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Properties

Label 26T25
26T25 1 3 1->3 5 1->5 12 1->12 16 1->16 2 4 2->4 10 2->10 11 2->11 20 2->20 3->2 3->5 3->10 24 3->24 6 4->6 7 4->7 9 4->9 15 4->15 5->7 8 5->8 5->12 19 5->19 6->4 6->7 6->8 23 6->23 7->9 7->9 14 7->14 8->1 8->10 18 8->18 9->6 9->11 22 9->22 10->11 10->12 26 10->26 11->3 13 11->13 17 11->17 12->1 12->8 21 12->21 13->2 25 13->25 14->12 14->18 14->21 15->2 15->16 15->19 16->5 16->20 16->24 17->8 17->19 17->21 18->11 18->14 18->22 19->1 19->22 19->23 20->4 20->17 20->24 21->7 21->25 21->25 22->10 22->20 22->26 23->13 23->14 23->15 24->15 24->23 25->6 25->16 25->18 26->9 26->17
Degree $26$
Order $2704$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}^2.C_2^2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(26, 25);
 

Group invariants

Abstract group:  $D_{13}^2.C_2^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $2704=2^{4} \cdot 13^{2}$
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 $t$:  $25$
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:  $(1,3,5,7,9,11,13,2,4,6,8,10,12)(14,18,22,26,17,21,25,16,20,24,15,19,23)$, $(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)$, $(1,5,12,8)(2,10,11,3)(4,7,9,6)(14,21,25,18)(15,16,24,23)(17,19,22,20)$, $(1,16,5,19)(2,20,4,15)(3,24)(6,23,13,25)(7,14,12,21)(8,18,11,17)(9,22,10,26)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$16$:  $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T25 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 $2^{13}$ $26$ $2$ $13$ $( 1,18)( 2,24)( 3,17)( 4,23)( 5,16)( 6,22)( 7,15)( 8,21)( 9,14)(10,20)(11,26)(12,19)(13,25)$
2B $2^{6},1^{14}$ $26$ $2$ $6$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,12)$
2C $2^{13}$ $26$ $2$ $13$ $( 1,18)( 2,22)( 3,26)( 4,17)( 5,21)( 6,25)( 7,16)( 8,20)( 9,24)(10,15)(11,19)(12,23)(13,14)$
2D $2^{12},1^{2}$ $169$ $2$ $12$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$
4A1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
4A-1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
4B $4^{6},2$ $338$ $4$ $19$ $( 1,22,10,25)( 2,18, 9,16)( 3,14, 8,20)( 4,23, 7,24)( 5,19, 6,15)(11,21,13,26)(12,17)$
4C $4^{6},2$ $338$ $4$ $19$ $( 1,18, 7,15)( 2,24, 6,22)( 3,17, 5,16)( 4,23)( 8,21,13,25)( 9,14,12,19)(10,20,11,26)$
4D $4^{6},1^{2}$ $338$ $4$ $18$ $( 1, 9, 8,13)( 2, 4, 7, 5)( 3,12, 6,10)(14,18,25,21)(15,23,24,16)(17,20,22,19)$
13A1 $13^{2}$ $8$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13A2 $13^{2}$ $8$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13A4 $13^{2}$ $8$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13B1 $13,1^{13}$ $8$ $13$ $12$ $(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13B2 $13,1^{13}$ $8$ $13$ $12$ $(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13B4 $13,1^{13}$ $8$ $13$ $12$ $(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13C1 $13^{2}$ $8$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13C2 $13^{2}$ $8$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13C4 $13^{2}$ $8$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13D1 $13^{2}$ $16$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13D2 $13^{2}$ $16$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13D4 $13^{2}$ $16$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13E1 $13^{2}$ $16$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13E2 $13^{2}$ $16$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13E4 $13^{2}$ $16$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
26A1 $26$ $104$ $26$ $25$ $( 1,24, 3,23, 5,22, 7,21, 9,20,11,19,13,18, 2,17, 4,16, 6,15, 8,14,10,26,12,25)$
26A3 $26$ $104$ $26$ $25$ $( 1,23, 7,20,13,17, 6,14,12,24, 5,21,11,18, 4,15,10,25, 3,22, 9,19, 2,16, 8,26)$
26A7 $26$ $104$ $26$ $25$ $( 1,21, 2,14, 3,20, 4,26, 5,19, 6,25, 7,18, 8,24, 9,17,10,23,11,16,12,22,13,15)$
26B1 $13,2^{6},1$ $104$ $26$ $18$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,12)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
26B3 $13,2^{6},1$ $104$ $26$ $18$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,12)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
26B7 $13,2^{6},1$ $104$ $26$ $18$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,12)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
26C1 $26$ $104$ $26$ $25$ $( 1,16,13,25,12,21,11,17,10,26, 9,22, 8,18, 7,14, 6,23, 5,19, 4,15, 3,24, 2,20)$
26C3 $26$ $104$ $26$ $25$ $( 1,25,11,26, 8,14, 5,15, 2,16,12,17, 9,18, 6,19, 3,20,13,21,10,22, 7,23, 4,24)$
26C7 $26$ $104$ $26$ $25$ $( 1,17, 7,15,13,26, 6,24,12,22, 5,20,11,18, 4,16,10,14, 3,25, 9,23, 2,21, 8,19)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

34 x 34 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed