,23, 2,24, 8,25)$
26B1 $13,2^{6},1$ $52$ $26$ $18$ $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
26B3 $13,2^{6},1$ $52$ $26$ $18$ $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
26B5 $13,2^{6},1$ $52$ $26$ $18$ $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
26B7 $13,2^{6},1$ $52$ $26$ $18$ $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
26B9 $13,2^{6},1$ $52$ $26$ $18$ $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
26B11 $13,2^{6},1$ $52$ $26$ $18$ $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)(14,17,20,23,26,16,19,22,25,15,18,21,24)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

44 x 44 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed