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Properties

Label 26T24
Order \(2028\)
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 $t$ :  $24$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,5)(3,8,10)(4,11,6)(9,13,12)(14,23,26)(15,19,16)(17,24,22)(18,20,25), (1,15,11,20)(2,22,10,26)(3,16,9,19)(4,23,8,25)(5,17,7,18)(6,24)(12,14,13,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

39T35 x 2

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 $ $1$ $1$ $()$
$ 13, 13 $ $12$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $12$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $12$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $12$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $12$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $12$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,16,18,20,22,24,26,15,17,19,21,23,25)$
$ 13, 13 $ $12$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$
$ 13, 13 $ $12$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $12$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $12$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,18,22,26,17,21,25,16,20,24,15,19,23)$
$ 13, 13 $ $12$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $12$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $338$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 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 $ $507$ $4$ $( 1,15,11,20)( 2,22,10,26)( 3,16, 9,19)( 4,23, 8,25)( 5,17, 7,18)( 6,24) (12,14,13,21)$
$ 4, 4, 4, 4, 4, 4, 2 $ $507$ $4$ $( 1,26,13,20)( 2,19,12,14)( 3,25,11,21)( 4,18,10,15)( 5,24, 9,22)( 6,17, 8,16) ( 7,23)$

Group invariants

Order:  $2028=2^{2} \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table:    
      2  2   .   .   .   .   .   .   .   .   .   .   .   .   .   .  1  2  1  2
      3  1   .   .   .   .   .   .   .   .   .   .   .   .   .   .  1  1  1  .
     13  2   2   2   2   2   2   2   2   2   2   2   2   2   2   2  .  .  .  .

        1a 13a 13b 13c 13d 13e 13f 13g 13h 13i 13j 13k 13l 13m 13n 3a 2a 6a 4a
     2P 1a 13b 13c 13d 13e 13f 13a 13l 13m 13j 13n 13h 13g 13k 13i 3a 1a 3a 2a
     3P 1a 13e 13f 13a 13b 13c 13d 13g 13m 13j 13n 13h 13l 13k 13i 1a 2a 2a 4b
     5P 1a 13d 13e 13f 13a 13b 13c 13l 13h 13i 13j 13k 13g 13m 13n 3a 2a 6a 4a
     7P 1a 13f 13a 13b 13c 13d 13e 13l 13k 13n 13i 13m 13g 13h 13j 3a 2a 6a 4b
    11P 1a 13b 13c 13d 13e 13f 13a 13l 13m 13j 13n 13h 13g 13k 13i 3a 2a 6a 4b
    13P 1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a 3a 2a 6a 4a

X.1      1   1   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  1  1 -1
X.3      1   1   1   1   1   1   1   1   1   1   1   1   1   1   1  1 -1 -1  R
X.4      1   1   1   1   1   1   1   1   1   1   1   1   1   1   1  1 -1 -1 -R
X.5      2   2   2   2   2   2   2   2   2   2   2   2   2   2   2 -1 -2  1  .
X.6      2   2   2   2   2   2   2   2   2   2   2   2   2   2   2 -1  2 -1  .
X.7     12   A  *A   A  *A   A  *A   K  -1  -1  -1  -1  *K  -1  -1  .  .  .  .
X.8     12  *A   A  *A   A  *A   A  *K  -1  -1  -1  -1   K  -1  -1  .  .  .  .
X.9     12   B   F   C   E   D   G  *A   I   J   I   J   A   H   H  .  .  .  .
X.10    12   C   E   D   G   B   F  *A   J   H   J   H   A   I   I  .  .  .  .
X.11    12   D   G   B   F   C   E  *A   H   I   H   I   A   J   J  .  .  .  .
X.12    12   E   D   G   B   F   C   A   I   J   I   J  *A   H   H  .  .  .  .
X.13    12   F   C   E   D   G   B   A   H   I   H   I  *A   J   J  .  .  .  .
X.14    12   G   B   F   C   E   D   A   J   H   J   H  *A   I   I  .  .  .  .
X.15    12   H   J   I   H   J   I  -1   L   P   O   M  -1   N   Q  .  .  .  .
X.16    12   I   H   J   I   H   J  -1   M   Q   P   N  -1   L   O  .  .  .  .
X.17    12   J   I   H   J   I   H  -1   N   O   Q   L  -1   M   P  .  .  .  .
X.18    12   H   J   I   H   J   I  -1   O   M   L   P  -1   Q   N  .  .  .  .
X.19    12   I   H   J   I   H   J  -1   P   N   M   Q  -1   O   L  .  .  .  .
X.20    12   J   I   H   J   I   H  -1   Q   L   N   O  -1   P   M  .  .  .  .

      2  2
      3  .
     13  .

        4b
     2P 2a
     3P 4a
     5P 4b
     7P 4a
    11P 4a
    13P 4b

X.1      1
X.2     -1
X.3     -R
X.4      R
X.5      .
X.6      .
X.7      .
X.8      .
X.9      .
X.10     .
X.11     .
X.12     .
X.13     .
X.14     .
X.15     .
X.16     .
X.17     .
X.18     .
X.19     .
X.20     .

A = 2*E(13)+2*E(13)^3+2*E(13)^4+2*E(13)^9+2*E(13)^10+2*E(13)^12
  = -1+Sqrt(13) = 2b13
B = -2*E(13)-2*E(13)^2-2*E(13)^4-E(13)^6-E(13)^7-2*E(13)^9-2*E(13)^11-2*E(13)^12
C = -E(13)^2-2*E(13)^3-2*E(13)^4-2*E(13)^5-2*E(13)^8-2*E(13)^9-2*E(13)^10-E(13)^11
D = -2*E(13)-2*E(13)^3-E(13)^5-2*E(13)^6-2*E(13)^7-E(13)^8-2*E(13)^10-2*E(13)^12
E = -2*E(13)^3-E(13)^4-2*E(13)^5-2*E(13)^6-2*E(13)^7-2*E(13)^8-E(13)^9-2*E(13)^10
F = -E(13)-2*E(13)^2-2*E(13)^4-2*E(13)^5-2*E(13)^8-2*E(13)^9-2*E(13)^11-E(13)^12
G = -2*E(13)-2*E(13)^2-E(13)^3-2*E(13)^6-2*E(13)^7-E(13)^10-2*E(13)^11-2*E(13)^12
H = E(13)^2+E(13)^3+2*E(13)^4+2*E(13)^6+2*E(13)^7+2*E(13)^9+E(13)^10+E(13)^11
I = E(13)+2*E(13)^2+2*E(13)^3+E(13)^5+E(13)^8+2*E(13)^10+2*E(13)^11+E(13)^12
J = 2*E(13)+E(13)^4+2*E(13)^5+E(13)^6+E(13)^7+2*E(13)^8+E(13)^9+2*E(13)^12
K = -6*E(13)-5*E(13)^2-6*E(13)^3-6*E(13)^4-5*E(13)^5-5*E(13)^6-5*E(13)^7-5*E(13)^8-6*E(13)^9-6*E(13)^10-5*E(13)^11-6*E(13)^12
  = (11-Sqrt(13))/2 = 5-b13
L = -2*E(13)-4*E(13)^2-4*E(13)^3-4*E(13)^4-2*E(13)^5-4*E(13)^6-4*E(13)^7-2*E(13)^8-4*E(13)^9-4*E(13)^10-4*E(13)^11-2*E(13)^12
M = -4*E(13)-4*E(13)^2-4*E(13)^3-2*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)^10-4*E(13)^11-4*E(13)^12
N = -4*E(13)-2*E(13)^2-2*E(13)^3-4*E(13)^4-4*E(13)^5-4*E(13)^6-4*E(13)^7-4*E(13)^8-4*E(13)^9-2*E(13)^10-2*E(13)^11-4*E(13)^12
O = 2*E(13)^2+2*E(13)^3+E(13)^4+E(13)^6+E(13)^7+E(13)^9+2*E(13)^10+2*E(13)^11
P = 2*E(13)+E(13)^2+E(13)^3+2*E(13)^5+2*E(13)^8+E(13)^10+E(13)^11+2*E(13)^12
Q = E(13)+2*E(13)^4+E(13)^5+2*E(13)^6+2*E(13)^7+E(13)^8+2*E(13)^9+E(13)^12
R = -E(4)
  = -Sqrt(-1) = -i