# Properties

 Label 14T30 Degree $14$ Order $1092$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSL(2,13)$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(14, 30);

## Group action invariants

 Degree $n$: $14$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $30$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $\PSL(2,13)$ CHM label: $L(14)=PSL(2,13)$ Parity: $1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,12)(2,6)(3,4)(7,11)(9,10)(13,14), (1,4,3,12,9,10)(2,8,6,11,5,7), (1,2,3,4,5,6,7,8,9,10,11,12,14) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 7: None

## Low degree siblings

28T120, 42T176

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 $1^{14}$ $1$ $1$ $()$ 2A $2^{6},1^{2}$ $91$ $2$ $( 1, 3)( 2, 8)( 5,11)( 6, 7)(10,12)(13,14)$ 3A $3^{4},1^{2}$ $182$ $3$ $( 1, 4, 2)( 5,14, 8)( 6, 7,13)( 9,12,11)$ 6A $6^{2},1^{2}$ $182$ $6$ $( 1, 6,10, 3, 7,12)( 2, 5,14, 8,11,13)$ 7A1 $7^{2}$ $156$ $7$ $( 1,12, 3, 5,14, 8,10)( 2, 7, 6,11, 4,13, 9)$ 7A2 $7^{2}$ $156$ $7$ $( 1, 3,14,10,12, 5, 8)( 2, 6, 4, 9, 7,11,13)$ 7A3 $7^{2}$ $156$ $7$ $( 1, 5,10, 3, 8,12,14)( 2,11, 9, 6,13, 7, 4)$ 13A1 $13,1$ $84$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,14, 8, 3,11, 6)$ 13A2 $13,1$ $84$ $13$ $( 1, 4, 7,10,14, 3, 6, 9,12, 2, 5, 8,11)$

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);