Properties

Label 26T5
Order \(78\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_{13}:C_3$

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Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $5$
Group :  $C_2\times C_{13}:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,4)(2,9,3)(5,19,16)(6,20,15)(7,12,22)(8,11,21)(17,24,26)(18,23,25), (1,2)(3,7,19,4,8,20)(5,13,11,6,14,12)(9,25,21,10,26,22)(15,17,23,16,18,24)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
39:  $C_{13}:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $C_{13}:C_3$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $13$ $3$ $( 3, 8,19)( 4, 7,20)( 5,14,11)( 6,13,12)( 9,26,21)(10,25,22)(15,18,23) (16,17,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $13$ $3$ $( 3,19, 8)( 4,20, 7)( 5,11,14)( 6,12,13)( 9,21,26)(10,22,25)(15,23,18) (16,24,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$
$ 6, 6, 6, 6, 2 $ $13$ $6$ $( 1, 2)( 3, 7,19, 4, 8,20)( 5,13,11, 6,14,12)( 9,25,21,10,26,22) (15,17,23,16,18,24)$
$ 6, 6, 6, 6, 2 $ $13$ $6$ $( 1, 2)( 3,20, 8, 4,19, 7)( 5,12,14, 6,11,13)( 9,22,26,10,21,25) (15,24,18,16,23,17)$
$ 26 $ $3$ $26$ $( 1, 3, 6, 8,10,11,13,16,18,19,22,24,25, 2, 4, 5, 7, 9,12,14,15,17,20,21,23,26 )$
$ 13, 13 $ $3$ $13$ $( 1, 4, 6, 7,10,12,13,15,18,20,22,23,25)( 2, 3, 5, 8, 9,11,14,16,17,19,21,24, 26)$
$ 26 $ $3$ $26$ $( 1, 5,10,14,18,21,25, 3, 7,11,15,19,23, 2, 6, 9,13,17,22,26, 4, 8,12,16,20,24 )$
$ 13, 13 $ $3$ $13$ $( 1, 6,10,13,18,22,25, 4, 7,12,15,20,23)( 2, 5, 9,14,17,21,26, 3, 8,11,16,19, 24)$
$ 26 $ $3$ $26$ $( 1, 9,18,26, 7,16,23, 5,13,21, 4,11,20, 2,10,17,25, 8,15,24, 6,14,22, 3,12,19 )$
$ 13, 13 $ $3$ $13$ $( 1,10,18,25, 7,15,23, 6,13,22, 4,12,20)( 2, 9,17,26, 8,16,24, 5,14,21, 3,11, 19)$
$ 13, 13 $ $3$ $13$ $( 1,15, 4,18, 6,20, 7,22,10,23,12,25,13)( 2,16, 3,17, 5,19, 8,21, 9,24,11,26, 14)$
$ 26 $ $3$ $26$ $( 1,16, 4,17, 6,19, 7,21,10,24,12,26,13, 2,15, 3,18, 5,20, 8,22, 9,23,11,25,14 )$

Group invariants

Order:  $78=2 \cdot 3 \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [78, 2]
Character table:   
      2  1  1  1  1   1   1   1   1   1   1   1   1   1   1
      3  1  1  1  1   1   1   .   .   .   .   .   .   .   .
     13  1  .  .  1   .   .   1   1   1   1   1   1   1   1

        1a 3a 3b 2a  6a  6b 26a 13a 26b 13b 26c 13c 13d 26d
     2P 1a 3b 3a 1a  3b  3a 13b 13b 13c 13c 13d 13d 13a 13a
     3P 1a 1a 1a 2a  2a  2a 26a 13a 26b 13b 26c 13c 13d 26d
     5P 1a 3b 3a 2a  6b  6a 26b 13b 26c 13c 26d 13d 13a 26a
     7P 1a 3a 3b 2a  6a  6b 26d 13d 26a 13a 26b 13b 13c 26c
    11P 1a 3b 3a 2a  6b  6a 26d 13d 26a 13a 26b 13b 13c 26c
    13P 1a 3a 3b 2a  6a  6b  2a  1a  2a  1a  2a  1a  1a  2a
    17P 1a 3b 3a 2a  6b  6a 26c 13c 26d 13d 26a 13a 13b 26b
    19P 1a 3a 3b 2a  6a  6b 26b 13b 26c 13c 26d 13d 13a 26a
    23P 1a 3b 3a 2a  6b  6a 26c 13c 26d 13d 26a 13a 13b 26b

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  A /A -1  -A -/A  -1   1  -1   1  -1   1   1  -1
X.4      1 /A  A -1 -/A  -A  -1   1  -1   1  -1   1   1  -1
X.5      1  A /A  1   A  /A   1   1   1   1   1   1   1   1
X.6      1 /A  A  1  /A   A   1   1   1   1   1   1   1   1
X.7      3  .  .  3   .   .   B   B  /C  /C  /B  /B   C   C
X.8      3  .  .  3   .   .   C   C   B   B  /C  /C  /B  /B
X.9      3  .  .  3   .   .  /C  /C  /B  /B   C   C   B   B
X.10     3  .  .  3   .   .  /B  /B   C   C   B   B  /C  /C
X.11     3  .  . -3   .   .  -B   B -/C  /C -/B  /B   C  -C
X.12     3  .  . -3   .   .  -C   C  -B   B -/C  /C  /B -/B
X.13     3  .  . -3   .   . -/C  /C -/B  /B  -C   C   B  -B
X.14     3  .  . -3   .   . -/B  /B  -C   C  -B   B  /C -/C

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(13)^7+E(13)^8+E(13)^11
C = E(13)^4+E(13)^10+E(13)^12