Galois Group: 25T46

• Label: 25T46
• Order: \(600\)
• n: \(25\)
• Cyclic: No
• Abelian: No
• Solvable: No
• Primitive: No
• $p$-group: No

Group action invariants

Degree $n$ :		$25$
Transitive number $t$ :		$46$
Parity:		$1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,2,3,4,5)(6,7,8,9,10)(11,22,18,14,25,16,12,23,19,15,21,17,13,24,20), (1,16,6,11,21)(2,20,7,15,22,5,17,10,12,25)(3,19,8,14,23,4,18,9,13,24)
$|\Aut(F/K)|$:		$1$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$
10:	$D_{5}$
60:	$A_5$
120:	$A_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$, $A_5$

Low degree siblings

30T128

Siblings are shown 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 $	$1$	$1$	$()$
$ 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 $	$2$	$5$	$( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $	$5$	$2$	$( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $	$15$	$2$	$( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 10, 10, 5 $	$30$	$10$	$( 1, 7, 3, 9, 5, 6, 2, 8, 4,10)(11,17,13,19,15,16,12,18,14,20)(21,22,23,24,25)$
$ 10, 10, 5 $	$30$	$10$	$( 1, 8, 5, 7, 4, 6, 3,10, 2, 9)(11,18,15,17,14,16,13,20,12,19)(21,23,25,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$75$	$2$	$( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,16)(12,20)(13,19)(14,18)(15,17)(22,25) (23,24)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$20$	$3$	$( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 15, 5, 5 $	$40$	$15$	$( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,17,18,19,20)(21,22,23,24,25)$
$ 15, 5, 5 $	$40$	$15$	$( 1, 8,15, 2, 9,11, 3,10,12, 4, 6,13, 5, 7,14)(16,18,20,17,19)(21,23,25,22,24)$
$ 6, 6, 3, 2, 2, 2, 2, 1, 1 $	$100$	$6$	$( 1, 6,11)( 2,10,12, 5, 7,15)( 3, 9,13, 4, 8,14)(17,20)(18,19)(22,25)(23,24)$
$ 5, 5, 5, 5, 5 $	$12$	$5$	$( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$
$ 5, 5, 5, 5, 5 $	$24$	$5$	$( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 5, 5, 5, 5, 5 $	$24$	$5$	$( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$
$ 10, 10, 5 $	$60$	$10$	$( 1, 6,11,16,21)( 2,10,12,20,22, 5, 7,15,17,25)( 3, 9,13,19,23, 4, 8,14,18,24)$
$ 5, 5, 5, 5, 5 $	$12$	$5$	$( 1, 6,11,21,16)( 2, 7,12,22,17)( 3, 8,13,23,18)( 4, 9,14,24,19) ( 5,10,15,25,20)$
$ 5, 5, 5, 5, 5 $	$24$	$5$	$( 1, 7,13,24,20)( 2, 8,14,25,16)( 3, 9,15,21,17)( 4,10,11,22,18) ( 5, 6,12,23,19)$
$ 5, 5, 5, 5, 5 $	$24$	$5$	$( 1, 8,15,22,19)( 2, 9,11,23,20)( 3,10,12,24,16)( 4, 6,13,25,17) ( 5, 7,14,21,18)$
$ 10, 10, 5 $	$60$	$10$	$( 1, 6,11,21,16)( 2,10,12,25,17, 5, 7,15,22,20)( 3, 9,13,24,18, 4, 8,14,23,19)$

Group invariants

Order:		$600=2^{3} \cdot 3 \cdot 5^{2}$
Cyclic:		No
Abelian:		No
Solvable:		No
GAP id:		[600, 146]

Character table:

2 3 2 2 3 3 2 2 3 1 . . 1 1 . . 1 1 . . 3 1 1 1 1 . . . . 1 1 1 1 . . . . . . . 5 2 2 2 1 1 1 1 . 1 1 1 . 2 2 2 1 2 2 2 1a 5a 5b 2a 2b 10a 10b 2c 3a 15a 15b 6a 5c 5d 5e 10c 5f 5g 5h 2P 1a 5b 5a 1a 1a 5b 5a 1a 3a 15b 15a 3a 5f 5h 5g 5f 5c 5e 5d 3P 1a 5b 5a 2a 2b 10b 10a 2c 1a 5b 5a 2a 5f 5h 5g 10d 5c 5e 5d 5P 1a 1a 1a 2a 2b 2b 2b 2c 3a 3a 3a 6a 1a 1a 1a 2a 1a 1a 1a 7P 1a 5b 5a 2a 2b 10b 10a 2c 3a 15b 15a 6a 5f 5h 5g 10d 5c 5e 5d 11P 1a 5a 5b 2a 2b 10a 10b 2c 3a 15a 15b 6a 5c 5d 5e 10c 5f 5g 5h 13P 1a 5b 5a 2a 2b 10b 10a 2c 3a 15b 15a 6a 5f 5h 5g 10d 5c 5e 5d X.1 1 1 1 1 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 1 -1 1 1 1 X.3 2 A *A . 2 A *A . 2 A *A . 2 A *A . 2 A *A X.4 2 *A A . 2 *A A . 2 *A A . 2 *A A . 2 *A A X.5 3 3 3 -3 -1 -1 -1 1 . . . . -A -A -A A -*A -*A -*A X.6 3 3 3 -3 -1 -1 -1 1 . . . . -*A -*A -*A *A -A -A -A X.7 3 3 3 3 -1 -1 -1 -1 . . . . -A -A -A -A -*A -*A -*A X.8 3 3 3 3 -1 -1 -1 -1 . . . . -*A -*A -*A -*A -A -A -A X.9 4 4 4 -4 . . . . 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 X.10 4 4 4 4 . . . . 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 X.11 5 5 5 -5 1 1 1 -1 -1 -1 -1 1 . . . . . . . X.12 5 5 5 5 1 1 1 1 -1 -1 -1 -1 . . . . . . . X.13 6 B *B . -2 -*A -A . . . . . E 1 *F . *E F 1 X.14 6 *B B . -2 -A -*A . . . . . *E 1 F . E *F 1 X.15 6 B *B . -2 -*A -A . . . . . *E F 1 . E 1 *F X.16 6 *B B . -2 -A -*A . . . . . E *F 1 . *E 1 F X.17 8 C *C . . . . . 2 A *A . -2 -A -*A . -2 -A -*A X.18 8 *C C . . . . . 2 *A A . -2 -*A -A . -2 -*A -A X.19 10 D *D . 2 A *A . -2 -A -*A . . . . . . . . X.20 10 *D D . 2 *A A . -2 -*A -A . . . . . . . . 2 1 3 . 5 1 10d 2P 5c 3P 10c 5P 2a 7P 10c 11P 10d 13P 10c X.1 1 X.2 -1 X.3 . X.4 . X.5 *A X.6 A X.7 -*A X.8 -A X.9 1 X.10 -1 X.11 . X.12 . X.13 . X.14 . X.15 . X.16 . X.17 . X.18 . X.19 . X.20 . A = E(5)+E(5)^4 = (-1+Sqrt(5))/2 = b5 B = 3*E(5)^2+3*E(5)^3 = (-3-3*Sqrt(5))/2 = -3-3b5 C = 4*E(5)+4*E(5)^4 = -2+2*Sqrt(5) = 4b5 D = 5*E(5)+5*E(5)^4 = (-5+5*Sqrt(5))/2 = 5b5 E = -2*E(5)-2*E(5)^4 = 1-Sqrt(5) = 1-r5 F = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4 = (-3-Sqrt(5))/2 = -2-b5