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Properties

Label 26T7
Order \(104\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times D_{13}.C_2$

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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)
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)
$|\Aut(F/K)|$:  $2$

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 TypeSizeOrderRepresentative
$ 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