# Properties

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

## Group action invariants

 Degree $n$ : $42$ Transitive number $t$ : $176$ Group : $\PSL(2,13)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) 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) $|\Aut(F/K)|$: $3$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: None

Degree 3: None

Degree 6: None

Degree 7: None

Degree 14: $\PSL(2,13)$

Degree 21: None

## Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

## Conjugacy Classes

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

## 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 1 2 1 . . . . . 3 1 1 1 1 . . . . . 7 1 . . . . . 1 1 1 13 1 . . . 1 1 . . . 1a 3a 2a 6a 13a 13b 7a 7b 7c 2P 1a 3a 1a 3a 13b 13a 7c 7a 7b 3P 1a 1a 2a 2a 13a 13b 7b 7c 7a 5P 1a 3a 2a 6a 13b 13a 7c 7a 7b 7P 1a 3a 2a 6a 13b 13a 1a 1a 1a 11P 1a 3a 2a 6a 13b 13a 7b 7c 7a 13P 1a 3a 2a 6a 1a 1a 7a 7b 7c X.1 1 1 1 1 1 1 1 1 1 X.2 7 1 -1 -1 A *A . . . X.3 7 1 -1 -1 *A A . . . X.4 12 . . . -1 -1 B C D X.5 12 . . . -1 -1 C D B X.6 12 . . . -1 -1 D B C X.7 13 1 1 1 . . -1 -1 -1 X.8 14 -1 2 -1 1 1 . . . X.9 14 -1 -2 1 1 1 . . . A = -E(13)-E(13)^3-E(13)^4-E(13)^9-E(13)^10-E(13)^12 = (1-Sqrt(13))/2 = -b13 B = -E(7)^3-E(7)^4 C = -E(7)^2-E(7)^5 D = -E(7)-E(7)^6