# Properties

 Label 26T8 Order $$156$$ n $$26$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $F_{13}$

## Group action invariants

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

## 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 2: $C_2$

Degree 13: $F_{13}$

## Low degree siblings

13T6, 39T11

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 3, 3, 1, 1$ $13$ $3$ $( 3, 7,19)( 4, 8,20)( 5,13,11)( 6,14,12)( 9,25,21)(10,26,22)(15,17,23) (16,18,24)$ $6, 6, 6, 6, 1, 1$ $13$ $6$ $( 3, 9, 7,25,19,21)( 4,10, 8,26,20,22)( 5,17,13,23,11,15)( 6,18,14,24,12,16)$ $3, 3, 3, 3, 3, 3, 3, 3, 1, 1$ $13$ $3$ $( 3,19, 7)( 4,20, 8)( 5,11,13)( 6,12,14)( 9,21,25)(10,22,26)(15,23,17) (16,24,18)$ $6, 6, 6, 6, 1, 1$ $13$ $6$ $( 3,21,19,25, 7, 9)( 4,22,20,26, 8,10)( 5,15,11,23,13,17)( 6,16,12,24,14,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $13$ $2$ $( 3,25)( 4,26)( 5,23)( 6,24)( 7,21)( 8,22)( 9,19)(10,20)(11,17)(12,18)(13,15) (14,16)$ $12, 12, 2$ $13$ $12$ $( 1, 2)( 3, 5, 9,17, 7,13,25,23,19,11,21,15)( 4, 6,10,18, 8,14,26,24,20,12,22, 16)$ $4, 4, 4, 4, 4, 4, 2$ $13$ $4$ $( 1, 2)( 3,11,25,17)( 4,12,26,18)( 5,21,23, 7)( 6,22,24, 8)( 9,15,19,13) (10,16,20,14)$ $12, 12, 2$ $13$ $12$ $( 1, 2)( 3,13,21,17,19, 5,25,15, 7,11, 9,23)( 4,14,22,18,20, 6,26,16, 8,12,10, 24)$ $12, 12, 2$ $13$ $12$ $( 1, 2)( 3,15,21,11,19,23,25,13, 7,17, 9, 5)( 4,16,22,12,20,24,26,14, 8,18,10, 6)$ $4, 4, 4, 4, 4, 4, 2$ $13$ $4$ $( 1, 2)( 3,17,25,11)( 4,18,26,12)( 5, 7,23,21)( 6, 8,24,22)( 9,13,19,15) (10,14,20,16)$ $12, 12, 2$ $13$ $12$ $( 1, 2)( 3,23, 9,11, 7,15,25, 5,19,17,21,13)( 4,24,10,12, 8,16,26, 6,20,18,22, 14)$ $13, 13$ $12$ $13$ $( 1, 3, 6, 7, 9,12,14,16,18,19,21,24,25)( 2, 4, 5, 8,10,11,13,15,17,20,22,23, 26)$

## 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 3a 6a 3b 6b 2a 12a 4a 12b 12c 4b 12d 13a 2P 1a 3b 3a 3a 3b 1a 6a 2a 6b 6b 2a 6a 13a 3P 1a 1a 2a 1a 2a 2a 4b 4b 4b 4a 4a 4a 13a 5P 1a 3b 6b 3a 6a 2a 12b 4a 12a 12d 4b 12c 13a 7P 1a 3a 6a 3b 6b 2a 12d 4b 12c 12b 4a 12a 13a 11P 1a 3b 6b 3a 6a 2a 12c 4b 12d 12a 4a 12b 13a 13P 1a 3a 6a 3b 6b 2a 12a 4a 12b 12c 4b 12d 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 -1 -1 B B B -B -B -B 1 X.4 1 1 -1 1 -1 -1 -B -B -B B B B 1 X.5 1 A -/A /A -A -1 C B -/C /C -B -C 1 X.6 1 A -/A /A -A -1 -C -B /C -/C B C 1 X.7 1 /A -A A -/A -1 -/C B C -C -B /C 1 X.8 1 /A -A A -/A -1 /C -B -C C B -/C 1 X.9 1 A /A /A A 1 -A -1 -/A -/A -1 -A 1 X.10 1 /A A A /A 1 -/A -1 -A -A -1 -/A 1 X.11 1 A /A /A A 1 A 1 /A /A 1 A 1 X.12 1 /A A A /A 1 /A 1 A A 1 /A 1 X.13 12 . . . . . . . . . . . -1 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = -E(4) = -Sqrt(-1) = -i C = -E(12)^11