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Properties

Label 26T18
26T18 1 12 1->12 13 1->13 22 1->22 2 4 2->4 2->12 17 2->17 3 9 3->9 11 3->11 25 3->25 4->1 10 4->10 20 4->20 5 6 5->6 5->9 15 5->15 8 6->8 6->11 23 6->23 7 7->3 18 7->18 26 8->26 9->13 21 9->21 10->5 16 10->16 11->10 24 11->24 12->2 19 12->19 13->7 14 13->14 14->8 14->15 14->19 15->3 15->24 16->11 16->26 17->6 17->21 17->25 18->1 18->24 18->26 19->9 19->18 19->23 20->22 20->23 21->12 21->15 22->7 22->20 23->2 23->25 24->10 24->17 25->5 25->22 26->13 26->14
Degree $26$
Order $1352$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}^2.C_2$

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Copy content magma:G := TransitiveGroup(26, 18);
 

Group invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$52$:  $C_{13}:C_4$ x 2
$104$:  26T7 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T18 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 $2^{13}$ $13$ $2$ $13$ $( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,26)(12,14)(13,15)$
2B $2^{13}$ $13$ $2$ $13$ $( 1,21)( 2,20)( 3,19)( 4,18)( 5,17)( 6,16)( 7,15)( 8,14)( 9,26)(10,25)(11,24)(12,23)(13,22)$
2C $2^{12},1^{2}$ $169$ $2$ $12$ $( 1, 5)( 2, 4)( 6,13)( 7,12)( 8,11)( 9,10)(14,23)(15,22)(16,21)(17,20)(18,19)(24,26)$
4A1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1, 6, 5,13)( 2,11, 4, 8)( 7,10,12, 9)(14,22,23,15)(16,19,21,18)(17,24,20,26)$
4A-1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1,13, 5, 6)( 2, 8, 4,11)( 7, 9,12,10)(14,15,23,22)(16,18,21,19)(17,26,20,24)$
4B1 $4^{6},2$ $169$ $4$ $19$ $( 1,23, 7,19)( 2,18, 6,24)( 3,26, 5,16)( 4,21)( 8,14,13,15)( 9,22,12,20)(10,17,11,25)$
4B-1 $4^{6},2$ $169$ $4$ $19$ $( 1,19, 7,23)( 2,24, 6,18)( 3,16, 5,26)( 4,21)( 8,15,13,14)( 9,20,12,22)(10,25,11,17)$
13A1 $13^{2}$ $4$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13A2 $13^{2}$ $4$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13A4 $13^{2}$ $4$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13B1 $13^{2}$ $4$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13B2 $13^{2}$ $4$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13B4 $13^{2}$ $4$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13C1 $13^{2}$ $8$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13C2 $13^{2}$ $8$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13C4 $13^{2}$ $8$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13D1 $13^{2}$ $8$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13D2 $13^{2}$ $8$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13D4 $13^{2}$ $8$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13E1 $13^{2}$ $8$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13E2 $13^{2}$ $8$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13E4 $13^{2}$ $8$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13F1 $13^{2}$ $8$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13F2 $13^{2}$ $8$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13F4 $13^{2}$ $8$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13G1 $13^{2}$ $8$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13G2 $13^{2}$ $8$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13G4 $13^{2}$ $8$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13H1 $13,1^{13}$ $8$ $13$ $12$ $(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13H2 $13,1^{13}$ $8$ $13$ $12$ $(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13H4 $13,1^{13}$ $8$ $13$ $12$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)$
26A1 $26$ $52$ $26$ $25$ $( 1,14,10,23, 6,19, 2,15,11,24, 7,20, 3,16,12,25, 8,21, 4,17,13,26, 9,22, 5,18)$
26A3 $26$ $52$ $26$ $25$ $( 1,23, 2,24, 3,25, 4,26, 5,14, 6,15, 7,16, 8,17, 9,18,10,19,11,20,12,21,13,22)$
26A7 $26$ $52$ $26$ $25$ $( 1,15,12,26,10,24, 8,22, 6,20, 4,18, 2,16,13,14,11,25, 9,23, 7,21, 5,19, 3,17)$
26B1 $26$ $52$ $26$ $25$ $( 1,15,13,16,12,17,11,18,10,19, 9,20, 8,21, 7,22, 6,23, 5,24, 4,25, 3,26, 2,14)$
26B3 $26$ $52$ $26$ $25$ $( 1,16,11,19, 8,22, 5,25, 2,15,12,18, 9,21, 6,24, 3,14,13,17,10,20, 7,23, 4,26)$
26B7 $26$ $52$ $26$ $25$ $( 1,18, 7,25,13,19, 6,26,12,20, 5,14,11,21, 4,15,10,22, 3,16, 9,23, 2,17, 8,24)$

Malle's constant $a(G)$:     $1/12$

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Character table

38 x 38 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed