Galois group: 25T4

• Label: 25T4
• Degree: $25$
• Order: $50$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $D_{25}$

Group action invariants

Degree $n$:		$25$
Transitive number $t$:		$4$
Group:		$D_{25}$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,17)(2,16)(3,20)(4,19)(5,18)(6,12)(7,11)(8,15)(9,14)(10,13)(21,22)(23,25), (1,24)(2,23)(3,22)(4,21)(5,25)(6,17)(7,16)(8,20)(9,19)(10,18)(11,12)(13,15)

Low degree resolvents

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

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

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{25}$	$1$	$1$	$0$	$()$
2A	$2^{12},1$	$25$	$2$	$12$	$( 2, 5)( 3, 4)( 6,24)( 7,23)( 8,22)( 9,21)(10,25)(11,17)(12,16)(13,20)(14,19)(15,18)$
5A1	$5^{5}$	$2$	$5$	$20$	$( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)$
5A2	$5^{5}$	$2$	$5$	$20$	$( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)$
25A1	$25$	$2$	$25$	$24$	$( 1,20, 8,24,15, 3,17,10,21,12, 5,19, 7,23,14, 2,16, 9,25,11, 4,18, 6,22,13)$
25A2	$25$	$2$	$25$	$24$	$( 1,19, 6,21,11, 3,16, 8,23,13, 5,18,10,25,15, 2,20, 7,22,12, 4,17, 9,24,14)$
25A3	$25$	$2$	$25$	$24$	$( 1,18, 9,23,12, 3,20, 6,25,14, 5,17, 8,22,11, 2,19,10,24,13, 4,16, 7,21,15)$
25A4	$25$	$2$	$25$	$24$	$( 1,22,18,11, 9, 2,23,19,12,10, 3,24,20,13, 6, 4,25,16,14, 7, 5,21,17,15, 8)$
25A6	$25$	$2$	$25$	$24$	$( 1,25,19,15, 6, 2,21,20,11, 7, 3,22,16,12, 8, 4,23,17,13, 9, 5,24,18,14,10)$
25A7	$25$	$2$	$25$	$24$	$( 1,16,10,22,14, 3,18, 7,24,11, 5,20, 9,21,13, 2,17, 6,23,15, 4,19, 8,25,12)$
25A8	$25$	$2$	$25$	$24$	$( 1,23,20,14, 8, 2,24,16,15, 9, 3,25,17,11,10, 4,21,18,12, 6, 5,22,19,13, 7)$
25A9	$25$	$2$	$25$	$24$	$( 1,17, 7,25,13, 3,19, 9,22,15, 5,16, 6,24,12, 2,18, 8,21,14, 4,20,10,23,11)$
25A11	$25$	$2$	$25$	$24$	$( 1,21,16,13,10, 2,22,17,14, 6, 3,23,18,15, 7, 4,24,19,11, 8, 5,25,20,12, 9)$
25A12	$25$	$2$	$25$	$24$	$( 1,24,17,12, 7, 2,25,18,13, 8, 3,21,19,14, 9, 4,22,20,15,10, 5,23,16,11, 6)$

Malle's constant $a(G)$:     $1/12$

Group invariants

Order:		$50=2 \cdot 5^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		50.1
Character table:

1A

2A

5A1

5A2

25A1

25A2

25A3

25A4

25A6

25A7

25A8

25A9

25A11

25A12

Size

1

25

2

2

2

2

2

2

2

2

2

2

2

2

2 P

1A

1A

5A2

5A1

25A4

25A6

25A9

25A8

25A3

25A11

25A2

25A1

25A7

25A12

5 P

1A

2A

1A

1A

5A2

5A2

5A2

5A1

5A1

5A2

5A1

5A2

5A1

5A1

Type

50.1.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

50.1.1b

R

$1$

$-1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

50.1.2a1

R

$2$

$0$

$2$

$2$

${\zeta }_5^{-2}+{\zeta }_5^2$

${\zeta }_5^{-1}+{\zeta }_5$

${\zeta }_5^{-1}+{\zeta }_5$

${\zeta }_5^{-2}+{\zeta }_5^2$

${\zeta }_5^{-2}+{\zeta }_5^2$

${\zeta }_5^{-1}+{\zeta }_5$

${\zeta }_5^{-1}+{\zeta }_5$

${\zeta }_5^{-2}+{\zeta }_5^2$

${\zeta }_5^{-2}+{\zeta }_5^2$

${\zeta }_5^{-1}+{\zeta }_5$

50.1.2a2

R

$2$

$0$

$2$

$2$

${\zeta }_5^{-1}+{\zeta }_5$

${\zeta }_5^{-2}+{\zeta }_5^2$

${\zeta }_5^{-2}+{\zeta }_5^2$

${\zeta }_5^{-1}+{\zeta }_5$

${\zeta }_5^{-1}+{\zeta }_5$

${\zeta }_5^{-2}+{\zeta }_5^2$

${\zeta }_5^{-2}+{\zeta }_5^2$

${\zeta }_5^{-1}+{\zeta }_5$

${\zeta }_5^{-1}+{\zeta }_5$

${\zeta }_5^{-2}+{\zeta }_5^2$

50.1.2b1

R

$2$

$0$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

50.1.2b2

R

$2$

$0$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

50.1.2b3

R

$2$

$0$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

50.1.2b4

R

$2$

$0$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

50.1.2b5

R

$2$

$0$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

50.1.2b6

R

$2$

$0$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

50.1.2b7

R

$2$

$0$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

50.1.2b8

R

$2$

$0$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

50.1.2b9

R

$2$

$0$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

50.1.2b10

R

$2$

$0$

${\zeta }_{25}^{-5}+{\zeta }_{25}^5$

${\zeta }_{25}^{-10}+{\zeta }_{25}^{10}$

${\zeta }_{25}^{-1}+{\zeta }_{25}$

${\zeta }_{25}^{-2}+{\zeta }_{25}^2$

${\zeta }_{25}^{-3}+{\zeta }_{25}^3$

${\zeta }_{25}^{-4}+{\zeta }_{25}^4$

${\zeta }_{25}^{-6}+{\zeta }_{25}^6$

${\zeta }_{25}^{-7}+{\zeta }_{25}^7$

${\zeta }_{25}^{-8}+{\zeta }_{25}^8$

${\zeta }_{25}^{-9}+{\zeta }_{25}^9$

${\zeta }_{25}^{-11}+{\zeta }_{25}^{11}$

${\zeta }_{25}^{-12}+{\zeta }_{25}^{12}$