Properties

Label 25T27
Order \(300\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_5^2:D_6$

Learn more about

Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $27$
Group :  $C_5^2:D_6$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,5,22,6,23)(2,12,24,21,11,4)(3,7,18,25,16,10)(8,13,19,20,15,9), (1,11)(2,17)(3,23)(5,10)(7,12)(8,18)(9,24)(14,19)(15,25)(16,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T18 x 2, 30T66 x 2, 30T72 x 2, 30T80 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 6,25)( 7,21)( 8,22)( 9,23)(10,24)(11,19)(12,20)(13,16)(14,17)(15,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)(22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $50$ $3$ $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$
$ 6, 6, 6, 6, 1 $ $50$ $6$ $( 2, 7, 6, 5,25,21)( 3,13,11, 4,19,16)( 8,12,10,24,20,22)( 9,18,15,23,14,17)$
$ 5, 5, 5, 5, 5 $ $6$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $30$ $10$ $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,20,13,17,15,19,12,16,14,18)$
$ 10, 10, 5 $ $30$ $10$ $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$
$ 10, 10, 5 $ $30$ $10$ $( 1, 2, 8, 9,15,11,17,18,24,25)( 3,14,10,16,12,23,19, 5,21, 7)( 4,20, 6,22,13)$
$ 5, 5, 5, 5, 5 $ $6$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 10, 10, 5 $ $30$ $10$ $( 1, 3,15,12,24,21, 8,10,17,19)( 2, 9,11,18,25)( 4,16,13, 5,22,14, 6,23,20, 7)$
$ 5, 5, 5, 5, 5 $ $6$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$
$ 5, 5, 5, 5, 5 $ $6$ $5$ $( 1, 9,12,20,23)( 2,10,13,16,24)( 3, 6,14,17,25)( 4, 7,15,18,21) ( 5, 8,11,19,22)$

Group invariants

Order:  $300=2^{2} \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [300, 25]
Character table:   
      2  2  2  2  2  1  1  1   1   1   1  1   1  1  1
      3  1  .  .  1  1  1  .   .   .   .  .   .  .  .
      5  2  1  1  .  .  .  2   1   1   1  2   1  2  2

        1a 2a 2b 2c 3a 6a 5a 10a 10b 10c 5b 10d 5c 5d
     2P 1a 1a 1a 1a 3a 3a 5b  5b  5a  5c 5a  5d 5d 5c
     3P 1a 2a 2b 2c 1a 2c 5b 10b 10a 10d 5a 10c 5d 5c
     5P 1a 2a 2b 2c 3a 6a 1a  2a  2a  2b 1a  2b 1a 1a
     7P 1a 2a 2b 2c 3a 6a 5b 10b 10a 10d 5a 10c 5d 5c

X.1      1  1  1  1  1  1  1   1   1   1  1   1  1  1
X.2      1 -1 -1  1  1  1  1  -1  -1  -1  1  -1  1  1
X.3      1 -1  1 -1  1 -1  1  -1  -1   1  1   1  1  1
X.4      1  1 -1 -1  1 -1  1   1   1  -1  1  -1  1  1
X.5      2  .  . -2 -1  1  2   .   .   .  2   .  2  2
X.6      2  .  .  2 -1 -1  2   .   .   .  2   .  2  2
X.7      6 -2  .  .  .  .  A   C  *C   . *A   . *B  B
X.8      6 -2  .  .  .  . *A  *C   C   .  A   .  B *B
X.9      6  . -2  .  .  .  B   .   .  *C *B   C  A *A
X.10     6  . -2  .  .  . *B   .   .   C  B  *C *A  A
X.11     6  .  2  .  .  .  B   .   . -*C *B  -C  A *A
X.12     6  .  2  .  .  . *B   .   .  -C  B -*C *A  A
X.13     6  2  .  .  .  .  A  -C -*C   . *A   . *B  B
X.14     6  2  .  .  .  . *A -*C  -C   .  A   .  B *B

A = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4
  = (-3-Sqrt(5))/2 = -2-b5
B = -2*E(5)-2*E(5)^4
  = 1-Sqrt(5) = 1-r5
C = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5