Properties

 Label 25T21 Order $$200$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $D_5\wr C_2$

Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $21$ Group : $D_5\wr C_2$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,22,3,24,5,21,2,23,4,25)(6,17,8,19,10,16,7,18,9,20)(11,12,13,14,15), (1,16,17,7,8,23,24,14,15,5)(2,6,18,22,9,13,25,4,11,20)(3,21,19,12,10) $|\Aut(F/K)|$: $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Degree 5: None

Low degree siblings

10T19, 10T21 x 2, 20T48 x 2, 20T50 x 2, 20T55, 20T57 x 2, 40T167 x 2, 40T170

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $10$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ $4, 4, 4, 4, 4, 4, 1$ $50$ $4$ $( 2,11, 5,16)( 3,21, 4, 6)( 7,13,25,19)( 8,23,24, 9)(10,18,22,14)(12,15,20,17)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $10$ $2$ $( 2,11)( 3,21)( 4, 6)( 5,16)( 7,14)( 8,24)(10,19)(13,22)(15,17)(18,25)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $10, 10, 5$ $20$ $10$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$ $10, 10, 5$ $20$ $10$ $( 1, 2,12,13,23,24, 9,10,20,16)( 3,22,14, 8,25,19, 6, 5,17,11)( 4, 7,15,18,21)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ $10, 10, 5$ $20$ $10$ $( 1, 3, 5, 2, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,18,15,17,14,16,13,20,12,19)$ $10, 10, 5$ $20$ $10$ $( 1, 3,23,25,20,17,12,14, 9, 6)( 2,13,24,10,16)( 4, 8,21, 5,18,22,15,19, 7,11)$ $5, 5, 5, 5, 5$ $8$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18) ( 5,11,22, 8,19)$

Group invariants

 Order: $200=2^{3} \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [200, 43]
 Character table:  2 3 2 3 2 2 1 1 1 1 1 1 . 1 1 5 2 1 . . 1 2 1 1 2 1 1 2 2 2 1a 2a 2b 4a 2c 5a 10a 10b 5b 10c 10d 5c 5d 5e 2P 1a 1a 1a 2b 1a 5b 5b 5e 5a 5a 5d 5c 5e 5d 3P 1a 2a 2b 4a 2c 5b 10c 10d 5a 10a 10b 5c 5e 5d 5P 1a 2a 2b 4a 2c 1a 2a 2c 1a 2a 2c 1a 1a 1a 7P 1a 2a 2b 4a 2c 5b 10c 10d 5a 10a 10b 5c 5e 5d X.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 X.3 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 1 1 X.4 1 1 1 -1 -1 1 1 -1 1 1 -1 1 1 1 X.5 2 . -2 . . 2 . . 2 . . 2 2 2 X.6 4 -2 . . . A C . *A *C . -1 *B B X.7 4 -2 . . . *A *C . A C . -1 B *B X.8 4 . . . -2 B . C *B . *C -1 A *A X.9 4 . . . -2 *B . *C B . C -1 *A A X.10 4 . . . 2 B . -C *B . -*C -1 A *A X.11 4 . . . 2 *B . -*C B . -C -1 *A A X.12 4 2 . . . A -C . *A -*C . -1 *B B X.13 4 2 . . . *A -*C . A -C . -1 B *B X.14 8 . . . . -2 . . -2 . . 3 -2 -2 A = -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4 = (3-Sqrt(5))/2 = 1-b5 B = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 C = -E(5)^2-E(5)^3 = (1+Sqrt(5))/2 = 1+b5