# Properties

 Label 25T19 Order $$200$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_5:F_5$

## Group action invariants

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

## 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:  $C_4\times C_2$
20:  $F_5$ x 2
40:  $F_{5}\times C_2$ x 2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $F_5$ x 2

## Low degree siblings

10T17 x 2, 20T54 x 2, 40T169 x 2

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

## Group invariants

 Order: $200=2^{3} \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [200, 42]
 Character table: 2 3 3 3 3 3 3 3 3 1 1 1 1 . . 5 2 1 . . . . 1 . 2 1 2 1 2 2 1a 2a 4a 4b 4c 4d 2b 2c 5a 10a 5b 10b 5c 5d 2P 1a 1a 2c 2c 2c 2c 1a 1a 5a 5a 5b 5b 5c 5d 3P 1a 2a 4d 4c 4b 4a 2b 2c 5a 10a 5b 10b 5c 5d 5P 1a 2a 4a 4b 4c 4d 2b 2c 1a 2a 1a 2b 1a 1a 7P 1a 2a 4d 4c 4b 4a 2b 2c 5a 10a 5b 10b 5c 5d X.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 X.3 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 X.5 1 -1 A -A A -A 1 -1 1 -1 1 1 1 1 X.6 1 -1 -A A -A A 1 -1 1 -1 1 1 1 1 X.7 1 1 A A -A -A -1 -1 1 1 1 -1 1 1 X.8 1 1 -A -A A A -1 -1 1 1 1 -1 1 1 X.9 4 . . . . . -4 . 4 . -1 1 -1 -1 X.10 4 . . . . . 4 . 4 . -1 -1 -1 -1 X.11 4 -4 . . . . . . -1 1 4 . -1 -1 X.12 4 4 . . . . . . -1 -1 4 . -1 -1 X.13 8 . . . . . . . -2 . -2 . -2 3 X.14 8 . . . . . . . -2 . -2 . 3 -2 A = -E(4) = -Sqrt(-1) = -i