# Properties

 Label 25T18 Degree $25$ Order $200$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_5\times F_5$

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $18$ Group: $D_5\times F_5$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,2)(3,5)(6,17,21,12)(7,16,22,11)(8,20,23,15)(9,19,24,14)(10,18,25,13), (1,3)(4,5)(6,23)(7,22)(8,21)(9,25)(10,24)(11,18)(12,17)(13,16)(14,20)(15,19), (1,7,13,19,25)(2,8,14,20,21)(3,9,15,16,22)(4,10,11,17,23)(5,6,12,18,24)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$10$:  $D_{5}$
$20$:  $F_5$, $D_{10}$
$40$:  $F_{5}\times C_2$, 20T6

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $D_{5}$, $F_5$

## Low degree siblings

20T51, 40T168

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $200=2^{3} \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [200, 41]
 Character table:  2 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 . . 5 2 1 1 1 1 . . . 2 1 1 1 2 1 1 1 2 1 2 2 1a 4a 4b 2a 2b 4c 4d 2c 5a 20a 20b 10a 5b 20c 20d 10b 5c 10c 5d 5e 2P 1a 2a 2a 1a 1a 2a 2a 1a 5b 10b 10b 5b 5a 10a 10a 5a 5c 5c 5e 5d 3P 1a 4b 4a 2a 2b 4d 4c 2c 5b 20d 20c 10b 5a 20b 20a 10a 5c 10c 5e 5d 5P 1a 4a 4b 2a 2b 4c 4d 2c 1a 4a 4b 2a 1a 4a 4b 2a 1a 2b 1a 1a 7P 1a 4b 4a 2a 2b 4d 4c 2c 5b 20d 20c 10b 5a 20b 20a 10a 5c 10c 5e 5d 11P 1a 4b 4a 2a 2b 4d 4c 2c 5a 20b 20a 10a 5b 20d 20c 10b 5c 10c 5d 5e 13P 1a 4a 4b 2a 2b 4c 4d 2c 5b 20c 20d 10b 5a 20a 20b 10a 5c 10c 5e 5d 17P 1a 4a 4b 2a 2b 4c 4d 2c 5b 20c 20d 10b 5a 20a 20b 10a 5c 10c 5e 5d 19P 1a 4b 4a 2a 2b 4d 4c 2c 5a 20b 20a 10a 5b 20d 20c 10b 5c 10c 5d 5e 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 A -A -1 -1 -A A 1 1 A -A -1 1 A -A -1 1 -1 1 1 X.6 1 -A A -1 -1 A -A 1 1 -A A -1 1 -A A -1 1 -1 1 1 X.7 1 A -A -1 1 A -A -1 1 A -A -1 1 A -A -1 1 1 1 1 X.8 1 -A A -1 1 -A A -1 1 -A A -1 1 -A A -1 1 1 1 1 X.9 2 -2 -2 2 . . . . C -C -C C *C -*C -*C *C 2 . C *C X.10 2 -2 -2 2 . . . . *C -*C -*C *C C -C -C C 2 . *C C X.11 2 2 2 2 . . . . C C C C *C *C *C *C 2 . C *C X.12 2 2 2 2 . . . . *C *C *C *C C C C C 2 . *C C X.13 2 B -B -2 . . . . C E -E -C *C F -F -*C 2 . C *C X.14 2 B -B -2 . . . . *C F -F -*C C E -E -C 2 . *C C X.15 2 -B B -2 . . . . C -E E -C *C -F F -*C 2 . C *C X.16 2 -B B -2 . . . . *C -F F -*C C -E E -C 2 . *C C X.17 4 . . . -4 . . . 4 . . . 4 . . . -1 1 -1 -1 X.18 4 . . . 4 . . . 4 . . . 4 . . . -1 -1 -1 -1 X.19 8 . . . . . . . D . . . *D . . . -2 . -C -*C X.20 8 . . . . . . . *D . . . D . . . -2 . -*C -C A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i C = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 D = 4*E(5)^2+4*E(5)^3 = -2-2*Sqrt(5) = -2-2r5 E = -E(20)^13-E(20)^17 F = -E(20)-E(20)^9