Properties

Label 28T120
Degree $28$
Order $1092$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\PSL(2,13)$

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Show commands: Magma

magma: G := TransitiveGroup(28, 120);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $120$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(2,13)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,26,11,15,17,13,10)(2,25,12,16,18,14,9)(3,7,5,28,20,22,23)(4,8,6,27,19,21,24), (1,14,9,17,28,3,20)(2,13,10,18,27,4,19)(5,7,26,15,23,22,11)(6,8,25,16,24,21,12)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: $\PSL(2,13)$

Low degree siblings

14T30, 42T176

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
1A $1^{28}$ $1$ $1$ $()$
2A $2^{14}$ $91$ $2$ $( 1, 2)( 3,14)( 4,13)( 5, 9)( 6,10)( 7,28)( 8,27)(11,22)(12,21)(15,26)(16,25)(17,23)(18,24)(19,20)$
3A $3^{8},1^{4}$ $182$ $3$ $( 1, 9,13)( 2,10,14)( 3,22,15)( 4,21,16)( 7,11,28)( 8,12,27)(17,20,26)(18,19,25)$
6A $6^{4},2^{2}$ $182$ $6$ $( 1, 2)( 3,22,15,14,11,26)( 4,21,16,13,12,25)( 5,24, 7, 9,18,28)( 6,23, 8,10,17,27)(19,20)$
7A1 $7^{4}$ $156$ $7$ $( 1,26, 8,27,14,19,12)( 2,25, 7,28,13,20,11)( 3,21,15, 6,17,10,23)( 4,22,16, 5,18, 9,24)$
7A2 $7^{4}$ $156$ $7$ $( 1, 8,14,12,26,27,19)( 2, 7,13,11,25,28,20)( 3,15,17,23,21, 6,10)( 4,16,18,24,22, 5, 9)$
7A3 $7^{4}$ $156$ $7$ $( 1,27,12, 8,19,26,14)( 2,28,11, 7,20,25,13)( 3, 6,23,15,10,21,17)( 4, 5,24,16, 9,22,18)$
13A1 $13^{2},1^{2}$ $84$ $13$ $( 1,19,10, 8,25, 5, 4,22,16,24,13,27,17)( 2,20, 9, 7,26, 6, 3,21,15,23,14,28,18)$
13A2 $13^{2},1^{2}$ $84$ $13$ $( 1,10,25, 4,16,13,17,19, 8, 5,22,24,27)( 2, 9,26, 3,15,14,18,20, 7, 6,21,23,28)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1092=2^{2} \cdot 3 \cdot 7 \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1092.25
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 6A 7A1 7A2 7A3 13A1 13A2
Size 1 91 182 182 156 156 156 84 84
2 P 1A 1A 3A 3A 7A2 7A3 7A1 13A2 13A1
3 P 1A 2A 1A 2A 7A3 7A1 7A2 13A1 13A2
7 P 1A 2A 3A 6A 1A 1A 1A 13A2 13A1
13 P 1A 2A 3A 6A 7A1 7A2 7A3 1A 1A
Type
1092.25.1a R 1 1 1 1 1 1 1 1 1
1092.25.7a1 R 7 1 1 1 0 0 0 ζ136+ζ135+ζ132+1+ζ132+ζ135+ζ136 ζ136ζ135ζ132ζ132ζ135ζ136
1092.25.7a2 R 7 1 1 1 0 0 0 ζ136ζ135ζ132ζ132ζ135ζ136 ζ136+ζ135+ζ132+1+ζ132+ζ135+ζ136
1092.25.12a1 R 12 0 0 0 ζ71ζ7 ζ72ζ72 ζ73ζ73 1 1
1092.25.12a2 R 12 0 0 0 ζ72ζ72 ζ73ζ73 ζ71ζ7 1 1
1092.25.12a3 R 12 0 0 0 ζ73ζ73 ζ71ζ7 ζ72ζ72 1 1
1092.25.13a R 13 1 1 1 1 1 1 0 0
1092.25.14a R 14 2 1 1 0 0 0 1 1
1092.25.14b R 14 2 1 1 0 0 0 1 1

magma: CharacterTable(G);