# Properties

 Label 42T176 Degree $42$ Order $1092$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $\PSL(2,13)$

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

 Label Cycle Type Size Order Representative 1A $1^{42}$ $1$ $1$ $()$ 2A $2^{18},1^{6}$ $91$ $2$ $( 1,39)( 2,38)( 3,37)( 4,14)( 5,15)( 6,13)(10,33)(11,32)(12,31)(16,21)(17,20)(18,19)(22,26)(23,25)(24,27)(28,40)(29,41)(30,42)$ 3A $3^{14}$ $182$ $3$ $( 1,32,39)( 2,31,38)( 3,33,37)( 4, 6, 5)( 7,21,42)( 8,20,40)( 9,19,41)(10,17,24)(11,18,23)(12,16,22)(13,25,36)(14,27,34)(15,26,35)(28,30,29)$ 6A $6^{6},3^{2}$ $182$ $6$ $( 1,15,42,39, 5,30)( 2,14,40,38, 4,28)( 3,13,41,37, 6,29)( 7, 9, 8)(10,16,26,33,21,22)(11,17,27,32,20,24)(12,18,25,31,19,23)(34,36,35)$ 7A1 $7^{6}$ $156$ $7$ $( 1,39,27, 7,33,35,20)( 2,38,25, 8,32,34,19)( 3,37,26, 9,31,36,21)( 4,42,15,28,23,12,18)( 5,41,13,30,24,11,17)( 6,40,14,29,22,10,16)$ 7A2 $7^{6}$ $156$ $7$ $( 1,27,33,20,39, 7,35)( 2,25,32,19,38, 8,34)( 3,26,31,21,37, 9,36)( 4,15,23,18,42,28,12)( 5,13,24,17,41,30,11)( 6,14,22,16,40,29,10)$ 7A3 $7^{6}$ $156$ $7$ $( 1, 7,20,27,35,39,33)( 2, 8,19,25,34,38,32)( 3, 9,21,26,36,37,31)( 4,28,18,15,12,42,23)( 5,30,17,13,11,41,24)( 6,29,16,14,10,40,22)$ 13A1 $13^{3},1^{3}$ $84$ $13$ $( 1,20, 5,23,10,18,30,27,39,40,36, 9,14)( 2,19, 4,22,11,16,28,25,38,41,35, 7,13)( 3,21, 6,24,12,17,29,26,37,42,34, 8,15)$ 13A2 $13^{3},1^{3}$ $84$ $13$ $( 1, 5,10,30,39,36,14,20,23,18,27,40, 9)( 2, 4,11,28,38,35,13,19,22,16,25,41, 7)( 3, 6,12,29,37,34,15,21,24,17,26,42, 8)$

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);