# Properties

 Label 26T7 Degree $26$ Order $104$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times D_{13}.C_2$

## Group action invariants

 Degree $n$: $26$ Transitive number $t$: $7$ Group: $C_2\times D_{13}.C_2$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,18,19,3)(2,17,20,4)(5,11,16,9)(6,12,15,10)(7,22,13,25)(8,21,14,26)(23,24), (1,2)(3,25)(4,26)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,15)(14,16)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$52$:  $C_{13}:C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 13: $C_{13}:C_4$

## Low degree siblings

26T7

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

## Group invariants

 Order: $104=2^{3} \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [104, 12]
 Character table:  2 3 3 3 3 3 3 3 3 1 1 1 1 1 1 13 1 . . . 1 . . . 1 1 1 1 1 1 1a 4a 4b 2a 2b 4c 4d 2c 26a 13a 26b 13b 13c 26c 2P 1a 2a 2a 1a 1a 2a 2a 1a 13b 13b 13c 13c 13a 13a 3P 1a 4b 4a 2a 2b 4d 4c 2c 26b 13b 26c 13c 13a 26a 5P 1a 4a 4b 2a 2b 4c 4d 2c 26a 13a 26b 13b 13c 26c 7P 1a 4b 4a 2a 2b 4d 4c 2c 26c 13c 26a 13a 13b 26b 11P 1a 4b 4a 2a 2b 4d 4c 2c 26b 13b 26c 13c 13a 26a 13P 1a 4a 4b 2a 2b 4c 4d 2c 2b 1a 2b 1a 1a 2b 17P 1a 4a 4b 2a 2b 4c 4d 2c 26c 13c 26a 13a 13b 26b 19P 1a 4b 4a 2a 2b 4d 4c 2c 26c 13c 26a 13a 13b 26b 23P 1a 4b 4a 2a 2b 4d 4c 2c 26b 13b 26c 13c 13a 26a X.1 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 -1 X.3 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 X.4 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 -1 X.5 1 A -A -1 -1 -A A 1 -1 1 -1 1 1 -1 X.6 1 -A A -1 -1 A -A 1 -1 1 -1 1 1 -1 X.7 1 A -A -1 1 A -A -1 1 1 1 1 1 1 X.8 1 -A A -1 1 -A A -1 1 1 1 1 1 1 X.9 4 . . . 4 . . . B B C C D D X.10 4 . . . 4 . . . C C D D B B X.11 4 . . . 4 . . . D D B B C C X.12 4 . . . -4 . . . -B B -C C D -D X.13 4 . . . -4 . . . -C C -D D B -B X.14 4 . . . -4 . . . -D D -B B C -C A = -E(4) = -Sqrt(-1) = -i B = E(13)^2+E(13)^3+E(13)^10+E(13)^11 C = E(13)^4+E(13)^6+E(13)^7+E(13)^9 D = E(13)+E(13)^5+E(13)^8+E(13)^12