Properties

Label 25T23
Order \(250\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:C_{10}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $D_{5}$, $C_{10}$
50:  $D_5\times C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Low degree siblings

25T24

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

Group invariants

Order:  $250=2 \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [250, 5]
Character table: Data not available.