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

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);