Properties

Label 25T17
Order \(200\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_5^2:Q_8$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $17$
Group :  $C_5^2:Q_8$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,24,18)(2,9,23,16)(3,6,22,19)(4,8,21,17)(5,10,25,20)(11,12,14,13), (1,7,11,10)(2,12,15,5)(3,17,14,25)(4,22,13,20)(8,16,9,21)(18,19,24,23)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

10T20 x 3, 20T47 x 3, 40T166 x 3

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 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 4, 4, 1 $ $50$ $4$ $( 2, 3, 5, 4)( 6,16,21,11)( 7,18,25,14)( 8,20,24,12)( 9,17,23,15)(10,19,22,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 4, 4, 4, 4, 4, 4, 1 $ $50$ $4$ $( 2, 6, 5,21)( 3,11, 4,16)( 7,10,25,22)( 8,15,24,17)( 9,20,23,12)(13,14,19,18)$
$ 4, 4, 4, 4, 4, 4, 1 $ $50$ $4$ $( 2,11, 5,16)( 3,21, 4, 6)( 7,13,25,19)( 8,23,24, 9)(10,18,22,14)(12,15,20,17)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [200, 44]
Character table:   
     2  3  2  3  2  2  .  .  .
     5  2  .  .  .  .  2  2  2

       1a 4a 2a 4b 4c 5a 5b 5c
    2P 1a 2a 1a 2a 2a 5a 5b 5c
    3P 1a 4a 2a 4b 4c 5a 5b 5c
    5P 1a 4a 2a 4b 4c 1a 1a 1a

X.1     1  1  1  1  1  1  1  1
X.2     1 -1  1 -1  1  1  1  1
X.3     1 -1  1  1 -1  1  1  1
X.4     1  1  1 -1 -1  1  1  1
X.5     2  . -2  .  .  2  2  2
X.6     8  .  .  .  .  3 -2 -2
X.7     8  .  .  .  . -2 -2  3
X.8     8  .  .  .  . -2  3 -2