Properties

Label 26T3
Order \(52\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{26}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
26:  $D_{13}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $D_{13}$

Low degree siblings

26T3

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

Group invariants

Order:  $52=2^{2} \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [52, 4]
Character table:   
      2  2  2  2  2   1   1   1   1   1   1   1   1   1   1   1   1
     13  1  .  1  .   1   1   1   1   1   1   1   1   1   1   1   1

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

X.1      1  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  -1   1
X.3      1 -1  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  -1   1
X.5      2  .  2  .   A   A   C   C   B   B   D   D   F   F   E   E
X.6      2  .  2  .   B   B   E   E   D   D   A   A   C   C   F   F
X.7      2  .  2  .   C   C   D   D   E   E   F   F   B   B   A   A
X.8      2  .  2  .   D   D   F   F   A   A   B   B   E   E   C   C
X.9      2  .  2  .   E   E   A   A   F   F   C   C   D   D   B   B
X.10     2  .  2  .   F   F   B   B   C   C   E   E   A   A   D   D
X.11     2  . -2  .   A  -A   C  -C   B  -B   D  -D   F  -F  -E   E
X.12     2  . -2  .   B  -B   E  -E   D  -D   A  -A   C  -C  -F   F
X.13     2  . -2  .   C  -C   D  -D   E  -E   F  -F   B  -B  -A   A
X.14     2  . -2  .   D  -D   F  -F   A  -A   B  -B   E  -E  -C   C
X.15     2  . -2  .   E  -E   A  -A   F  -F   C  -C   D  -D  -B   B
X.16     2  . -2  .   F  -F   B  -B   C  -C   E  -E   A  -A  -D   D

A = E(13)^3+E(13)^10
B = E(13)^4+E(13)^9
C = E(13)^6+E(13)^7
D = E(13)+E(13)^12
E = E(13)^5+E(13)^8
F = E(13)^2+E(13)^11