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Properties

Label 15T19
Order \(300\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_5^2 : C_4):C_3$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $19$
Group :  $(C_5^2 : C_4):C_3$
CHM label :  $[5^{2}:4]3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,10,7,4)(2,5,8,11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (1,7,4,13)(2,14,8,11)(3,6,12,9)
$|\Aut(F/K)|$:  $1$

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 3: $C_3$

Degree 5: None

Low degree siblings

15T19, 25T26, 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $25$ $2$ $( 3, 6)( 4,13)( 5,14)( 7,10)( 8,11)( 9,15)$
$ 4, 4, 4, 1, 1, 1 $ $25$ $4$ $( 3, 9, 6,15)( 4, 7,13,10)( 5, 8,14,11)$
$ 4, 4, 4, 1, 1, 1 $ $25$ $4$ $( 3,15, 6, 9)( 4,10,13, 7)( 5,11,14, 8)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 3, 3, 3, 3, 3 $ $25$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$
$ 6, 6, 3 $ $25$ $6$ $( 1, 2, 3, 4,14, 6)( 5,15, 7,11, 9,13)( 8,12,10)$
$ 12, 3 $ $25$ $12$ $( 1, 2, 3, 7,14,12,10, 5, 9, 4, 8,15)( 6,13,11)$
$ 12, 3 $ $25$ $12$ $( 1, 2, 3,13, 8, 6, 7, 5,12,10,14, 9)( 4,11,15)$
$ 3, 3, 3, 3, 3 $ $25$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)$
$ 6, 6, 3 $ $25$ $6$ $( 1, 3, 5,13, 6, 2)( 4,15, 8,10, 9,14)( 7,12,11)$
$ 12, 3 $ $25$ $12$ $( 1, 3, 8,13,12,11, 4, 9, 5, 7,15, 2)( 6,14,10)$
$ 12, 3 $ $25$ $12$ $( 1, 3,14, 7, 6, 8, 4,12,11,13, 9, 2)( 5,10,15)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$

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 2a 4a 4b 5a 3a  6a 12a 12b 3b  6b 12c 12d 5b
     2P 1a 1a 2a 2a 5a 3b  3b  6b  6b 3a  3a  6a  6a 5b
     3P 1a 2a 4b 4a 5a 1a  2a  4b  4a 1a  2a  4b  4a 5b
     5P 1a 2a 4a 4b 1a 3b  6b 12c 12d 3a  6a 12a 12b 1a
     7P 1a 2a 4b 4a 5a 3a  6a 12b 12a 3b  6b 12d 12c 5b
    11P 1a 2a 4b 4a 5a 3b  6b 12d 12c 3a  6a 12b 12a 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  A -A  1  1  -1   A  -A  1  -1   A  -A  1
X.4      1 -1 -A  A  1  1  -1  -A   A  1  -1  -A   A  1
X.5      1 -1  A -A  1  B  -B   C  -C /B -/B -/C  /C  1
X.6      1 -1  A -A  1 /B -/B -/C  /C  B  -B   C  -C  1
X.7      1 -1 -A  A  1  B  -B  -C   C /B -/B  /C -/C  1
X.8      1 -1 -A  A  1 /B -/B  /C -/C  B  -B  -C   C  1
X.9      1  1 -1 -1  1  B   B  -B  -B /B  /B -/B -/B  1
X.10     1  1 -1 -1  1 /B  /B -/B -/B  B   B  -B  -B  1
X.11     1  1  1  1  1  B   B   B   B /B  /B  /B  /B  1
X.12     1  1  1  1  1 /B  /B  /B  /B  B   B   B   B  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