Properties

Label 15T17
Order \(300\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_5^2 : C_3):C_4$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $17$
Group :  $(C_5^2 : C_3):C_4$
CHM label :  $1/2[5^{2}:4]S(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11), (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T17, 25T28, 30T71 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, 2, 1 $ $75$ $4$ $( 2, 3, 8,15)( 4, 7,13,10)( 5, 9)( 6,14,12,11)$
$ 4, 4, 4, 2, 1 $ $75$ $4$ $( 2, 3,14, 9)( 4,10,13, 7)( 5,12,11,15)( 6, 8)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 3, 3, 3, 3, 3 $ $50$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$
$ 6, 6, 3 $ $50$ $6$ $( 1, 2, 3, 4,14, 6)( 5,15, 7,11, 9,13)( 8,12,10)$
$ 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, 23]
Character table:   
     2  2  2  2  2  .  1  1  .
     3  1  1  .  .  .  1  1  .
     5  2  .  .  .  2  .  .  2

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

X.1     1  1  1  1  1  1  1  1
X.2     1  1 -1 -1  1  1  1  1
X.3     1 -1  A -A  1  1 -1  1
X.4     1 -1 -A  A  1  1 -1  1
X.5     2 -2  .  .  2 -1  1  2
X.6     2  2  .  .  2 -1 -1  2
X.7    12  .  .  .  2  .  . -3
X.8    12  .  .  . -3  .  .  2

A = -E(4)
  = -Sqrt(-1) = -i