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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 invariants

Abstract group:  $\PSL(2,13)$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $120$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(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 TypeSizeOrderIndexRepresentative
1A $1^{28}$ $1$ $1$ $0$ $()$
2A $2^{14}$ $91$ $2$ $14$ $( 1, 2)( 3,10)( 4, 9)( 5,25)( 6,26)( 7, 8)(11,16)(12,15)(13,20)(14,19)(17,27)(18,28)(21,23)(22,24)$
3A $3^{8},1^{4}$ $182$ $3$ $16$ $( 3,21,19)( 4,22,20)( 5,27,12)( 6,28,11)( 9,24,13)(10,23,14)(15,25,17)(16,26,18)$
6A $6^{4},2^{2}$ $182$ $6$ $22$ $( 1, 2)( 3,14,21,10,19,23)( 4,13,22, 9,20,24)( 5,15,27,25,12,17)( 6,16,28,26,11,18)( 7, 8)$
7A1 $7^{4}$ $156$ $7$ $24$ $( 1,27,25, 7,14,24,10)( 2,28,26, 8,13,23, 9)( 3,16, 5,21,17,11,19)( 4,15, 6,22,18,12,20)$
7A2 $7^{4}$ $156$ $7$ $24$ $( 1,25,14,10,27, 7,24)( 2,26,13, 9,28, 8,23)( 3, 5,17,19,16,21,11)( 4, 6,18,20,15,22,12)$
7A3 $7^{4}$ $156$ $7$ $24$ $( 1, 7,10,25,24,27,14)( 2, 8, 9,26,23,28,13)( 3,21,19, 5,11,16,17)( 4,22,20, 6,12,15,18)$
13A1 $13^{2},1^{2}$ $84$ $13$ $24$ $( 1,19,23, 6,13,18,12,26, 4,28,21,10, 8)( 2,20,24, 5,14,17,11,25, 3,27,22, 9, 7)$
13A2 $13^{2},1^{2}$ $84$ $13$ $24$ $( 1,23,13,12, 4,21, 8,19, 6,18,26,28,10)( 2,24,14,11, 3,22, 7,20, 5,17,25,27, 9)$

Malle's constant $a(G)$:     $1/14$

magma: ConjugacyClasses(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 ζ13−6+ζ13−5+ζ13−2+1+ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2−ζ132−ζ135−ζ136
1092.25.7a2 R 7 −1 1 −1 0 0 0 −ζ13−6−ζ13−5−ζ13−2−ζ132−ζ135−ζ136 ζ13−6+ζ13−5+ζ13−2+1+ζ132+ζ135+ζ136
1092.25.12a1 R 12 0 0 0 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −1 −1
1092.25.12a2 R 12 0 0 0 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −1 −1
1092.25.12a3 R 12 0 0 0 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −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);
 

Regular extensions

Data not computed