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Properties

Label 26T35
26T35 1 15 1->15 16 1->16 2 2->15 21 2->21 3 14 3->14 3->14 4 20 4->20 26 4->26 5 25 5->25 5->26 6 19 6->19 24 6->24 7 23 7->23 7->25 8 18 8->18 22 8->22 9 9->21 9->24 10 17 10->17 10->20 11 11->19 11->23 12 12->16 12->18 13 13->17 13->22 14->13 14->13 15->1 15->2 16->2 16->4 17->3 17->6 18->4 19->5 19->10 20->6 20->12 21->1 21->7 22->3 23->5 23->9 24->7 24->10 25->9 25->11 26->11 26->12
Degree $26$
Order $4056$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:C_3:Q_8$

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Show commands: Magma

magma: G := TransitiveGroup(26, 35);
 

Group invariants

Abstract group:  $C_{13}^2:C_3:Q_8$
magma: IdentifyGroup(G);
 
Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,15,2,21)(3,14,13,22)(4,20,12,16)(5,26,11,23)(6,19,10,17)(7,25,9,24)(8,18)$, $(1,16,2,15)(3,14,13,17)(4,26,12,18)(5,25,11,19)(6,24,10,20)(7,23,9,21)(8,22)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $Q_8$
$12$:  $D_{6}$
$24$:  24T5

