# Properties

 Label 25T25 Degree $25$ Order $250$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{25}:C_5$

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $25$ Group: $D_{25}:C_5$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,17,5,18,4,19,3,20,2,16)(6,14,9,11,7,13,10,15,8,12)(22,25)(23,24), (1,19,10,24,13,4,17,8,22,11,2,20,6,25,14,5,18,9,23,12,3,16,7,21,15)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $D_{5}$, $C_{10}$
$50$:  $D_5\times C_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $D_{5}$

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

## Group invariants

 Order: $250=2 \cdot 5^{3}$ Cyclic: no Abelian: no Solvable: yes GAP id: [250, 6]
 Character table: not available.