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Properties

Label 26T45
26T45 1 19 1->19 21 1->21 2 16 2->16 23 2->23 3 25 3->25 26 3->26 4 14 4->14 4->23 5 5->16 20 5->20 6 17 6->17 18 6->18 7 7->14 7->20 8 22 8->22 24 8->24 9 9->21 9->24 10 10->18 10->26 11 15 11->15 11->15 12 12->17 12->25 13 13->19 13->22 14->9 14->12 15->7 15->10 16->11 17->10 18->5 18->13 19->1 19->13 20->2 20->8 21->3 21->3 22->4 22->11 23->5 23->6 24->1 24->6 25->7 25->9 26->4 26->8
Degree $26$
Order $8112$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}^2.C_{12}$

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Copy content magma:G := TransitiveGroup(26, 45);
 

Group invariants

Abstract group:  $D_{13}^2.C_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $8112=2^{4} \cdot 3 \cdot 13^{2}$
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 $t$:  $45$
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:  $(1,19,13,22,11,15,7,14,12,25,9,21,3,26,4,23,6,17,10,18,5,20,8,24)(2,16)$, $(1,21,3,25,7,20,2,23,5,16,11,15,10,26,8,22,4,14,9,24,6,18,13,19)(12,17)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $C_6$ x 3
$8$:  $C_4\times C_2$
$12$:  $C_{12}$ x 2, $C_6\times C_2$
$16$:  $C_8:C_2$
$24$:  24T2
$48$:  24T16

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

There are no siblings 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 $2^{6},1^{14}$ $26$ $2$ $6$ $( 1, 3)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$
2B $2^{12},1^{2}$ $169$ $2$ $12$ $( 1,13)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(14,19)(15,18)(16,17)(20,26)(21,25)(22,24)$
3A1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 5, 2)( 3,10, 8)( 4, 6,11)( 9,12,13)(14,20,22)(15,16,25)(17,21,18)(19,26,24)$
3A-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 2, 5)( 3, 8,10)( 4,11, 6)( 9,13,12)(14,22,20)(15,25,16)(17,18,21)(19,24,26)$
4A1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1, 3,13,11)( 2, 8,12, 6)( 4, 5,10, 9)(14,17,19,16)(15,22,18,24)(20,21,26,25)$
4A-1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1,11,13, 3)( 2, 6,12, 8)( 4, 9,10, 5)(14,16,19,17)(15,24,18,22)(20,25,26,21)$
4B $4^{6},1^{2}$ $338$ $4$ $18$ $( 1,13, 8, 9)( 2, 5, 7, 4)( 3,10, 6,12)(14,17,15,25)(16,20,26,22)(18,23,24,19)$
6A1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1,12, 5,13, 2, 9)( 3, 6,10,11, 8, 4)(14,24,20,19,22,26)(15,21,16,18,25,17)$
6A-1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1, 9, 2,13, 5,12)( 3, 4, 8,11,10, 6)(14,26,22,19,20,24)(15,17,25,18,16,21)$
6B1 $6^{2},3^{4},1^{2}$ $338$ $6$ $18$ $( 1, 5, 6, 3,12,11)( 4, 9, 7,13, 8,10)(14,25,19)(16,18,24)(17,21,20)(22,23,26)$
6B-1 $6^{2},3^{4},1^{2}$ $338$ $6$ $18$ $( 1,11,12, 3, 6, 5)( 4,10, 8,13, 7, 9)(14,19,25)(16,24,18)(17,20,21)(22,26,23)$
8A1 $8^{3},2$ $338$ $8$ $22$ $( 1,20, 3,21,13,26,11,25)( 2,14, 8,17,12,19, 6,16)( 4,15, 5,22,10,18, 9,24)( 7,23)$
8A-1 $8^{3},2$ $338$ $8$ $22$ $( 1,25,11,26,13,21, 3,20)( 2,16, 6,19,12,17, 8,14)( 4,24, 9,18,10,22, 5,15)( 7,23)$
8B1 $8^{3},2$ $338$ $8$ $22$ $( 1,23, 4,15, 2,16,12,24)( 3,22, 7,20,13,17, 9,19)( 5,21,10,25,11,18, 6,14)( 8,26)$
8B-1 $8^{3},2$ $338$ $8$ $22$ $( 1,24,12,16, 2,15, 4,23)( 3,19, 9,17,13,20, 7,22)( 5,14, 6,18,11,25,10,21)( 8,26)$
12A1 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 4,12, 3, 5, 6,13,10, 2,11, 9, 8)(14,25,24,17,20,15,19,21,22,16,26,18)$
12A-1 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 8, 9,11, 2,10,13, 6, 5, 3,12, 4)(14,18,26,16,22,21,19,15,20,17,24,25)$
12A5 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 6, 9, 3, 2, 4,13, 8, 5,11,12,10)(14,15,26,17,22,25,19,18,20,16,24,21)$
12A-5 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1,10,12,11, 5, 8,13, 4, 2, 3, 9, 6)(14,21,24,16,20,18,19,25,22,17,26,15)$
12B1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 6, 2,13,12, 5, 8, 3, 7, 9,10, 4)(14,18,16,17,23,20,15,24,26,25,19,22)$
12B-1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 4,10, 9, 7, 3, 8, 5,12,13, 2, 6)(14,22,19,25,26,24,15,20,23,17,16,18)$
13A $13,1^{13}$ $24$ $13$ $12$ $(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13B $13^{2}$ $48$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13C $13^{2}$ $48$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13D $13^{2}$ $48$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
24A1 $24,2$ $338$ $24$ $24$ $( 1,17, 4,20,12,15, 3,19, 5,21, 6,22,13,16,10,26, 2,18,11,14, 9,25, 8,24)( 7,23)$
24A-1 $24,2$ $338$ $24$ $24$ $( 1,24, 8,25, 9,14,11,18, 2,26,10,16,13,22, 6,21, 5,19, 3,15,12,20, 4,17)( 7,23)$
24A5 $24,2$ $338$ $24$ $24$ $( 1,15, 6,26, 9,17, 3,22, 2,25, 4,19,13,18, 8,20, 5,16,11,24,12,21,10,14)( 7,23)$
24A-5 $24,2$ $338$ $24$ $24$ $( 1,14,10,21,12,24,11,16, 5,20, 8,18,13,19, 4,25, 2,22, 3,17, 9,26, 6,15)( 7,23)$
24B1 $24,2$ $338$ $24$ $24$ $( 1,19, 5,23, 3,21, 4,22,10,15, 7,25, 2,20,11,16,13,18,12,17, 6,24, 9,14)( 8,26)$
24B-1 $24,2$ $338$ $24$ $24$ $( 1,14, 9,24, 6,17,12,18,13,16,11,20, 2,25, 7,15,10,22, 4,21, 3,23, 5,19)( 8,26)$
24B5 $24,2$ $338$ $24$ $24$ $( 1,21, 7,16, 6,19, 4,25,13,24, 5,22, 2,18, 9,23,10,20,12,14, 3,15,11,17)( 8,26)$
24B-5 $24,2$ $338$ $24$ $24$ $( 1,17,11,15, 3,14,12,20,10,23, 9,18, 2,22, 5,24,13,25, 4,19, 6,16, 7,21)( 8,26)$
26A $13,2^{6},1$ $312$ $26$ $18$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(14,22,17,25,20,15,23,18,26,21,16,24,19)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

35 x 35 character table

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Regular extensions

Data not computed