# Properties

 Label 39T11 Order $$156$$ n $$39$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $F_{13}$

## Group action invariants

 Degree $n$ : $39$ Transitive number $t$ : $11$ Group : $F_{13}$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,21,34)(2,19,35)(3,20,36)(4,30,22)(5,29,24)(6,28,23)(7,39,11)(8,37,10)(9,38,12)(13,17,26)(14,18,27)(15,16,25)(31,32,33), (1,32,13,17,24,34,20,28,8,6,38,27)(2,33,14,18,23,35,21,30,9,4,39,25)(3,31,15,16,22,36,19,29,7,5,37,26)(10,12,11) $|\Aut(F/K)|$: $3$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $C_{12}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 13: $F_{13}$

## Low degree siblings

13T6, 26T8

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1$ $13$ $4$ $( 4,18,38,26)( 5,16,39,27)( 6,17,37,25)( 7,32,36,10)( 8,33,34,12)( 9,31,35,11) (13,23,30,19)(14,22,29,20)(15,24,28,21)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1$ $13$ $4$ $( 4,26,38,18)( 5,27,39,16)( 6,25,37,17)( 7,10,36,32)( 8,12,34,33)( 9,11,35,31) (13,19,30,23)(14,20,29,22)(15,21,28,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $13$ $2$ $( 4,38)( 5,39)( 6,37)( 7,36)( 8,34)( 9,35)(10,32)(11,31)(12,33)(13,30)(14,29) (15,28)(16,27)(17,25)(18,26)(19,23)(20,22)(21,24)$ $12, 12, 12, 3$ $13$ $12$ $( 1, 2, 3)( 4, 8,14,26,12,20,38,34,29,18,33,22)( 5, 9,15,27,11,21,39,35,28,16, 31,24)( 6, 7,13,25,10,19,37,36,30,17,32,23)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 2, 3)( 4,12,29)( 5,11,28)( 6,10,30)( 7,19,17)( 8,20,18)( 9,21,16) (13,37,32)(14,38,33)(15,39,31)(22,26,34)(23,25,36)(24,27,35)$ $6, 6, 6, 6, 6, 6, 3$ $13$ $6$ $( 1, 2, 3)( 4,33,29,38,12,14)( 5,31,28,39,11,15)( 6,32,30,37,10,13) ( 7,23,17,36,19,25)( 8,22,18,34,20,26)( 9,24,16,35,21,27)$ $12, 12, 12, 3$ $13$ $12$ $( 1, 2, 3)( 4,34,14,18,12,22,38, 8,29,26,33,20)( 5,35,15,16,11,24,39, 9,28,27, 31,21)( 6,36,13,17,10,23,37, 7,30,25,32,19)$ $6, 6, 6, 6, 6, 6, 3$ $13$ $6$ $( 1, 3, 2)( 4,14,12,38,29,33)( 5,15,11,39,28,31)( 6,13,10,37,30,32) ( 7,25,19,36,17,23)( 8,26,20,34,18,22)( 9,27,21,35,16,24)$ $12, 12, 12, 3$ $13$ $12$ $( 1, 3, 2)( 4,20,33,26,29, 8,38,22,12,18,14,34)( 5,21,31,27,28, 9,39,24,11,16, 15,35)( 6,19,32,25,30, 7,37,23,10,17,13,36)$ $12, 12, 12, 3$ $13$ $12$ $( 1, 3, 2)( 4,22,33,18,29,34,38,20,12,26,14, 8)( 5,24,31,16,28,35,39,21,11,27, 15, 9)( 6,23,32,17,30,36,37,19,10,25,13, 7)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 3, 2)( 4,29,12)( 5,28,11)( 6,30,10)( 7,17,19)( 8,18,20)( 9,16,21) (13,32,37)(14,33,38)(15,31,39)(22,34,26)(23,36,25)(24,35,27)$ $13, 13, 13$ $12$ $13$ $( 1, 6, 9,11,14,17,20,22,25,29,31,35,37)( 2, 4, 7,10,15,18,21,24,26,28,32,36, 38)( 3, 5, 8,12,13,16,19,23,27,30,33,34,39)$

## Group invariants

 Order: $156=2^{2} \cdot 3 \cdot 13$ Cyclic: No Abelian: No Solvable: Yes GAP id: [156, 7]
 Character table: 2 2 2 2 2 2 2 2 2 2 2 2 2 . 3 1 1 1 1 1 1 1 1 1 1 1 1 . 13 1 . . . . . . . . . . . 1 1a 4a 4b 2a 12a 3a 6a 12b 6b 12c 12d 3b 13a 2P 1a 2a 2a 1a 6b 3b 3b 6b 3a 6a 6a 3a 13a 3P 1a 4b 4a 2a 4b 1a 2a 4a 2a 4b 4a 1a 13a 5P 1a 4a 4b 2a 12c 3b 6b 12d 6a 12a 12b 3a 13a 7P 1a 4b 4a 2a 12b 3a 6a 12a 6b 12d 12c 3b 13a 11P 1a 4b 4a 2a 12d 3b 6b 12c 6a 12b 12a 3a 13a 13P 1a 4a 4b 2a 12a 3a 6a 12b 6b 12c 12d 3b 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 X.3 1 -1 -1 1 B -B -B B -/B /B /B -/B 1 X.4 1 -1 -1 1 /B -/B -/B /B -B B B -B 1 X.5 1 1 1 1 -/B -/B -/B -/B -B -B -B -B 1 X.6 1 1 1 1 -B -B -B -B -/B -/B -/B -/B 1 X.7 1 A -A -1 A 1 -1 -A -1 A -A 1 1 X.8 1 -A A -1 -A 1 -1 A -1 -A A 1 1 X.9 1 A -A -1 C -B B -C /B -/C /C -/B 1 X.10 1 A -A -1 -/C -/B /B /C B C -C -B 1 X.11 1 -A A -1 /C -/B /B -/C B -C C -B 1 X.12 1 -A A -1 -C -B B C /B /C -/C -/B 1 X.13 12 . . . . . . . . . . . -1 A = -E(4) = -Sqrt(-1) = -i B = -E(3) = (1-Sqrt(-3))/2 = -b3 C = -E(12)^7