# Properties

 Label 25T35 Order $$500$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5.C_5:F_5$

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## Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $35$ Group : $C_5.C_5:F_5$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,18,14,21)(2,19,15,22)(3,20,11,23)(4,16,12,24)(5,17,13,25), (1,24,18,15,9)(2,25,19,11,10)(3,21,20,12,6)(4,22,16,13,7)(5,23,17,14,8) $|\Aut(F/K)|$: $5$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
20:  $F_5$ x 2
100:  $C_5^2 : C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $F_5$

## Low degree siblings

25T35

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

## Group invariants

 Order: $500=2^{2} \cdot 5^{3}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [500, 25]
 Character table: Data not available.