Properties

Label 25T34
Order \(500\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:C_4:C_5$

Learn more about

Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $34$
Group :  $C_5^2:C_4:C_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,18,6,5,12,19,9,4,14,20,7,3,11,16,10,2,13,17,8)(21,24,23,25), (1,17,9,23,15)(2,18,10,24,11)(3,19,6,25,12)(4,20,7,21,13)(5,16,8,22,14)
$|\Aut(F/K)|$:  $1$

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: $F_5$

Low degree siblings

25T37

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

Group invariants

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