Properties

Label 15T21
Order \(360\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $\GL(2,4):C_2$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $21$
Group :  $\GL(2,4):C_2$
CHM label :  $3S_{5}(15)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,2,15)(4,5,6)(8,9,10)(12,13,14), (1,4,10)(2,5,8)(3,7,11)(6,9,15)(12,14,13), (1,4)(2,6)(3,7)(5,15)(8,9)(12,13)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
120:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $S_5$

Low degree siblings

15T21, 15T22, 18T146, 30T89, 30T93 x 2, 30T101, 36T554, 45T45

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $30$ $2$ $( 3,15)( 4, 9)( 5,11)( 6, 8)( 7,10)(13,14)$
$ 3, 3, 3, 3, 1, 1, 1 $ $40$ $3$ $( 2, 3,15)( 4, 7, 5)( 9,11,10)(12,13,14)$
$ 4, 4, 4, 2, 1 $ $90$ $4$ $( 2, 5,12,11)( 3, 9,10,14)( 4,15,13, 7)( 6, 8)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $15$ $2$ $( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$
$ 15 $ $24$ $15$ $( 1, 2, 5, 3,12, 8, 9,15,11,14, 6, 4,10, 7,13)$
$ 6, 6, 3 $ $60$ $6$ $( 1, 2, 6, 9, 8, 4)( 3,10,12,11,15,14)( 5,13, 7)$
$ 6, 6, 3 $ $30$ $6$ $( 1, 2, 6, 4, 8, 9)( 3,14, 7,12,11,13)( 5,15,10)$
$ 5, 5, 5 $ $24$ $5$ $( 1, 2, 7,14, 5)( 3,13,15, 8, 9)( 4,11,12,10, 6)$
$ 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 2,12)( 3,11, 7)( 4,13, 6)( 5,10,15)( 8, 9,14)$
$ 15 $ $24$ $15$ $( 1, 2,13, 5,11, 6, 4,14,10, 3, 8, 9,12,15, 7)$
$ 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 6, 8)( 2, 4, 9)( 3, 7,11)( 5,10,15)(12,13,14)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [360, 120]
Character table:   
      2  3  2  .  2  3   .  1  2  .  1   .  2
      3  2  1  2  .  1   1  1  1  1  2   1  2
      5  1  .  .  .  .   1  .  .  1  .   1  1

        1a 2a 3a 4a 2b 15a 6a 6b 5a 3b 15b 3c
     2P 1a 1a 3a 2b 1a 15a 3b 3c 5a 3b 15b 3c
     3P 1a 2a 1a 4a 2b  5a 2a 2b 5a 1a  5a 1a
     5P 1a 2a 3a 4a 2b  3c 6a 6b 1a 3b  3c 3c
     7P 1a 2a 3a 4a 2b 15b 6a 6b 5a 3b 15a 3c
    11P 1a 2a 3a 4a 2b 15b 6a 6b 5a 3b 15a 3c
    13P 1a 2a 3a 4a 2b 15b 6a 6b 5a 3b 15a 3c

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

A = -E(15)-E(15)^2-E(15)^4-E(15)^8
  = (-1-Sqrt(-15))/2 = -1-b15