Properties

Label 25T22
Order \(250\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $He_5:C_2$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $22$
Group :  $He_5:C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,9)(3,10)(4,6)(5,7)(11,22)(12,23)(13,24)(14,25)(15,21), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15), (1,3,5,2,4)(6,21,10,25,9,24,8,23,7,22)(11,18,15,17,14,16,13,20,12,19)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
10:  $D_{5}$ x 6
50:  25T5

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Low degree siblings

25T22 x 5

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 $ $10$ $5$ $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ $10$ $5$ $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)(21,24,22,25,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $25$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,20)(12,16)(13,17)(14,18)(15,19)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $25$ $10$ $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,19,13,16,15,18,12,20,14,17)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 10, 10, 5 $ $25$ $10$ $( 1, 3, 5, 2, 4)( 6,21,10,25, 9,24, 8,23, 7,22)(11,18,15,17,14,16,13,20,12,19)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$
$ 10, 10, 5 $ $25$ $10$ $( 1, 4, 2, 5, 3)( 6,21, 7,22, 8,23, 9,24,10,25)(11,17,12,18,13,19,14,20,15,16)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$
$ 10, 10, 5 $ $25$ $10$ $( 1, 5, 4, 3, 2)( 6,21, 9,24, 7,22,10,25, 8,23)(11,16,14,19,12,17,15,20,13,18)$
$ 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, 8]
Character table: Data not available.