Properties

Label 25T48
Order \(625\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Learn more about

Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $48$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,20,22,12,9)(2,16,23,13,10)(3,17,24,14,6)(4,18,25,15,7)(5,19,21,11,8), (1,8,12,21,17)(2,9,13,22,18)(3,10,14,23,19)(4,6,15,24,20)(5,7,11,25,16)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
5:  $C_5$ x 6
25:  25T2
125:  25T14

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

Low degree siblings

25T48 x 4

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

Group invariants

Order:  $625=5^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [625, 7]
Character table: Data not available.