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Properties

Label 26T21
Degree $26$
Order $2028$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}:F_{13}$

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Show commands: Magma

magma: G := TransitiveGroup(26, 21);
 

Group invariants

Abstract group:  $C_{13}:F_{13}$
magma: IdentifyGroup(G);
 
Order:  $2028=2^{2} \cdot 3 \cdot 13^{2}$
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 $t$:  $21$
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:  $(1,15,2,21)(3,14,13,22)(4,20,12,16)(5,26,11,23)(6,19,10,17)(7,25,9,24)(8,18)$, $(1,4,3,12,9,10)(2,8,6,11,5,7)(14,26,22,19,20,24)(15,17,25,18,16,21)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$156$:  $F_{13}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T21 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 $2^{12},1^{2}$ $169$ $2$ $12$ $( 1, 2)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)$
3A1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 4,13)( 2, 7, 9)( 3,10, 5)( 8,12,11)(14,19,21)(15,22,17)(16,25,26)(20,24,23)$
3A-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1,13, 4)( 2, 9, 7)( 3, 5,10)( 8,11,12)(14,21,19)(15,17,22)(16,26,25)(20,23,24)$
4A1 $4^{6},2$ $169$ $4$ $19$ $( 1,21, 2,15)( 3,22,13,14)( 4,16,12,20)( 5,23,11,26)( 6,17,10,19)( 7,24, 9,25)( 8,18)$
4A-1 $4^{6},2$ $169$ $4$ $19$ $( 1,15, 2,21)( 3,14,13,22)( 4,20,12,16)( 5,26,11,23)( 6,19,10,17)( 7,25, 9,24)( 8,18)$
6A1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1,12, 4,11,13, 8)( 2, 3, 7,10, 9, 5)(14,15,19,22,21,17)(16,23,25,20,26,24)$
6A-1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1, 8,13,11, 4,12)( 2, 5, 9,10, 7, 3)(14,17,21,22,19,15)(16,24,26,20,25,23)$
12A1 $12^{2},2$ $169$ $12$ $23$ $( 1,17,12,14, 4,15,11,19,13,22, 8,21)( 2,25, 3,20, 7,26,10,24, 9,16, 5,23)( 6,18)$
12A-1 $12^{2},2$ $169$ $12$ $23$ $( 1,19,12,22, 4,21,11,17,13,14, 8,15)( 2,24, 3,16, 7,23,10,25, 9,20, 5,26)( 6,18)$
12A5 $12^{2},2$ $169$ $12$ $23$ $( 1,21, 8,22,13,19,11,15, 4,14,12,17)( 2,23, 5,16, 9,24,10,26, 7,20, 3,25)( 6,18)$
12A-5 $12^{2},2$ $169$ $12$ $23$ $( 1,15, 8,14,13,17,11,21, 4,22,12,19)( 2,26, 5,20, 9,25,10,23, 7,16, 3,24)( 6,18)$
13A $13,1^{13}$ $12$ $13$ $12$ $(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13B $13^{2}$ $12$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13C1 $13^{2}$ $12$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13C2 $13,1^{13}$ $12$ $13$ $12$ $(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13D1 $13^{2}$ $12$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13D2 $13^{2}$ $12$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13E1 $13^{2}$ $12$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13E2 $13^{2}$ $12$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13F1 $13^{2}$ $12$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13F2 $13^{2}$ $12$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13G1 $13^{2}$ $12$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13G2 $13^{2}$ $12$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13H1 $13^{2}$ $12$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13H2 $13^{2}$ $12$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,17,20,23,26,16,19,22,25,15,18,21,24)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5 13A 13B 13C1 13C2 13D1 13D2 13E1 13E2 13F1 13F2 13G1 13G2 13H1 13H2
Size 1 169 169 169 169 169 169 169 169 169 169 169 12 12 12 12 12 12 12 12 12 12 12 12 12 12
2 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 6A1 6A1 6A-1 6A-1 13F2 13A 13B 13F1 13H1 13H2 13C2 13C1 13D2 13D1 13G1 13E2 13E1 13G2
3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 4A-1 4A1 4A1 4A-1 13F1 13A 13B 13F2 13H2 13H1 13C1 13C2 13D1 13D2 13G2 13E1 13E2 13G1
13 P 1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A-5 12A1 12A5 12A-1 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A
Type

magma: CharacterTable(G);
 

Regular extensions

Data not computed