# Properties

 Label 25T9 Degree $25$ Order $100$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5^2:C_4$

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $9$ Group: $C_5^2:C_4$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,14,24,6,17)(2,15,25,7,18)(3,11,21,8,19)(4,12,22,9,20)(5,13,23,10,16), (1,25,19,12,10)(2,21,20,13,6)(3,22,16,14,7)(4,23,17,15,8)(5,24,18,11,9), (1,19,25,10)(2,16,24,8)(3,18,23,6)(4,20,22,9)(5,17,21,7)(11,15,13,14)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$ x 6

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $F_5$ x 6

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

## Group invariants

 Order: $100=2^{2} \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [100, 11]
 Character table:  2 2 2 2 2 . . . . . . 5 2 . . . 2 2 2 2 2 2 1a 4a 4b 2a 5a 5b 5c 5d 5e 5f 2P 1a 2a 2a 1a 5a 5b 5c 5d 5e 5f 3P 1a 4b 4a 2a 5a 5b 5c 5d 5e 5f 5P 1a 4a 4b 2a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 1 1 X.3 1 A -A -1 1 1 1 1 1 1 X.4 1 -A A -1 1 1 1 1 1 1 X.5 4 . . . 4 -1 -1 -1 -1 -1 X.6 4 . . . -1 4 -1 -1 -1 -1 X.7 4 . . . -1 -1 4 -1 -1 -1 X.8 4 . . . -1 -1 -1 4 -1 -1 X.9 4 . . . -1 -1 -1 -1 -1 4 X.10 4 . . . -1 -1 -1 -1 4 -1 A = -E(4) = -Sqrt(-1) = -i