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Properties

Label 26T36
26T36 1 20 1->20 25 1->25 2 21 2->21 2->25 3 17 3->17 3->17 4 22 4->22 26 4->26 5 14 5->14 5->22 6 18 6->18 19 6->19 7 7->14 24 7->24 8 16 8->16 23 8->23 9 9->19 9->21 10 15 10->15 10->26 11 11->18 11->24 12 12->20 12->23 13 13->15 13->16 14->1 14->2 15->4 15->10 16->5 16->7 17->10 17->13 18->8 18->13 19->3 19->3 20->6 20->11 21->6 21->9 22->1 22->12 23->2 23->9 24->4 24->5 25->8 25->12 26->7 26->11
Degree $26$
Order $4056$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:D_{12}$

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Copy content magma:G := TransitiveGroup(26, 36);
 

Group invariants

Abstract group:  $C_{13}^2:D_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
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 $t$:  $36$
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:  $(1,25,8,23,2,21,9,19,3,17,10,15,4,26,11,24,5,22,12,20,6,18,13,16,7,14)$, $(1,20,11,18,8,16,5,14,2,25,12,23,9,21,6,19,3,17,13,15,10,26,7,24,4,22)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$24$:  $D_{12}$

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 $2^{13}$ $78$ $2$ $13$ $( 1,18)( 2,25)( 3,19)( 4,26)( 5,20)( 6,14)( 7,21)( 8,15)( 9,22)(10,16)(11,23)(12,17)(13,24)$
2B $2^{13}$ $78$ $2$ $13$ $( 1,23)( 2,24)( 3,25)( 4,26)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)(10,19)(11,20)(12,21)(13,22)$
2C $2^{12},1^{2}$ $169$ $2$ $12$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,12)(14,24)(15,23)(16,22)(17,21)(18,20)(25,26)$
3A $3^{8},1^{2}$ $338$ $3$ $16$ $( 1,12, 7)( 2, 8,10)( 3, 4,13)( 5, 9, 6)(14,17,26)(15,20,22)(16,23,18)(21,25,24)$
4A $4^{6},1^{2}$ $338$ $4$ $18$ $( 1,13, 8, 9)( 2, 5, 7, 4)( 3,10, 6,12)(14,18,24,20)(15,26,23,25)(16,21,22,17)$
6A $6^{4},1^{2}$ $338$ $6$ $20$ $( 1, 2,12, 8, 7,10)( 3, 9, 4, 6,13, 5)(14,25,17,24,26,21)(15,16,20,23,22,18)$
12A1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 6, 2,13,12, 5, 8, 3, 7, 9,10, 4)(14,22,25,18,17,15,24,16,26,20,21,23)$
12A5 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 5,10,13, 7, 6, 8, 4,12, 9, 2, 3)(14,15,21,18,26,22,24,23,17,20,25,16)$
13A1 $13^{2}$ $12$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13A2 $13^{2}$ $12$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13A3 $13^{2}$ $12$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13A4 $13^{2}$ $12$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13A5 $13^{2}$ $12$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13A6 $13^{2}$ $12$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13B1 $13^{2}$ $12$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13B2 $13^{2}$ $12$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13B3 $13^{2}$ $12$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13B4 $13^{2}$ $12$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13B5 $13^{2}$ $12$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13B6 $13^{2}$ $12$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13C $13,1^{13}$ $24$ $13$ $12$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)$
26A1 $26$ $156$ $26$ $25$ $( 1,25, 3,26, 5,14, 7,15, 9,16,11,17,13,18, 2,19, 4,20, 6,21, 8,22,10,23,12,24)$
26A3 $26$ $156$ $26$ $25$ $( 1,26, 7,16,13,19, 6,22,12,25, 5,15,11,18, 4,21,10,24, 3,14, 9,17, 2,20, 8,23)$
26A5 $26$ $156$ $26$ $25$ $( 1,14,11,19, 8,24, 5,16, 2,21,12,26, 9,18, 6,23, 3,15,13,20,10,25, 7,17, 4,22)$
26A7 $26$ $156$ $26$ $25$ $( 1,15, 2,22, 3,16, 4,23, 5,17, 6,24, 7,18, 8,25, 9,19,10,26,11,20,12,14,13,21)$
26A9 $26$ $156$ $26$ $25$ $( 1,16, 6,25,11,21, 3,17, 8,26,13,22, 5,18,10,14, 2,23, 7,19,12,15, 4,24, 9,20)$
26A11 $26$ $156$ $26$ $25$ $( 1,17,10,15, 6,26, 2,24,11,22, 7,20, 3,18,12,16, 8,14, 4,25,13,23, 9,21, 5,19)$
26B1 $26$ $156$ $26$ $25$ $( 1,16,13,15,12,14,11,26,10,25, 9,24, 8,23, 7,22, 6,21, 5,20, 4,19, 3,18, 2,17)$
26B3 $26$ $156$ $26$ $25$ $( 1,15,11,25, 8,22, 5,19, 2,16,12,26, 9,23, 6,20, 3,17,13,14,10,24, 7,21, 4,18)$
26B5 $26$ $156$ $26$ $25$ $( 1,14, 9,22, 4,17,12,25, 7,20, 2,15,10,23, 5,18,13,26, 8,21, 3,16,11,24, 6,19)$
26B7 $26$ $156$ $26$ $25$ $( 1,26, 7,19,13,25, 6,18,12,24, 5,17,11,23, 4,16,10,22, 3,15, 9,21, 2,14, 8,20)$
26B9 $26$ $156$ $26$ $25$ $( 1,25, 5,16, 9,20,13,24, 4,15, 8,19,12,23, 3,14, 7,18,11,22, 2,26, 6,17,10,21)$
26B11 $26$ $156$ $26$ $25$ $( 1,24, 3,26, 5,15, 7,17, 9,19,11,21,13,23, 2,25, 4,14, 6,16, 8,18,10,20,12,22)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

34 x 34 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed