Properties

Label 26T42
Order \(7800\)
n \(26\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,25)$

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Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $42$
Group :  $\PSL(2,25)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,26,4,17,2,6,22,18,5,21,12,16)(3,8,7,14,23,13,24,10,25,9,11,15,20), (1,21,5,9,4,2,10,16,12,18,14,25,3)(6,23,7,24,17,11,8,20,26,15,22,19,13)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $650$ $3$ $( 1, 6,11)( 2,13,17)( 3,24, 5)( 4,15,25)( 7,19, 8)( 9,23,10)(12,18,14) (16,26,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $325$ $2$ $( 1,16)( 2,25)( 3,23)( 4,13)( 5, 9)( 6,26)( 7,18)( 8,12)(10,24)(11,21)(14,19) (15,17)$
$ 6, 6, 6, 6, 1, 1 $ $650$ $6$ $( 1,21, 6,16,11,26)( 2,15,13,25,17, 4)( 3, 9,24,23, 5,10)( 7,12,19,18, 8,14)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $650$ $4$ $( 1,25,16, 2)( 3,19,23,14)( 4,26,13, 6)( 5, 7, 9,18)( 8,10,12,24)(11,15,21,17)$
$ 12, 12, 1, 1 $ $650$ $12$ $( 1,13,21,25, 6,17,16, 4,11, 2,26,15)( 3,12, 9,19,24,18,23, 8, 5,14,10, 7)$
$ 12, 12, 1, 1 $ $650$ $12$ $( 1, 4,21, 2, 6,15,16,13,11,25,26,17)( 3, 8, 9,14,24, 7,23,12, 5,19,10,18)$
$ 13, 13 $ $600$ $13$ $( 1, 2,19, 4,25,26, 3,24, 9,21,22,15,13)( 5, 8,14,20,23,16, 6,11,10,18,12,17, 7)$
$ 13, 13 $ $600$ $13$ $( 1, 9, 4,15, 3, 2,21,25,13,24,19,22,26)( 5,10,20,17, 6, 8,18,23, 7,11,14,12, 16)$
$ 13, 13 $ $600$ $13$ $( 1,21,26, 2,22, 3,19,15,24, 4,13, 9,25)( 5,18,16, 8,12, 6,14,17,11,20, 7,10, 23)$
$ 13, 13 $ $600$ $13$ $( 1,24, 2, 9,19,21, 4,22,25,15,26,13, 3)( 5,11, 8,10,14,18,20,12,23,17,16, 7, 6)$
$ 13, 13 $ $600$ $13$ $( 1, 4, 3,21,13,19,26, 9,15, 2,25,24,22)( 5,20, 6,18, 7,14,16,10,17, 8,23,11, 12)$
$ 13, 13 $ $600$ $13$ $( 1,15,21,24,26, 4, 2,13,22, 9, 3,25,19)( 5,17,18,11,16,20, 8, 7,12,10, 6,23, 14)$
$ 5, 5, 5, 5, 5, 1 $ $312$ $5$ $( 1,26,20, 9,12)( 2,17, 3,16,21)( 4,19,25, 5,18)( 6, 8,24,11,14) ( 7,10,22,13,15)$
$ 5, 5, 5, 5, 5, 1 $ $312$ $5$ $( 1,17,22, 6, 5)( 2,10,14,25,12)( 3,13, 8,18,26)( 4,20,16,15,24) ( 7,11,19, 9,21)$

Group invariants

Order:  $7800=2^{3} \cdot 3 \cdot 5^{2} \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2  3  3  2  2  2   2   2   .   .   .   .   .   .  .  .
      3  1  1  1  1  1   1   1   .   .   .   .   .   .  .  .
      5  2  .  .  .  .   .   .   .   .   .   .   .   .  2  2
     13  1  .  .  .  .   .   .   1   1   1   1   1   1  .  .

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

X.1      1  1  1  1  1   1   1   1   1   1   1   1   1  1  1
X.2     13  1  1 -1  1  -1  -1   .   .   .   .   .   .  3 -2
X.3     13  1  1 -1  1  -1  -1   .   .   .   .   .   . -2  3
X.4     24  .  .  .  .   .   .   B   G   F   C   D   E -1 -1
X.5     24  .  .  .  .   .   .   C   F   E   D   G   B -1 -1
X.6     24  .  .  .  .   .   .   D   E   B   G   F   C -1 -1
X.7     24  .  .  .  .   .   .   E   D   G   B   C   F -1 -1
X.8     24  .  .  .  .   .   .   F   C   D   E   B   G -1 -1
X.9     24  .  .  .  .   .   .   G   B   C   F   E   D -1 -1
X.10    25  1  1  1  1   1   1  -1  -1  -1  -1  -1  -1  .  .
X.11    26  2 -1 -2 -1   1   1   .   .   .   .   .   .  1  1
X.12    26  2 -1  2 -1  -1  -1   .   .   .   .   .   .  1  1
X.13    26 -2  2  . -2   .   .   .   .   .   .   .   .  1  1
X.14    26 -2 -1  .  1   A  -A   .   .   .   .   .   .  1  1
X.15    26 -2 -1  .  1  -A   A   .   .   .   .   .   .  1  1

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