Galois group: 25T28

• Label: 25T28
• Degree: $25$
• Order: $300$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_5^2:C_3:C_4$

Group action invariants

Degree $n$:		$25$
Transitive number $t$:		$28$
Group:		$C_5^2:C_3:C_4$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,10,13,9)(2,17,12,22)(3,4,11,15)(5,23,14,16)(6,25,8,19)(18,24,21,20), (1,7,25,19)(2,22,24,4)(3,12,23,14)(5,17,21,9)(6,10,20,16)(8,15,18,11)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$6$:	$S_3$
$12$:	$C_3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T17 x 2, 30T71 x 2

Siblings are shown 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	$2^{12},1$	$25$	$2$	$12$	$( 1, 9)( 2, 8)( 3, 7)( 4, 6)( 5,10)(11,24)(12,23)(13,22)(14,21)(15,25)(16,19)(17,18)$
3A	$3^{8},1$	$50$	$3$	$16$	$( 1,24,20)( 2, 4,14)( 3, 9, 8)( 5,19,21)( 6,18,16)( 7,23,15)(10,13,22)(11,12,17)$
4A1	$4^{6},1$	$75$	$4$	$18$	$( 1,10,13, 9)( 2,17,12,22)( 3, 4,11,15)( 5,23,14,16)( 6,25, 8,19)(18,24,21,20)$
4A-1	$4^{6},1$	$75$	$4$	$18$	$( 2,19, 5,13)( 3, 7, 4,25)( 6,11,21,16)( 8,17,24,15)( 9,10,23,22)(12,14,20,18)$
5A	$5^{5}$	$12$	$5$	$20$	$( 1,24,17,15, 8)( 2,25,18,11, 9)( 3,21,19,12,10)( 4,22,20,13, 6)( 5,23,16,14, 7)$
5B	$5^{5}$	$12$	$5$	$20$	$( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24)( 5,10,15,20,25)$
6A	$6^{4},1$	$50$	$6$	$20$	$( 1,11,23,25,15, 3)( 2, 6,17,24,20, 9)( 4,21,10,22, 5,16)( 7,12,18,19,14, 8)$

Malle's constant $a(G)$:     $1/12$

Group invariants

Order:		$300=2^{2} \cdot 3 \cdot 5^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		300.23
Character table:

		1A	2A	3A	4A1	4A-1	5A	5B	6A
	Size	1	25	50	75	75	12	12	50
	2 P	1A	1A	3A	2A	2A	5A	5B	3A
	3 P	1A	2A	1A	4A-1	4A1	5A	5B	2A
	5 P	1A	2A	3A	4A1	4A-1	1A	1A	6A
	Type
300.23.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
300.23.1b	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
300.23.1c1	C	$1$	$-1$	$1$	$-i$	$i$	$1$	$1$	$-1$
300.23.1c2	C	$1$	$-1$	$1$	$i$	$-i$	$1$	$1$	$-1$
300.23.2a	R	$2$	$2$	$-1$	$0$	$0$	$2$	$2$	$-1$
300.23.2b	S	$2$	$-2$	$-1$	$0$	$0$	$2$	$2$	$1$
300.23.12a	R	$12$	$0$	$0$	$0$	$0$	$-3$	$2$	$0$
300.23.12b	R	$12$	$0$	$0$	$0$	$0$	$2$	$-3$	$0$