Properties

Label 25T29
Order \(300\)
n \(25\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_5\times A_5$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $29$
Group :  $C_5\times A_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,3,25,5,22,2,24,4,21)(6,12,8,14,10,11,7,13,9,15)(16,17,18,19,20), (1,20,13,6,24)(2,16,14,7,25)(3,17,15,8,21)(4,18,11,9,22)(5,19,12,10,23)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
5:  $C_5$
60:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$, $A_5$

Low degree siblings

30T69

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

Group invariants

Order:  $300=2^{2} \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [300, 22]
Character table: Data not available.