Properties

Label 25T37
Order \(500\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:C_4:C_5$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $37$
Group :  $C_5^2:C_4:C_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,15,17,9,4,21,11,19,6,3,23,14,20,7,5,24,13,18,10)(2,25,12,16,8), (1,8,16,11,22,5,10,18,12,21,3,9,17,14,24,4,7,20,13,25)(2,6,19,15,23)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
5:  $C_5$
10:  $C_{10}$
20:  $F_5$, 20T1
100:  20T29

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

Low degree siblings

25T34

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

Order:  $500=2^{2} \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [500, 17]
Character table: Data not available.