# Properties

 Label 25T5 Order $$50$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5:D_5$

## Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $5$ Group : $C_5:D_5$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,16)(2,20)(3,19)(4,18)(5,17)(6,11)(7,15)(8,14)(9,13)(10,12)(22,25)(23,24), (1,21,16,11,6)(2,22,17,12,7)(3,23,18,13,8)(4,24,19,14,9)(5,25,20,15,10), (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) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
10:  $D_{5}$ x 6

Resolvents shown for degrees $\leq 47$

## Subfields

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

## Group invariants

 Order: $50=2 \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [50, 4]
 Character table:  2 1 1 . . . . . . . . . . . . 5 2 . 2 2 2 2 2 2 2 2 2 2 2 2 1a 2a 5a 5b 5c 5d 5e 5f 5g 5h 5i 5j 5k 5l 2P 1a 1a 5b 5a 5h 5j 5l 5i 5k 5c 5f 5d 5g 5e 3P 1a 2a 5b 5a 5h 5j 5l 5i 5k 5c 5f 5d 5g 5e 5P 1a 2a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 X.3 2 . 2 2 A A A A A *A *A *A *A *A X.4 2 . 2 2 *A *A *A *A *A A A A A A X.5 2 . A *A 2 A *A *A A 2 A *A *A A X.6 2 . *A A 2 *A A A *A 2 *A A A *A X.7 2 . A *A A 2 A *A *A *A A 2 A *A X.8 2 . *A A *A 2 *A A A A *A 2 *A A X.9 2 . A *A A *A *A A 2 *A *A A 2 A X.10 2 . *A A *A A A *A 2 A A *A 2 *A X.11 2 . A *A *A A 2 A *A A *A *A A 2 X.12 2 . *A A A *A 2 *A A *A A A *A 2 X.13 2 . A *A *A *A A 2 A A 2 A *A *A X.14 2 . *A A A A *A 2 *A *A 2 *A A A A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5