# Properties

 Label 25T17 Degree $25$ Order $200$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_5^2:Q_8$

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $17$ Group: $C_5^2:Q_8$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,7,24,18)(2,9,23,16)(3,6,22,19)(4,8,21,17)(5,10,25,20)(11,12,14,13), (1,7,11,10)(2,12,15,5)(3,17,14,25)(4,22,13,20)(8,16,9,21)(18,19,24,23)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $Q_8$

Resolvents shown for degrees $\leq 47$

Degree 5: None

## Low degree siblings

10T20 x 3, 20T47 x 3, 40T166 x 3

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

## Group invariants

 Order: $200=2^{3} \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [200, 44]
 Character table:  2 3 2 3 2 2 . . . 5 2 . . . . 2 2 2 1a 4a 2a 4b 4c 5a 5b 5c 2P 1a 2a 1a 2a 2a 5a 5b 5c 3P 1a 4a 2a 4b 4c 5a 5b 5c 5P 1a 4a 2a 4b 4c 1a 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 1 1 X.3 1 -1 1 1 -1 1 1 1 X.4 1 1 1 -1 -1 1 1 1 X.5 2 . -2 . . 2 2 2 X.6 8 . . . . 3 -2 -2 X.7 8 . . . . -2 -2 3 X.8 8 . . . . -2 3 -2