Properties

Label 25T40
Order \(500\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{25}.C_{10}$

Related objects

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $40$
Group :  $D_{25}.C_{10}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,5,4)(6,18,21,11,7,16,25,13,8,19,24,15,9,17,23,12,10,20,22,14), (1,10,12,17,23,2,6,13,18,24,3,7,14,19,25,4,8,15,20,21,5,9,11,16,22)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
5:  $C_5$
10:  $C_{10}$
20:  $F_5$, 20T1
100:  20T29

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Low degree siblings

There are no siblings 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 $ $5$ $5$ $( 6,10, 9, 8, 7)(11,14,12,15,13)(16,18,20,17,19)(21,22,23,24,25)$
$ 20, 4, 1 $ $25$ $20$ $( 2, 3, 5, 4)( 6,11,23,19,10,14,24,16, 9,12,25,18, 8,15,21,20, 7,13,22,17)$
$ 20, 4, 1 $ $25$ $20$ $( 2, 3, 5, 4)( 6,12,22,16, 7,14,21,19, 8,11,25,17, 9,13,24,20,10,15,23,18)$
$ 20, 4, 1 $ $25$ $20$ $( 2, 3, 5, 4)( 6,13,21,18, 9,14,23,17, 7,15,25,16,10,11,22,20, 8,12,24,19)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,14,25,20)( 7,11,24,18)( 8,13,23,16)( 9,15,22,19)(10,12,21,17)$
$ 20, 4, 1 $ $25$ $20$ $( 2, 3, 5, 4)( 6,15,24,17, 8,14,22,18,10,13,25,19, 7,12,23,20, 9,11,21,16)$
$ 20, 4, 1 $ $25$ $20$ $( 2, 4, 5, 3)( 6,16,21,11, 9,20,23,12, 7,19,25,13,10,18,22,14, 8,17,24,15)$
$ 20, 4, 1 $ $25$ $20$ $( 2, 4, 5, 3)( 6,17,22,13, 7,20,21,15, 8,18,25,12, 9,16,24,14,10,19,23,11)$
$ 20, 4, 1 $ $25$ $20$ $( 2, 4, 5, 3)( 6,18,23,15,10,20,24,13, 9,17,25,11, 8,19,21,14, 7,16,22,12)$
$ 20, 4, 1 $ $25$ $20$ $( 2, 4, 5, 3)( 6,19,24,12, 8,20,22,11,10,16,25,15, 7,17,23,14, 9,18,21,13)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,20,25,14)( 7,18,24,11)( 8,16,23,13)( 9,19,22,15)(10,17,21,12)$
$ 10, 10, 2, 2, 1 $ $25$ $10$ $( 2, 5)( 3, 4)( 6,21, 9,23, 7,25,10,22, 8,24)(11,20,12,19,13,18,14,17,15,16)$
$ 10, 10, 2, 2, 1 $ $25$ $10$ $( 2, 5)( 3, 4)( 6,22, 7,21, 8,25, 9,24,10,23)(11,17,13,20,15,18,12,16,14,19)$
$ 10, 10, 2, 2, 1 $ $25$ $10$ $( 2, 5)( 3, 4)( 6,23,10,24, 9,25, 8,21, 7,22)(11,19,14,16,12,18,15,20,13,17)$
$ 10, 10, 2, 2, 1 $ $25$ $10$ $( 2, 5)( 3, 4)( 6,24, 8,22,10,25, 7,23, 9,21)(11,16,15,17,14,18,13,19,12,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,25)( 7,24)( 8,23)( 9,22)(10,21)(11,18)(12,17)(13,16)(14,20) (15,19)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 25 $ $20$ $25$ $( 1, 6,11,16,24, 2, 7,12,17,25, 3, 8,13,18,21, 4, 9,14,19,22, 5,10,15,20,23)$
$ 25 $ $20$ $25$ $( 1, 6,12,19,25, 2, 7,13,20,21, 3, 8,14,16,22, 4, 9,15,17,23, 5,10,11,18,24)$
$ 25 $ $20$ $25$ $( 1, 6,13,17,21, 2, 7,14,18,22, 3, 8,15,19,23, 4, 9,11,20,24, 5,10,12,16,25)$
$ 25 $ $20$ $25$ $( 1, 6,14,20,22, 2, 7,15,16,23, 3, 8,11,17,24, 4, 9,12,18,25, 5,10,13,19,21)$
$ 25 $ $20$ $25$ $( 1, 6,15,18,23, 2, 7,11,19,24, 3, 8,12,20,25, 4, 9,13,16,21, 5,10,14,17,22)$

Group invariants

Order:  $500=2^{2} \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [500, 18]
Character table: Data not available.