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 $2^{12},1^{2}$ $169$ $2$ $12$ $( 1, 5)( 2, 4)( 6,13)( 7,12)( 8,11)( 9,10)(14,16)(17,26)(18,25)(19,24)(20,23)(21,22)$
3A $3^{8},1^{2}$ $338$ $3$ $16$ $( 1,10,11)( 2,13, 7)( 4, 6,12)( 5, 9, 8)(14,19,25)(16,24,18)(17,20,21)(22,26,23)$
4A $4^{6},1^{2}$ $338$ $4$ $18$ $( 1, 6, 5,13)( 2,11, 4, 8)( 7,10,12, 9)(14,20,16,23)(17,18,26,25)(19,21,24,22)$
4B $4^{6},2$ $1014$ $4$ $19$ $( 1,14, 4,15)( 2,23, 3,19)( 5,24,13,18)( 6,20,12,22)( 7,16,11,26)( 8,25,10,17)( 9,21)$
4C $4^{6},2$ $1014$ $4$ $19$ $( 1,25, 9,22)( 2,23, 8,24)( 3,21, 7,26)( 4,19, 6,15)( 5,17)(10,20,13,14)(11,18,12,16)$
6A $6^{4},1^{2}$ $338$ $6$ $20$ $( 1, 8,10, 5,11, 9)( 2,12,13, 4, 7, 6)(14,18,19,16,25,24)(17,22,20,26,21,23)$
12A1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 7, 8, 6,10, 2, 5,12,11,13, 9, 4)(14,22,18,20,19,26,16,21,25,23,24,17)$
12A5 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 2, 9, 6,11, 7, 5, 4,10,13, 8,12)(14,26,24,20,25,22,16,17,19,23,18,21)$
13A $13,1^{13}$ $24$ $13$ $12$ $(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13B1 $13^{2}$ $24$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13B2 $13^{2}$ $24$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13B4 $13^{2}$ $24$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13C1 $13^{2}$ $24$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13C2 $13^{2}$ $24$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13C4 $13^{2}$ $24$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,15,16,17,18,19,20,21,22,23,24,25,26)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 3A 4A 4B 4C 6A 12A1 12A5 13A 13B1 13B2 13B4 13C1 13C2 13C4
Size 1 169 338 338 1014 1014 338 338 338 24 24 24 24 24 24 24
2 P 1A 1A 3A 2A 2A 2A 3A 6A 6A 13A 13B2 13B4 13B1 13C2 13C4 13C1
3 P 1A 2A 1A 4A 4B 4C 2A 4A 4A 13A 13B2 13B4 13B1 13C2 13C4 13C1
13 P 1A 2A 3A 4A 4B 4C 6A 12A1 12A5 1A 1A 1A 1A 1A 1A 1A
Type
4056.bg.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4056.bg.1b R 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1
4056.bg.1c R 1 1 1 −1 1 −1 1 −1 −1 1 1 1 1 1 1 1
4056.bg.1d R 1 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1
4056.bg.2a R 2 2 −1 2 0 0 −1 −1 −1 2 2 2 2 2 2 2
4056.bg.2b R 2 2 −1 −2 0 0 −1 1 1 2 2 2 2 2 2 2
4056.bg.2c S 2 −2 2 0 0 0 −2 0 0 2 2 2 2 2 2 2
4056.bg.2d1 S 2 −2 −1 0 0 0 1 −ζ12−1−ζ12 ζ12−1+ζ12 2 2 2 2 2 2 2
4056.bg.2d2 S 2 −2 −1 0 0 0 1 ζ12−1+ζ12 −ζ12−1−ζ12 2 2 2 2 2 2 2
4056.bg.24a R 24 0 0 0 0 0 0 0 0 −2 −2 −2 −2 −2 −2 11
4056.bg.24b1 R 24 0 0 0 0 0 0 0 0 4ζ13−6+4ζ13−4+2ζ13−3+2ζ13−2+2ζ132+2ζ133+4ζ134+4ζ136 −ζ13−3−ζ13−2+2−ζ132−ζ133 −ζ13−6−ζ13−4+2−ζ134−ζ136 −ζ13−5−ζ13−1+2−ζ13−ζ135 −2ζ13−6−2ζ13−4+2ζ13−3+2ζ13−2−2+2ζ132+2ζ133−2ζ134−2ζ136 −2ζ13−6−2ζ13−4−4ζ13−3−4ζ13−2−4−4ζ132−4ζ133−2ζ134−2ζ136 −2
4056.bg.24b2 R 24 0 0 0 0 0 0 0 0 −2ζ13−6−2ζ13−4−4ζ13−3−4ζ13−2−4−4ζ132−4ζ133−2ζ134−2ζ136 −ζ13−6−ζ13−4+2−ζ134−ζ136 −ζ13−5−ζ13−1+2−ζ13−ζ135 −ζ13−3−ζ13−2+2−ζ132−ζ133 4ζ13−6+4ζ13−4+2ζ13−3+2ζ13−2+2ζ132+2ζ133+4ζ134+4ζ136 −2ζ13−6−2ζ13−4+2ζ13−3+2ζ13−2−2+2ζ132+2ζ133−2ζ134−2ζ136 −2
4056.bg.24b3 R 24 0 0 0 0 0 0 0 0 −2ζ13−6−2ζ13−4+2ζ13−3+2ζ13−2−2+2ζ132+2ζ133−2ζ134−2ζ136 −ζ13−5−ζ13−1+2−ζ13−ζ135 −ζ13−3−ζ13−2+2−ζ132−ζ133 −ζ13−6−ζ13−4+2−ζ134−ζ136 −2ζ13−6−2ζ13−4−4ζ13−3−4ζ13−2−4−4ζ132−4ζ133−2ζ134−2ζ136 4ζ13−6+4ζ13−4+2ζ13−3+2ζ13−2+2ζ132+2ζ133+4ζ134+4ζ136 −2
4056.bg.24c1 R 24 0 0 0 0 0 0 0 0 −ζ13−3−ζ13−2+2−ζ132−ζ133 4ζ13−6+4ζ13−4+2ζ13−3+2ζ13−2+2ζ132+2ζ133+4ζ134+4ζ136 −2ζ13−6−2ζ13−4−4ζ13−3−4ζ13−2−4−4ζ132−4ζ133−2ζ134−2ζ136 −2ζ13−6−2ζ13−4+2ζ13−3+2ζ13−2−2+2ζ132+2ζ133−2ζ134−2ζ136 −ζ13−5−ζ13−1+2−ζ13−ζ135 −ζ13−6−ζ13−4+2−ζ134−ζ136 −2
4056.bg.24c2 R 24 0 0 0 0 0 0 0 0 −ζ13−5−ζ13−1+2−ζ13−ζ135 −2ζ13−6−2ζ13−4+2ζ13−3+2ζ13−2−2+2ζ132+2ζ133−2ζ134−2ζ136 4ζ13−6+4ζ13−4+2ζ13−3+2ζ13−2+2ζ132+2ζ133+4ζ134+4ζ136 −2ζ13−6−2ζ13−4−4ζ13−3−4ζ13−2−4−4ζ132−4ζ133−2ζ134−2ζ136 −ζ13−6−ζ13−4+2−ζ134−ζ136 −ζ13−3−ζ13−2+2−ζ132−ζ133 −2
4056.bg.24c3 R 24 0 0 0 0 0 0 0 0 −ζ13−6−ζ13−4+2−ζ134−ζ136 −2ζ13−6−2ζ13−4−4ζ13−3−4ζ13−2−4−4ζ132−4ζ133−2ζ134−2ζ136 −2ζ13−6−2ζ13−4+2ζ13−3+2ζ13−2−2+2ζ132+2ζ133−2ζ134−2ζ136 4ζ13−6+4ζ13−4+2ζ13−3+2ζ13−2+2ζ132+2ζ133+4ζ134+4ζ136 −ζ13−3−ζ13−2+2−ζ132−ζ133 −ζ13−5−ζ13−1+2−ζ13−ζ135 −2

magma: CharacterTable(G);
 

Regular extensions

Data not computed