# Properties

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

## Group action invariants

 Degree $n$ : $28$ Transitive number $t$ : $120$ Group : $\PSL(2,13)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) 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) $|\Aut(F/K)|$: $2$

## 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

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

## Group invariants

 Order: $1092=2^{2} \cdot 3 \cdot 7 \cdot 13$ Cyclic: No Abelian: No Solvable: No GAP id: [1092, 25]
 Character table:  2 2 2 1 1 . . . . . 3 1 1 1 1 . . . . . 7 1 . . . 1 1 1 . . 13 1 . . . . . . 1 1 1a 2a 3a 6a 7a 7b 7c 13a 13b 2P 1a 1a 3a 3a 7c 7a 7b 13b 13a 3P 1a 2a 1a 2a 7b 7c 7a 13a 13b 5P 1a 2a 3a 6a 7c 7a 7b 13b 13a 7P 1a 2a 3a 6a 1a 1a 1a 13b 13a 11P 1a 2a 3a 6a 7b 7c 7a 13b 13a 13P 1a 2a 3a 6a 7a 7b 7c 1a 1a X.1 1 1 1 1 1 1 1 1 1 X.2 7 -1 1 -1 . . . D *D X.3 7 -1 1 -1 . . . *D D X.4 12 . . . A B C -1 -1 X.5 12 . . . B C A -1 -1 X.6 12 . . . C A B -1 -1 X.7 13 1 1 1 -1 -1 -1 . . X.8 14 2 -1 -1 . . . 1 1 X.9 14 -2 -1 1 . . . 1 1 A = -E(7)^3-E(7)^4 B = -E(7)^2-E(7)^5 C = -E(7)-E(7)^6 D = -E(13)-E(13)^3-E(13)^4-E(13)^9-E(13)^10-E(13)^12 = (1-Sqrt(13))/2 = -b13