Properties

Label 25T6
Order \(75\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_5^2:C_3$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $6$
Group :  $C_5^2:C_3$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23), (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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T9 x 2

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

Group invariants

Order:  $75=3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [75, 2]
Character table:   
      3  1  1  1  .  .  .  .  .  .  .  .
      5  2  .  .  2  2  2  2  2  2  2  2

        1a 3a 3b 5a 5b 5c 5d 5e 5f 5g 5h
     2P 1a 3b 3a 5b 5d 5a 5c 5f 5g 5h 5e
     3P 1a 1a 1a 5c 5a 5d 5b 5h 5e 5f 5g
     5P 1a 3b 3a 1a 1a 1a 1a 1a 1a 1a 1a

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1  A /A  1  1  1  1  1  1  1  1
X.3      1 /A  A  1  1  1  1  1  1  1  1
X.4      3  .  .  B /C  C /B  D *D  D *D
X.5      3  .  .  C  B /B /C *D  D *D  D
X.6      3  .  . /C /B  B  C *D  D *D  D
X.7      3  .  . /B  C /C  B  D *D  D *D
X.8      3  .  .  D *D *D  D  C  B /C /B
X.9      3  .  .  D *D *D  D /C /B  C  B
X.10     3  .  . *D  D  D *D /B  C  B /C
X.11     3  .  . *D  D  D *D  B /C /B  C

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(5)+E(5)^3
C = 2*E(5)^3+E(5)^4
D = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5