# Properties

 Label 28T120 Degree $28$ Order $1092$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $\PSL(2,13)$

# Learn more

Show commands: Magma

magma: G := TransitiveGroup(28, 120);

## Group action invariants

 Degree $n$: $28$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $120$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $\PSL(2,13)$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ 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

 Label Cycle Type Size Order Representative 1A $1^{28}$ $1$ $1$ $()$ 2A $2^{14}$ $91$ $2$ $( 1, 2)( 3,14)( 4,13)( 5, 9)( 6,10)( 7,28)( 8,27)(11,22)(12,21)(15,26)(16,25)(17,23)(18,24)(19,20)$ 3A $3^{8},1^{4}$ $182$ $3$ $( 1, 9,13)( 2,10,14)( 3,22,15)( 4,21,16)( 7,11,28)( 8,12,27)(17,20,26)(18,19,25)$ 6A $6^{4},2^{2}$ $182$ $6$ $( 1, 2)( 3,22,15,14,11,26)( 4,21,16,13,12,25)( 5,24, 7, 9,18,28)( 6,23, 8,10,17,27)(19,20)$ 7A1 $7^{4}$ $156$ $7$ $( 1,26, 8,27,14,19,12)( 2,25, 7,28,13,20,11)( 3,21,15, 6,17,10,23)( 4,22,16, 5,18, 9,24)$ 7A2 $7^{4}$ $156$ $7$ $( 1, 8,14,12,26,27,19)( 2, 7,13,11,25,28,20)( 3,15,17,23,21, 6,10)( 4,16,18,24,22, 5, 9)$ 7A3 $7^{4}$ $156$ $7$ $( 1,27,12, 8,19,26,14)( 2,28,11, 7,20,25,13)( 3, 6,23,15,10,21,17)( 4, 5,24,16, 9,22,18)$ 13A1 $13^{2},1^{2}$ $84$ $13$ $( 1,19,10, 8,25, 5, 4,22,16,24,13,27,17)( 2,20, 9, 7,26, 6, 3,21,15,23,14,28,18)$ 13A2 $13^{2},1^{2}$ $84$ $13$ $( 1,10,25, 4,16,13,17,19, 8, 5,22,24,27)( 2, 9,26, 3,15,14,18,20, 7, 6,21,23,28)$

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);