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Properties

Label 25T26
Order \(300\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_5^2:C_{12}$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $26$
Group :  $C_5^2:C_{12}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,9,22,21,11,10,7,2,14,15,25)(3,24,16,23,6,17,8,12,20,13,5,19), (1,2)(3,5)(6,22)(7,21)(8,25)(9,24)(10,23)(11,17)(12,16)(13,20)(14,19)(15,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $C_{12}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T19 x 2, 30T78 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 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$
$ 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 $ $25$ $3$ $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$
$ 6, 6, 6, 6, 1 $ $25$ $6$ $( 2, 7, 6, 5,25,21)( 3,13,11, 4,19,16)( 8,12,10,24,20,22)( 9,18,15,23,14,17)$
$ 12, 12, 1 $ $25$ $12$ $( 2,11, 7, 4, 6,19, 5,16,25, 3,21,13)( 8,14,12,17,10, 9,24,18,20,15,22,23)$
$ 12, 12, 1 $ $25$ $12$ $( 2,13,21, 3,25,16, 5,19, 6, 4, 7,11)( 8,23,22,15,20,18,24, 9,10,17,12,14)$
$ 12, 12, 1 $ $25$ $12$ $( 2,16, 7, 3, 6,13, 5,11,25, 4,21,19)( 8,18,12,15,10,23,24,14,20,17,22, 9)$
$ 12, 12, 1 $ $25$ $12$ $( 2,19,21, 4,25,11, 5,13, 6, 3, 7,16)( 8, 9,22,17,20,14,24,23,10,15,12,18)$
$ 6, 6, 6, 6, 1 $ $25$ $6$ $( 2,21,25, 5, 6, 7)( 3,16,19, 4,11,13)( 8,22,20,24,10,12)( 9,17,14,23,15,18)$
$ 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 $ $12$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$

Group invariants

Order:  $300=2^{2} \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [300, 24]
Character table:    
      2  2  2  2  2  2   2   2   2   2   2   2  2  .  .
      3  1  1  1  1  1   1   1   1   1   1   1  1  .  .
      5  2  .  .  .  .   .   .   .   .   .   .  .  2  2

        1a 4a 4b 2a 3a  6a 12a 12b 12c 12d  6b 3b 5a 5b
     2P 1a 2a 2a 1a 3b  3a  6a  6b  6a  6b  3b 3a 5a 5b
     3P 1a 4b 4a 2a 1a  2a  4b  4a  4a  4b  2a 1a 5a 5b
     5P 1a 4a 4b 2a 3b  6b 12d 12c 12b 12a  6a 3a 1a 1a
     7P 1a 4b 4a 2a 3a  6a 12c 12d 12a 12b  6b 3b 5a 5b
    11P 1a 4b 4a 2a 3b  6b 12b 12a 12d 12c  6a 3a 5a 5b

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

A = -E(4)
  = -Sqrt(-1) = -i
B = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
C = -E(12)^11