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Properties

Label 25T50
Degree $25$
Order $800$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no

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Group action invariants

Degree $n$:  $25$
Transitive number $t$:  $50$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (2,23,4,12,5,9,3,20)(6,17,16,24,21,15,11,8)(7,14,19,10,25,18,13,22), (2,15,6,23)(3,24,11,20)(4,8,16,12)(5,17,21,9)(10,14,22,18), (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)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

10T33, 20T155, 20T161, 20T167, 20T169, 40T874, 40T875, 40T876, 40T877, 40T878, 40T879, 40T880, 40T881, 40T882, 40T883

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5 $ $16$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1,14,22,10,18)( 2,15,23, 6,19)( 3,11,24, 7,20)( 4,12,25, 8,16) ( 5,13,21, 9,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$
$ 8, 8, 8, 1 $ $100$ $8$ $( 2,23, 4,12, 5, 9, 3,20)( 6,17,16,24,21,15,11, 8)( 7,14,19,10,25,18,13,22)$
$ 8, 8, 8, 1 $ $100$ $8$ $( 2,12, 3,23, 5,20, 4, 9)( 6,24,11,17,21, 8,16,15)( 7,10,13,14,25,22,19,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $10$ $2$ $( 2, 6)( 3,11)( 4,16)( 5,21)( 8,12)( 9,17)(10,22)(14,18)(15,23)(20,24)$
$ 10, 10, 5 $ $40$ $10$ $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$
$ 4, 4, 4, 4, 4, 4, 1 $ $50$ $4$ $( 2,16, 5,11)( 3, 6, 4,21)( 7,19,25,13)( 8, 9,24,23)(10,14,22,18)(12,17,20,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $20$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 10, 10, 5 $ $80$ $10$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$
$ 4, 4, 4, 4, 4, 4, 1 $ $100$ $4$ $( 2, 4, 5, 3)( 6,11,21,16)( 7,14,25,18)( 8,12,24,20)( 9,15,23,17)(10,13,22,19)$
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ $10$ $4$ $( 2,23, 6,15)( 3,20,11,24)( 4,12,16, 8)( 5, 9,21,17)(10,18,22,14)$
$ 20, 5 $ $40$ $20$ $( 1, 2,24, 4,13,14, 6,11,25,21,18,23, 7, 8, 5,10,19,20,12,17)( 3,16, 9,22,15)$
$ 4, 4, 4, 4, 4, 2, 2, 1 $ $50$ $4$ $( 2, 9, 6,17)( 3,12,11, 8)( 4,20,16,24)( 5,23,21,15)( 7,25)(10,14,22,18) (13,19)$
$ 4, 4, 4, 4, 4, 2, 2, 1 $ $50$ $4$ $( 2,12,21,24)( 3,23,16,17)( 4, 9,11,15)( 5,20, 6, 8)( 7,19,25,13)(10,22) (14,18)$
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ $10$ $4$ $( 2,20,21, 8)( 3, 9,16,15)( 4,23,11,17)( 5,12, 6,24)( 7,13,25,19)$
$ 20, 5 $ $40$ $20$ $( 1, 2,16,11,18,19, 8, 3,10, 6,25,20,22,23,12, 7,14,15, 4,24)( 5,13,21, 9,17)$

Group invariants

Order:  $800=2^{5} \cdot 5^{2}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [800, 1191]
Character table:    
      2  5  1  2  5  5  5  3  3  4   2  4  3   1  3   4   2   4   4   4   2
      5  2  2  2  .  .  .  .  .  1   1  .  1   1  .   1   1   .   .   1   1

        1a 5a 5b 2a 4a 4b 8a 8b 2b 10a 4c 2c 10b 4d  4e 20a  4f  4g  4h 20b
     2P 1a 5a 5b 1a 2a 2a 4a 4b 1a  5b 2a 1a  5a 2a  2b 10a  2b  2b  2b 10a
     3P 1a 5a 5b 2a 4b 4a 8b 8a 2b 10a 4c 2c 10b 4d  4h 20b  4g  4f  4e 20a
     5P 1a 1a 1a 2a 4a 4b 8a 8b 2b  2b 4c 2c  2c 4d  4e  4e  4f  4g  4h  4h
     7P 1a 5a 5b 2a 4b 4a 8b 8a 2b 10a 4c 2c 10b 4d  4h 20b  4g  4f  4e 20a
    11P 1a 5a 5b 2a 4b 4a 8b 8a 2b 10a 4c 2c 10b 4d  4h 20b  4g  4f  4e 20a
    13P 1a 5a 5b 2a 4a 4b 8a 8b 2b 10a 4c 2c 10b 4d  4e 20a  4f  4g  4h 20b
    17P 1a 5a 5b 2a 4a 4b 8a 8b 2b 10a 4c 2c 10b 4d  4e 20a  4f  4g  4h 20b
    19P 1a 5a 5b 2a 4b 4a 8b 8a 2b 10a 4c 2c 10b 4d  4h 20b  4g  4f  4e 20a

X.1      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   1
X.3      1  1  1  1  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  -1  -1  -1  -1
X.5      1  1  1  1 -1 -1  B -B -1  -1  1 -1  -1  1  -B  -B  -B   B   B   B
X.6      1  1  1  1 -1 -1 -B  B -1  -1  1 -1  -1  1   B   B   B  -B  -B  -B
X.7      1  1  1  1 -1 -1  B -B -1  -1  1  1   1 -1   B   B   B  -B  -B  -B
X.8      1  1  1  1 -1 -1 -B  B -1  -1  1  1   1 -1  -B  -B  -B   B   B   B
X.9      2  2  2  2  2  2  .  . -2  -2 -2  .   .  .   .   .   .   .   .   .
X.10     2  2  2  2 -2 -2  .  .  2   2 -2  .   .  .   .   .   .   .   .   .
X.11     2  2  2 -2  A -A  .  .  .   .  .  .   .  .   C   C  -C -/C  /C  /C
X.12     2  2  2 -2 -A  A  .  .  .   .  .  .   .  .  /C  /C -/C  -C   C   C
X.13     2  2  2 -2 -A  A  .  .  .   .  .  .   .  . -/C -/C  /C   C  -C  -C
X.14     2  2  2 -2  A -A  .  .  .   .  .  .   .  .  -C  -C   C  /C -/C -/C
X.15     8 -2  3  .  .  .  .  .  4  -1  .  .   .  .  -4   1   .   .  -4   1
X.16     8 -2  3  .  .  .  .  .  4  -1  .  .   .  .   4  -1   .   .   4  -1
X.17     8 -2  3  .  .  .  .  . -4   1  .  .   .  .   D  -B   .   .  -D   B
X.18     8 -2  3  .  .  .  .  . -4   1  .  .   .  .  -D   B   .   .   D  -B
X.19    16  1 -4  .  .  .  .  .  .   .  . -4   1  .   .   .   .   .   .   .
X.20    16  1 -4  .  .  .  .  .  .   .  .  4  -1  .   .   .   .   .   .   .

A = -2*E(4)
  = -2*Sqrt(-1) = -2i
B = -E(4)
  = -Sqrt(-1) = -i
C = 1-E(4)
  = 1-Sqrt(-1) = 1-i
D = -4*E(4)
  = -4*Sqrt(-1) = -4i