Properties

Label 45T40
Order \(360\)
n \(45\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_3\times A_5$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $40$
Group :  $S_3\times A_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,30,38,33,2,4,28,37,32)(3,6,29,39,31)(7,17,11,45,40,9,16,10,43,42)(8,18,12,44,41)(13,35,19,22,26,14,36,21,24,25)(15,34,20,23,27), (1,18,24,3,16,23)(2,17,22)(4,20,30,6,19,29)(5,21,28)(7,15,26,8,13,27)(9,14,25)(10,32,45)(11,31,43,12,33,44)(34,40,39,36,41,37)(35,42,38)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
60:  $A_5$
120:  $A_5\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $A_5$

Degree 9: None

Degree 15: $A_5$, $A_5 \times S_3$

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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, 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, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 4,13)( 5,14)( 6,15)( 7,37)( 8,39)( 9,38)(10,28)(11,30)(12,29)(19,40)(20,41) (21,42)(25,45)(26,43)(27,44)(31,34)(32,35)(33,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 5, 6)( 8, 9)(10,12)(14,15)(17,18)(20,21)(22,23)(25,27)(28,29)(31,32) (34,35)(38,39)(41,42)(44,45)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2, 3)( 4,13)( 5,15)( 6,14)( 7,37)( 8,38)( 9,39)(10,29)(11,30)(12,28)(17,18) (19,40)(20,42)(21,41)(22,23)(25,44)(26,43)(27,45)(31,35)(32,34)(33,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,14,15)(16,17,18)(19,21,20) (22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,38,39)(40,42,41) (43,45,44)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $30$ $6$ $( 1, 2, 3)( 4,14, 6,13, 5,15)( 7,38, 8,37, 9,39)(10,29,11,28,12,30)(16,17,18) (19,42,20,40,21,41)(22,23,24)(25,44,26,45,27,43)(31,36,32,34,33,35)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 4,19,40,13)( 2, 5,21,42,14)( 3, 6,20,41,15)( 7,37,43,24,26) ( 8,39,44,23,27)( 9,38,45,22,25)(10,32,35,28,17)(11,33,36,30,16) (12,31,34,29,18)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 4,30,37,33)( 2, 5,28,38,32)( 3, 6,29,39,31)( 7,16,11,43,40) ( 8,18,12,44,41)( 9,17,10,45,42)(13,36,19,24,26)(14,35,21,22,25) (15,34,20,23,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 4,36)( 2, 5,35)( 3, 6,34)( 7,33,19)( 8,31,20)( 9,32,21)(10,38,17) (11,37,16)(12,39,18)(13,30,43)(14,28,45)(15,29,44)(22,25,42)(23,27,41) (24,26,40)$
$ 10, 10, 10, 5, 5, 5 $ $36$ $10$ $( 1, 4,19,40,13)( 2, 6,21,41,14, 3, 5,20,42,15)( 7,37,43,24,26) ( 8,38,44,22,27, 9,39,45,23,25)(10,31,35,29,17,12,32,34,28,18)(11,33,36,30,16)$
$ 10, 10, 10, 5, 5, 5 $ $36$ $10$ $( 1, 4,30,37,33)( 2, 6,28,39,32, 3, 5,29,38,31)( 7,16,11,43,40) ( 8,17,12,45,41, 9,18,10,44,42)(13,36,19,24,26)(14,34,21,23,25,15,35,20,22,27)$
$ 6, 6, 6, 6, 6, 3, 3, 3, 3, 3 $ $60$ $6$ $( 1, 4,36)( 2, 6,35, 3, 5,34)( 7,33,19)( 8,32,20, 9,31,21)(10,39,17,12,38,18) (11,37,16)(13,30,43)(14,29,45,15,28,44)(22,27,42,23,25,41)(24,26,40)$
$ 15, 15, 15 $ $24$ $15$ $( 1, 5,20,40,14, 3, 4,21,41,13, 2, 6,19,42,15)( 7,38,44,24,25, 8,37,45,23,26, 9,39,43,22,27)(10,31,36,28,18,11,32,34,30,17,12,33,35,29,16)$
$ 15, 15, 15 $ $24$ $15$ $( 1, 5,29,37,32, 3, 4,28,39,33, 2, 6,30,38,31)( 7,17,12,43,42, 8,16,10,44,40, 9,18,11,45,41)(13,35,20,24,25,15,36,21,23,26,14,34,19,22,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 5,34)( 2, 6,36)( 3, 4,35)( 7,32,20)( 8,33,21)( 9,31,19)(10,39,16) (11,38,18)(12,37,17)(13,28,44)(14,29,43)(15,30,45)(22,27,40)(23,26,42) (24,25,41)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [360, 121]
Character table:   
      2  3  3  3  3  2  2  1  1  1   1   1  1   .   .  .
      3  2  1  1  .  2  1  1  1  2   .   .  1   1   1  2
      5  1  .  1  .  1  .  1  1  .   1   1  .   1   1  .

        1a 2a 2b 2c 3a 6a 5a 5b 3b 10a 10b 6b 15a 15b 3c
     2P 1a 1a 1a 1a 3a 3a 5b 5a 3b  5b  5a 3b 15b 15a 3c
     3P 1a 2a 2b 2c 1a 2a 5b 5a 1a 10b 10a 2b  5b  5a 1a
     5P 1a 2a 2b 2c 3a 6a 1a 1a 3b  2b  2b 6b  3a  3a 3c
     7P 1a 2a 2b 2c 3a 6a 5b 5a 3b 10b 10a 6b 15b 15a 3c
    11P 1a 2a 2b 2c 3a 6a 5a 5b 3b 10a 10b 6b 15a 15b 3c
    13P 1a 2a 2b 2c 3a 6a 5b 5a 3b 10b 10a 6b 15b 15a 3c

X.1      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  1
X.3      2  2  .  . -1 -1  2  2  2   .   .  .  -1  -1 -1
X.4      3 -1 -3  1  3 -1  A *A  .  -A -*A  .   A  *A  .
X.5      3 -1 -3  1  3 -1 *A  A  . -*A  -A  .  *A   A  .
X.6      3 -1  3 -1  3 -1  A *A  .   A  *A  .   A  *A  .
X.7      3 -1  3 -1  3 -1 *A  A  .  *A   A  .  *A   A  .
X.8      4  . -4  .  4  . -1 -1  1   1   1 -1  -1  -1  1
X.9      4  .  4  .  4  . -1 -1  1  -1  -1  1  -1  -1  1
X.10     5  1 -5 -1  5  1  .  . -1   .   .  1   .   . -1
X.11     5  1  5  1  5  1  .  . -1   .   . -1   .   . -1
X.12     6 -2  .  . -3  1  B *B  .   .   .  .  -A -*A  .
X.13     6 -2  .  . -3  1 *B  B  .   .   .  . -*A  -A  .
X.14     8  .  .  . -4  . -2 -2  2   .   .  .   1   1 -1
X.15    10  2  .  . -5 -1  .  . -2   .   .  .   .   .  1

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