Properties

 Label 25T20 Order $$200$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $C_5:D_5.C_4$

Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $20$ Group : $C_5:D_5.C_4$ Parity: $-1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (2,23,4,12,5,9,3,20)(6,17,16,24,21,15,11,8)(7,14,19,10,25,18,13,22), (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) $|\Aut(F/K)|$: $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 47$

Degree 5: None

Low degree siblings

10T18 x 3, 20T56 x 3, 40T171 x 3

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

Group invariants

 Order: $200=2^{3} \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [200, 40]
 Character table:  2 3 3 3 3 3 3 3 3 . . . 5 2 . . . . . . . 2 2 2 1a 4a 4b 2a 8a 8b 8c 8d 5a 5b 5c 2P 1a 2a 2a 1a 4b 4a 4a 4b 5a 5b 5c 3P 1a 4b 4a 2a 8c 8d 8a 8b 5a 5b 5c 5P 1a 4a 4b 2a 8d 8c 8b 8a 1a 1a 1a 7P 1a 4b 4a 2a 8b 8a 8d 8c 5a 5b 5c X.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 X.3 1 -1 -1 1 A -A -A A 1 1 1 X.4 1 -1 -1 1 -A A A -A 1 1 1 X.5 1 A -A -1 B /B -/B -B 1 1 1 X.6 1 A -A -1 -B -/B /B B 1 1 1 X.7 1 -A A -1 -/B -B B /B 1 1 1 X.8 1 -A A -1 /B B -B -/B 1 1 1 X.9 8 . . . . . . . 3 -2 -2 X.10 8 . . . . . . . -2 -2 3 X.11 8 . . . . . . . -2 3 -2 A = -E(4) = -Sqrt(-1) = -i B = -E(8)