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Properties

Label 26T40
26T40 1 19 1->19 1->19 2 17 2->17 21 2->21 3 15 3->15 23 3->23 4 25 4->25 26 4->26 5 14 5->14 24 5->24 6 16 6->16 22 6->22 7 18 7->18 20 7->20 8 8->18 8->20 9 9->16 9->22 10 10->14 10->24 11 11->25 11->26 12 12->15 12->23 13 13->17 13->21 14->1 14->9 15->3 16->5 16->10 17->4 17->7 18->9 18->11 19->5 19->11 20->12 20->13 21->2 21->6 22->4 22->13 23->6 23->7 24->1 24->8 25->8 25->10 26->2 26->12
Degree $26$
Order $6084$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:(C_6\times S_3)$

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Copy content magma:G := TransitiveGroup(26, 40);
 

Group invariants

Abstract group:  $C_{13}^2:(C_6\times S_3)$
Copy content magma:IdentifyGroup(G);
 
Order:  $6084=2^{2} \cdot 3^{2} \cdot 13^{2}$
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 $t$:  $40$
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:  $(1,19,5,14,9,22,13,17,4,25,8,20,12,15,3,23,7,18,11,26,2,21,6,16,10,24)$, $(1,19,11,25,10,14)(2,17,7,20,13,21)(3,15)(4,26,12,23,6,22)(5,24,8,18,9,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

39T46 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 $2^{13}$ $39$ $2$ $13$ $( 1,14)( 2,19)( 3,24)( 4,16)( 5,21)( 6,26)( 7,18)( 8,23)( 9,15)(10,20)(11,25)(12,17)(13,22)$
2B $2^{13}$ $39$ $2$ $13$ $( 1,15)( 2,22)( 3,16)( 4,23)( 5,17)( 6,24)( 7,18)( 8,25)( 9,19)(10,26)(11,20)(12,14)(13,21)$
2C $2^{12},1^{2}$ $169$ $2$ $12$ $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,13)(14,19)(15,18)(16,17)(20,26)(21,25)(22,24)$
3A1 $3^{4},1^{14}$ $26$ $3$ $8$ $( 1, 6,12)( 3,11, 5)( 4, 7, 8)( 9,13,10)$
3A-1 $3^{4},1^{14}$ $26$ $3$ $8$ $( 1,12, 6)( 3, 5,11)( 4, 8, 7)( 9,10,13)$
3B1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 9, 7)( 2,12, 3)( 4, 5, 8)( 6,11,13)(14,23,24)(15,26,20)(17,19,25)(18,22,21)$
3B-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 7, 9)( 2, 3,12)( 4, 8, 5)( 6,13,11)(14,24,23)(15,20,26)(17,25,19)(18,21,22)$
3C $3^{8},1^{2}$ $338$ $3$ $16$ $( 1,10,11)( 2,13, 7)( 4, 6,12)( 5, 9, 8)(14,24,23)(15,20,26)(17,25,19)(18,21,22)$
6A1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1,12, 4,11,13, 8)( 2, 3, 7,10, 9, 5)(14,26,22,19,20,24)(15,17,25,18,16,21)$
6A-1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1, 8,13,11, 4,12)( 2, 5, 9,10, 7, 3)(14,24,20,19,22,26)(15,21,16,18,25,17)$
6B $6^{4},1^{2}$ $338$ $6$ $20$ $( 1, 8,10, 5,11, 9)( 2,12,13, 4, 7, 6)(14,22,24,18,23,21)(15,19,20,17,26,25)$
6C1 $6^{2},2^{6},1^{2}$ $338$ $6$ $16$ $( 1, 5, 6, 3,12,11)( 4, 9, 7,13, 8,10)(14,24)(15,23)(16,22)(17,21)(18,20)(25,26)$
6C-1 $6^{2},2^{6},1^{2}$ $338$ $6$ $16$ $( 1,11,12, 3, 6, 5)( 4,10, 8,13, 7, 9)(14,24)(15,23)(16,22)(17,21)(18,20)(25,26)$
6D1 $6^{4},2$ $507$ $6$ $21$ $( 1,23, 9,24, 7,14)( 2,15,12,26, 3,20)( 4,25, 5,17, 8,19)( 6,22,11,21,13,18)(10,16)$
6D-1 $6^{4},2$ $507$ $6$ $21$ $( 1,14, 7,24, 9,23)( 2,20, 3,26,12,15)( 4,19, 8,17, 5,25)( 6,18,13,21,11,22)(10,16)$
6E1 $6^{4},2$ $507$ $6$ $21$ $( 1,22, 5,15, 2,17)( 3,25,10,16, 8,26)( 4,20, 6,23,11,24)( 7,18)( 9,21,12,19,13,14)$
6E-1 $6^{4},2$ $507$ $6$ $21$ $( 1,17, 2,15, 5,22)( 3,26, 8,16,10,25)( 4,24,11,23, 6,20)( 7,18)( 9,14,13,19,12,21)$
13A1 $13,1^{13}$ $12$ $13$ $12$ $(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13A2 $13,1^{13}$ $12$ $13$ $12$ $(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13B1 $13^{2}$ $18$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13B2 $13^{2}$ $18$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13C1 $13^{2}$ $18$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13C2 $13^{2}$ $18$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13D1 $13^{2}$ $36$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13D2 $13^{2}$ $36$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
26A1 $26$ $234$ $26$ $25$ $( 1,23, 2,15, 3,20, 4,25, 5,17, 6,22, 7,14, 8,19, 9,24,10,16,11,21,12,26,13,18)$
26A5 $26$ $234$ $26$ $25$ $( 1,20, 6,19,11,18, 3,17, 8,16,13,15, 5,14,10,26, 2,25, 7,24,12,23, 4,22, 9,21)$
26B1 $26$ $234$ $26$ $25$ $( 1,14, 6,15,11,16, 3,17, 8,18,13,19, 5,20,10,21, 2,22, 7,23,12,24, 4,25, 9,26)$
26B5 $26$ $234$ $26$ $25$ $( 1,16,13,21,12,26,11,18,10,23, 9,15, 8,20, 7,25, 6,17, 5,22, 4,14, 3,19, 2,24)$
39A1 $13,3^{4},1$ $156$ $39$ $20$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
39A-1 $13,3^{4},1$ $156$ $39$ $20$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
39A2 $13,3^{4},1$ $156$ $39$ $20$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
39A-2 $13,3^{4},1$ $156$ $39$ $20$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(14,21,15,22,16,23,17,24,18,25,19,26,20)$

Malle's constant $a(G)$:     $1/8$

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Character table

34 x 34 character table

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Regular extensions

Data not computed