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Properties

Label 42T176
Degree $42$
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(42, 176);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $176$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(2,13)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,21,38,36,13,29,27)(2,20,37,35,15,30,25)(3,19,39,34,14,28,26)(4,16,22,8,31,42,11)(5,18,23,7,32,41,10)(6,17,24,9,33,40,12), (1,7,12,14,19,37)(2,8,10,13,21,39)(3,9,11,15,20,38)(4,17,40,35,26,23)(5,16,42,36,25,24)(6,18,41,34,27,22)(28,29,30)(31,32,33)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 6: None

Degree 7: None

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

Degree 21: None

Low degree siblings

14T30, 28T120

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 7, 7, 7, 7, 7, 7 $ $156$ $7$ $( 1,10,15,28, 6, 7,17)( 2,11,14,29, 5, 8,18)( 3,12,13,30, 4, 9,16) (19,27,39,32,40,36,22)(20,26,37,33,42,34,23)(21,25,38,31,41,35,24)$
$ 7, 7, 7, 7, 7, 7 $ $156$ $7$ $( 1, 6,10, 7,15,17,28)( 2, 5,11, 8,14,18,29)( 3, 4,12, 9,13,16,30) (19,40,27,36,39,22,32)(20,42,26,34,37,23,33)(21,41,25,35,38,24,31)$
$ 7, 7, 7, 7, 7, 7 $ $156$ $7$ $( 1,15, 6,17,10,28, 7)( 2,14, 5,18,11,29, 8)( 3,13, 4,16,12,30, 9) (19,39,40,22,27,32,36)(20,37,42,23,26,33,34)(21,38,41,24,25,31,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $91$ $2$ $( 1,21)( 2,20)( 3,19)( 4, 8)( 5, 7)( 6, 9)(10,32)(11,31)(12,33)(16,22)(17,24) (18,23)(25,37)(26,39)(27,38)(28,34)(29,36)(30,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $182$ $3$ $( 1,11,39)( 2,12,38)( 3,10,37)( 4,34,24)( 5,35,22)( 6,36,23)( 7,30,16) ( 8,28,17)( 9,29,18)(13,14,15)(19,32,25)(20,33,27)(21,31,26)(40,41,42)$
$ 6, 6, 6, 6, 6, 6, 3, 3 $ $182$ $6$ $( 1,26,11,21,39,31)( 2,27,12,20,38,33)( 3,25,10,19,37,32)( 4,17,34, 8,24,28) ( 5,16,35, 7,22,30)( 6,18,36, 9,23,29)(13,15,14)(40,42,41)$
$ 13, 13, 13, 1, 1, 1 $ $84$ $13$ $( 1,20,16, 9,37, 6,35,33,11,15,29,23,41)( 2,19,17, 7,39, 5,34,32,12,14,30,22, 42)( 3,21,18, 8,38, 4,36,31,10,13,28,24,40)$
$ 13, 13, 13, 1, 1, 1 $ $84$ $13$ $( 1,11, 9,23,35,20,15,37,41,33,16,29, 6)( 2,12, 7,22,34,19,14,39,42,32,17,30, 5)( 3,10, 8,24,36,21,13,38,40,31,18,28, 4)$

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 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);