# Properties

 Label 25T32 Degree $25$ Order $400$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $F_5^2$

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $32$ Group: $F_5^2$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,23,11,18)(2,22,12,17)(3,21,13,16)(4,25,14,20)(5,24,15,19)(6,8)(9,10), (1,23,4,22)(2,21,3,24)(5,25)(6,18,9,17)(7,16,8,19)(10,20)(11,13,14,12)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 6, $C_2^2$
$8$:  $C_4\times C_2$ x 3
$16$:  $C_4^2$
$20$:  $F_5$ x 2
$40$:  $F_{5}\times C_2$ x 2
$80$:  20T20 x 2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $F_5$ x 2

## Low degree siblings

20T102, 40T396

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

## Group invariants

 Order: $400=2^{4} \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [400, 205]
 Character table: not available.