Properties

Label 25T49
Order \(625\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $49$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,15,17,8,21,2,11,18,9,22,3,12,19,10,23,4,13,20,6,24,5,14,16,7,25), (1,25,7,19,11,5,24,6,18,15,4,23,10,17,14,3,22,9,16,13,2,21,8,20,12)
$|\Aut(F/K)|$:  $5$

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

25T49 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 $ $1$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 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 $ $1$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$
$ 25 $ $25$ $25$ $( 1,15,17, 8,21, 2,11,18, 9,22, 3,12,19,10,23, 4,13,20, 6,24, 5,14,16, 7,25)$
$ 25 $ $25$ $25$ $( 1,17,21,11, 9, 3,19,23,13, 6, 5,16,25,15, 8, 2,18,22,12,10, 4,20,24,14, 7)$
$ 25 $ $25$ $25$ $( 1,21, 9,19,13, 5,25, 8,18,12, 4,24, 7,17,11, 3,23, 6,16,15, 2,22,10,20,14)$
$ 25 $ $25$ $25$ $( 1, 8,11,22,19, 4, 6,14,25,17, 2, 9,12,23,20, 5, 7,15,21,18, 3,10,13,24,16)$
$ 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 $ $5$ $5$ $(11,13,15,12,14)(16,18,20,17,19)(21,22,23,24,25)$
$ 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, 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 $ $5$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(21,25,24,23,22)$
$ 25 $ $25$ $25$ $( 1,15,19, 7,25, 2,11,20, 8,21, 3,12,16, 9,22, 4,13,17,10,23, 5,14,18, 6,24)$
$ 25 $ $25$ $25$ $( 1,17,23,14, 9, 3,19,25,11, 6, 5,16,22,13, 8, 2,18,24,15,10, 4,20,21,12, 7)$
$ 25 $ $25$ $25$ $( 1,21,10,20,11, 5,25, 9,19,15, 4,24, 8,18,14, 3,23, 7,17,13, 2,22, 6,16,12)$
$ 25 $ $25$ $25$ $( 1, 8,11,24,17, 4, 6,14,22,20, 2, 9,12,25,18, 5, 7,15,23,16, 3,10,13,21,19)$
$ 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, 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 $ $5$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,14,12,15,13)(16,19,17,20,18) (21,22,23,24,25)$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ $5$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,13,15,12,14)(16,18,20,17,19)$
$ 25 $ $25$ $25$ $( 1,15,16, 6,24, 2,11,17, 7,25, 3,12,18, 8,21, 4,13,19, 9,22, 5,14,20,10,23)$
$ 25 $ $25$ $25$ $( 1,17,25,12, 9, 3,19,22,14, 6, 5,16,24,11, 8, 2,18,21,13,10, 4,20,23,15, 7)$
$ 25 $ $25$ $25$ $( 1,21, 6,16,14, 5,25,10,20,13, 4,24, 9,19,12, 3,23, 8,18,11, 2,22, 7,17,15)$
$ 25 $ $25$ $25$ $( 1, 8,11,21,20, 4, 6,14,24,18, 2, 9,12,22,16, 5, 7,15,25,19, 3,10,13,23,17)$
$ 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, 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 $ $5$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,13,15,12,14)(16,18,20,17,19) (21,24,22,25,23)$
$ 5, 5, 5, 5, 5 $ $5$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,12,13,14,15)(16,17,18,19,20) (21,23,25,22,24)$
$ 25 $ $25$ $25$ $( 1,15,20, 9,22, 2,11,16,10,23, 3,12,17, 6,24, 4,13,18, 7,25, 5,14,19, 8,21)$
$ 25 $ $25$ $25$ $( 1,17,24,13, 9, 3,19,21,15, 6, 5,16,23,12, 8, 2,18,25,14,10, 4,20,22,11, 7)$
$ 25 $ $25$ $25$ $( 1,21, 8,18,15, 5,25, 7,17,14, 4,24, 6,16,13, 3,23,10,20,12, 2,22, 9,19,11)$
$ 25 $ $25$ $25$ $( 1, 8,11,25,16, 4, 6,14,23,19, 2, 9,12,21,17, 5, 7,15,24,20, 3,10,13,22,18)$
$ 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, 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, 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 $ $5$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,15,14,13,12)(16,20,19,18,17) (21,22,23,24,25)$
$ 25 $ $25$ $25$ $( 1,15,18,10,23, 2,11,19, 6,24, 3,12,20, 7,25, 4,13,16, 8,21, 5,14,17, 9,22)$
$ 25 $ $25$ $25$ $( 1,17,22,15, 9, 3,19,24,12, 6, 5,16,21,14, 8, 2,18,23,11,10, 4,20,25,13, 7)$
$ 25 $ $25$ $25$ $( 1,21, 7,17,12, 5,25, 6,16,11, 4,24,10,20,15, 3,23, 9,19,14, 2,22, 8,18,13)$
$ 25 $ $25$ $25$ $( 1, 8,11,23,18, 4, 6,14,21,16, 2, 9,12,24,19, 5, 7,15,22,17, 3,10,13,25,20)$

Group invariants

Order:  $625=5^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [625, 8]
Character table: Data not available.