Properties

Label 25T41
Order \(600\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $41$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,25,10,21,2,17)(3,14,22,24,13,5)(4,6,18,23,16,9)(7,15,19,20,12,8), (1,25,13,19)(2,11,12,3)(4,18,15,21)(5,9,14,10)(6,16,8,23)(17,24,22,20)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
12:  $A_4$
24:  $\SL(2,3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

30T126

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, 5 $ $24$ $5$ $( 1,24,17,15, 8)( 2,25,18,11, 9)( 3,21,19,12,10)( 4,22,20,13, 6) ( 5,23,16,14, 7)$
$ 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 $ $150$ $4$ $( 2, 4, 5, 3)( 6,14,21,18)( 7,12,25,20)( 8,15,24,17)( 9,13,23,19)(10,11,22,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $100$ $3$ $( 2,22, 9)( 3,18,12)( 4,14,20)( 5,10,23)( 6,19, 8)( 7,15,11)(13,24,21) (16,25,17)$
$ 6, 6, 6, 6, 1 $ $100$ $6$ $( 2,10, 9, 5,22,23)( 3,14,12, 4,18,20)( 6,13, 8,21,19,24)( 7,17,11,25,15,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $100$ $3$ $( 2, 9,22)( 3,12,18)( 4,20,14)( 5,23,10)( 6, 8,19)( 7,11,15)(13,21,24) (16,17,25)$
$ 6, 6, 6, 6, 1 $ $100$ $6$ $( 2,23,22, 5, 9,10)( 3,20,18, 4,12,14)( 6,24,19,21, 8,13)( 7,16,15,25,11,17)$

Group invariants

Order:  $600=2^{3} \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [600, 150]
Character table:   
     2  3  .  3  2   1  1   1  1
     3  1  .  1  .   1  1   1  1
     5  2  2  .  .   .  .   .  .

       1a 5a 2a 4a  3a 6a  3b 6b
    2P 1a 5a 1a 2a  3b 3b  3a 3a
    3P 1a 5a 2a 4a  1a 2a  1a 2a
    5P 1a 1a 2a 4a  3b 6b  3a 6a

X.1     1  1  1  1   1  1   1  1
X.2     1  1  1  1   A  A  /A /A
X.3     1  1  1  1  /A /A   A  A
X.4     2  2 -2  .  -1  1  -1  1
X.5     2  2 -2  .  -A  A -/A /A
X.6     2  2 -2  . -/A /A  -A  A
X.7     3  3  3 -1   .  .   .  .
X.8    24 -1  .  .   .  .   .  .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3