Galois group: 25T43

• Label: 25T43
• Degree: $25$
• Order: $600$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• pp-group: no
• Group: $C_5^2:(C_4\times S_3)$

Group action invariants

Degree $n$:		$25$
Transitive number $t$:		$43$
Group:		$C_5^2:(C_4\times S_3)$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,4,3)(6,20,24,15)(7,17,23,13)(8,19,22,11)(9,16,21,14)(10,18,25,12), (1,11)(2,17)(3,23)(5,10)(7,12)(8,18)(9,24)(14,19)(15,25)(16,21)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$6$:	$S_3$
$8$:	$C_4\times C_2$
$12$:	$D_{6}$
$24$:	$S_3 \times C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T27 x 2, 30T150 x 2, 30T153 x 2, 30T155 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	$2^{10},1^{5}$	$15$	$2$	$10$	$( 1,11)( 2,17)( 3,23)( 5,10)( 7,12)( 8,18)( 9,24)(14,19)(15,25)(16,21)$
2B	$2^{10},1^{5}$	$15$	$2$	$10$	$( 1,24)( 2,18)( 3,12)( 4, 6)( 7,23)( 8,17)( 9,11)(13,22)(14,16)(19,21)$
2C	$2^{12},1$	$25$	$2$	$12$	$( 1, 2)( 3, 5)( 6,22)( 7,21)( 8,25)( 9,24)(10,23)(11,17)(12,16)(13,20)(14,19)(15,18)$
3A	$3^{8},1$	$50$	$3$	$16$	$( 1, 3,13)( 2, 8, 7)( 4,18,25)( 5,23,19)( 6,22,14)( 9,12,21)(10,17,20)(11,16,15)$
4A1	$4^{6},1$	$25$	$4$	$18$	$( 1, 3, 2, 5)( 6,13,22,20)( 7,15,21,18)( 8,12,25,16)( 9,14,24,19)(10,11,23,17)$
4A-1	$4^{6},1$	$25$	$4$	$18$	$( 1, 5, 2, 3)( 6,20,22,13)( 7,18,21,15)( 8,16,25,12)( 9,19,24,14)(10,17,23,11)$
4B1	$4^{6},1$	$75$	$4$	$18$	$( 1,19,18, 5)( 2, 7,17,12)( 3,25,16,24)( 4,13,20, 6)( 8,10,11,14)( 9,23,15,21)$
4B-1	$4^{6},1$	$75$	$4$	$18$	$( 1, 7, 9, 3)( 2,19, 8,16)( 4,13, 6,22)( 5,25,10,15)(11,12,24,23)(14,18,21,17)$
5A	$5^{5}$	$12$	$5$	$20$	$( 1,20, 9,23,12)( 2,16,10,24,13)( 3,17, 6,25,14)( 4,18, 7,21,15)( 5,19, 8,22,11)$
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	$6^{4},1$	$50$	$6$	$20$	$( 1, 5,10,11,12, 7)( 2,25, 4,15,17,13)( 3,20,23,14,22,19)( 8,21,24, 9,16,18)$
10A	$10^{2},5$	$60$	$10$	$22$	$( 1,18,15, 2,24,11, 8,25,17, 9)( 3, 5,12,14,21,23,10, 7,19,16)( 4, 6,13,20,22)$
10B	$10^{2},5$	$60$	$10$	$22$	$( 1,20,11, 5,21,15, 6,25,16,10)( 2,14,12,24,22, 9, 7,19,17, 4)( 3, 8,13,18,23)$
12A1	$12^{2},1$	$50$	$12$	$22$	$( 2,16, 7, 3, 6,13, 5,11,25, 4,21,19)( 8,18,12,15,10,23,24,14,20,17,22, 9)$
12A-1	$12^{2},1$	$50$	$12$	$22$	$( 1, 2,12,18,16,21,14,13, 3,22,24,19)( 4, 7, 5,17, 6,20,11, 8,15,23, 9,25)$

Malle's constant $a(G)$:     $1/10$

Group invariants

Order:		$600=2^{3} \cdot 3 \cdot 5^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		600.151
Character table:

		1A	2A	2B	2C	3A	4A1	4A-1	4B1	4B-1	5A	5B	6A	10A	10B	12A1	12A-1
	Size	1	15	15	25	50	25	25	75	75	12	12	50	60	60	50	50
	2 P	1A	1A	1A	1A	3A	2C	2C	2C	2C	5A	5B	3A	5A	5B	6A	6A
	3 P	1A	2A	2B	2C	1A	4A-1	4A1	4B-1	4B1	5A	5B	2C	10A	10B	4A1	4A-1
	5 P	1A	2A	2B	2C	3A	4A1	4A-1	4B1	4B-1	1A	1A	6A	2A	2B	12A1	12A-1
	Type
600.151.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
600.151.1b	R	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
600.151.1c	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
600.151.1d	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
600.151.1e1	C	$1$	$-1$	$1$	$-1$	$1$	$-i$	$i$	$-i$	$i$	$1$	$1$	$-1$	$-1$	$1$	$i$	$-i$
600.151.1e2	C	$1$	$-1$	$1$	$-1$	$1$	$i$	$-i$	$i$	$-i$	$1$	$1$	$-1$	$-1$	$1$	$-i$	$i$
600.151.1f1	C	$1$	$1$	$-1$	$-1$	$1$	$-i$	$i$	$i$	$-i$	$1$	$1$	$-1$	$1$	$-1$	$i$	$-i$
600.151.1f2	C	$1$	$1$	$-1$	$-1$	$1$	$i$	$-i$	$-i$	$i$	$1$	$1$	$-1$	$1$	$-1$	$-i$	$i$
600.151.2a	R	$2$	$0$	$0$	$2$	$-1$	$2$	$2$	$0$	$0$	$2$	$2$	$-1$	$0$	$0$	$-1$	$-1$
600.151.2b	R	$2$	$0$	$0$	$2$	$-1$	$-2$	$-2$	$0$	$0$	$2$	$2$	$-1$	$0$	$0$	$1$	$1$
600.151.2c1	C	$2$	$0$	$0$	$-2$	$-1$	$-2i$	$2i$	$0$	$0$	$2$	$2$	$1$	$0$	$0$	$-i$	$i$
600.151.2c2	C	$2$	$0$	$0$	$-2$	$-1$	$2i$	$-2i$	$0$	$0$	$2$	$2$	$1$	$0$	$0$	$i$	$-i$
600.151.12a	R	$12$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$-3$	$0$	$0$	$-1$	$0$	$0$
600.151.12b	R	$12$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-3$	$2$	$0$	$-1$	$0$	$0$	$0$
600.151.12c	R	$12$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-3$	$2$	$0$	$1$	$0$	$0$	$0$
600.151.12d	R	$12$	$0$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$-3$	$0$	$0$	$1$	$0$	$0$