Properties

 Label 25T3 Degree $25$ Order $50$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5\times D_5$

Related objects

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $D_{5}$, $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$, $D_{5}$

Low degree siblings

10T6 x 2

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

Group invariants

 Order: $50=2 \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [50, 3]
 Character table:  2 1 1 1 1 1 1 1 1 1 1 . . . . . . . . . . 5 2 1 2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 1a 2a 5a 10a 5b 10b 5c 10c 5d 10d 5e 5f 5g 5h 5i 5j 5k 5l 5m 5n 2P 1a 1a 5b 5b 5d 5d 5a 5a 5c 5c 5k 5m 5j 5l 5n 5f 5h 5e 5g 5i 3P 1a 2a 5c 10c 5a 10a 5d 10d 5b 10b 5l 5j 5m 5k 5n 5g 5e 5h 5f 5i 5P 1a 2a 1a 2a 1a 2a 1a 2a 1a 2a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 7P 1a 2a 5b 10b 5d 10d 5a 10a 5c 10c 5k 5m 5j 5l 5n 5f 5h 5e 5g 5i X.1 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 1 X.3 1 -1 A -A B -B /B -/B /A -/A A B /B /A 1 A B /B /A 1 X.4 1 -1 B -B /A -/A A -A /B -/B B /A A /B 1 B /A A /B 1 X.5 1 -1 /B -/B A -A /A -/A B -B /B A /A B 1 /B A /A B 1 X.6 1 -1 /A -/A /B -/B B -B A -A /A /B B A 1 /A /B B A 1 X.7 1 1 A A B B /B /B /A /A A B /B /A 1 A B /B /A 1 X.8 1 1 B B /A /A A A /B /B B /A A /B 1 B /A A /B 1 X.9 1 1 /B /B A A /A /A B B /B A /A B 1 /B A /A B 1 X.10 1 1 /A /A /B /B B B A A /A /B B A 1 /A /B B A 1 X.11 2 . 2 . 2 . 2 . 2 . E E E E E *E *E *E *E *E X.12 2 . 2 . 2 . 2 . 2 . *E *E *E *E *E E E E E E X.13 2 . C . D . /D . /C . F H /H /F E I G /G /I *E X.14 2 . /C . /D . D . C . /F /H H F E /I /G G I *E X.15 2 . D . /C . C . /D . G /I I /G *E H /F F /H E X.16 2 . /D . C . /C . D . /G I /I G *E /H F /F H E X.17 2 . D . /C . C . /D . H /F F /H E G /I I /G *E X.18 2 . /D . C . /C . D . /H F /F H E /G I /I G *E X.19 2 . C . D . /D . /C . I G /G /I *E F H /H /F E X.20 2 . /C . /D . D . C . /I /G G I *E /F /H H F E A = E(5)^4 B = E(5)^3 C = 2*E(5)^4 D = 2*E(5)^3 E = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 F = E(5)+E(5)^2 G = E(5)^2+E(5)^4 H = -E(5)^2-E(5)^3-E(5)^4 I = -E(5)-E(5)^2-E(5)^4