# Properties

 Label 25T7 Degree $25$ Order $100$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5\times F_5$

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $7$ Group: $C_5\times F_5$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $5$ Generators: (1,20,21,8)(2,16,22,9)(3,17,23,10)(4,18,24,6)(5,19,25,7), (1,2,3,4,5)(6,19,21,15,10,18,25,14,9,17,24,13,8,16,23,12,7,20,22,11)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $C_{10}$
$20$:  $F_5$, 20T1

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $C_5$, $F_5$

## Low degree siblings

20T29

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

## Group invariants

 Order: $100=2^{2} \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [100, 9]
 Character table: not available.