Properties

Label 26T35
Order \(4056\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $35$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,2,21)(3,14,13,22)(4,20,12,16)(5,26,11,23)(6,19,10,17)(7,25,9,24)(8,18), (1,16,2,15)(3,14,13,17)(4,26,12,18)(5,25,11,19)(6,24,10,20)(7,23,9,21)(8,22)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $Q_8$
12:  $D_{6}$
24:  24T5

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

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 $ $1$ $1$ $()$
$ 13, 13 $ $24$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $24$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $24$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $13$ $(14,26,25,24,23,22,21,20,19,18,17,16,15)$
$ 13, 13 $ $24$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $24$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $24$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $338$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 6, 6, 6, 6, 1, 1 $ $338$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 4, 4, 4, 4, 4, 4, 2 $ $1014$ $4$ $( 1,15, 2,21)( 3,14,13,22)( 4,20,12,16)( 5,26,11,23)( 6,19,10,17)( 7,25, 9,24) ( 8,18)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,20,24,22,23,16,26,21,17,19,18,25)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,21,24,19,23,25,26,20,17,22,18,16)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $338$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 4, 4, 4, 4, 4, 4, 2 $ $1014$ $4$ $( 1,15,12,26)( 2,16,11,25)( 3,17,10,24)( 4,18, 9,23)( 5,19, 8,22)( 6,20, 7,21) (13,14)$

Group invariants

Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table:   
      2  3   .   .   .   .   .   .   .  3  2  2  2   2   2  2  2
      3  1   .   .   .   .   .   .   .  1  1  1  .   1   1  1  .
     13  2   2   2   2   2   2   2   2  .  .  .  .   .   .  .  .

        1a 13a 13b 13c 13d 13e 13f 13g 2a 3a 6a 4a 12a 12b 4b 4c
     2P 1a 13b 13c 13a 13d 13g 13e 13f 1a 3a 3a 2a  6a  6a 2a 2a
     3P 1a 13b 13c 13a 13d 13g 13e 13f 2a 1a 2a 4a  4b  4b 4b 4c
     5P 1a 13a 13b 13c 13d 13e 13f 13g 2a 3a 6a 4a 12b 12a 4b 4c
     7P 1a 13c 13a 13b 13d 13f 13g 13e 2a 3a 6a 4a 12b 12a 4b 4c
    11P 1a 13b 13c 13a 13d 13g 13e 13f 2a 3a 6a 4a 12a 12b 4b 4c
    13P 1a  1a  1a  1a  1a  1a  1a  1a 2a 3a 6a 4a 12a 12b 4b 4c

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   2   2   2   2   2   2  2 -1 -1  .  -1  -1  2  .
X.6      2   2   2   2   2   2   2   2  2 -1 -1  .   1   1 -2  .
X.7      2   2   2   2   2   2   2   2 -2  2 -2  .   .   .  .  .
X.8      2   2   2   2   2   2   2   2 -2 -1  1  .   G  -G  .  .
X.9      2   2   2   2   2   2   2   2 -2 -1  1  .  -G   G  .  .
X.10    24  -2  -2  -2  11  -2  -2  -2  .  .  .  .   .   .  .  .
X.11    24   A   C   B  -2   D   E   F  .  .  .  .   .   .  .  .
X.12    24   B   A   C  -2   E   F   D  .  .  .  .   .   .  .  .
X.13    24   C   B   A  -2   F   D   E  .  .  .  .   .   .  .  .
X.14    24   D   F   E  -2   B   C   A  .  .  .  .   .   .  .  .
X.15    24   E   D   F  -2   C   A   B  .  .  .  .   .   .  .  .
X.16    24   F   E   D  -2   A   B   C  .  .  .  .   .   .  .  .

A = 2*E(13)+4*E(13)^2+4*E(13)^3+2*E(13)^5+2*E(13)^8+4*E(13)^10+4*E(13)^11+2*E(13)^12
B = 4*E(13)+2*E(13)^4+4*E(13)^5+2*E(13)^6+2*E(13)^7+4*E(13)^8+2*E(13)^9+4*E(13)^12
C = 2*E(13)^2+2*E(13)^3+4*E(13)^4+4*E(13)^6+4*E(13)^7+4*E(13)^9+2*E(13)^10+2*E(13)^11
D = -2*E(13)-3*E(13)^2-3*E(13)^3-2*E(13)^4-2*E(13)^5-2*E(13)^6-2*E(13)^7-2*E(13)^8-2*E(13)^9-3*E(13)^10-3*E(13)^11-2*E(13)^12
E = -3*E(13)-2*E(13)^2-2*E(13)^3-2*E(13)^4-3*E(13)^5-2*E(13)^6-2*E(13)^7-3*E(13)^8-2*E(13)^9-2*E(13)^10-2*E(13)^11-3*E(13)^12
F = -2*E(13)-2*E(13)^2-2*E(13)^3-3*E(13)^4-2*E(13)^5-3*E(13)^6-3*E(13)^7-2*E(13)^8-3*E(13)^9-2*E(13)^10-2*E(13)^11-2*E(13)^12
G = -E(12)^7+E(12)^11
  = Sqrt(3) = r3