Properties

Label 45T16
Order \(180\)
n \(45\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_3\times A_5$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $16$
Group :  $C_3\times A_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,18)(2,23,17)(3,24,16)(4,38,45)(5,39,43)(6,37,44)(7,20,33)(8,19,32)(9,21,31)(10,42,15)(11,40,13)(12,41,14)(25,35,30)(26,36,29)(27,34,28), (1,44,32,34,25,2,45,31,36,27,3,43,33,35,26)(4,16,13,7,37,5,18,14,8,38,6,17,15,9,39)(10,24,29,20,41,11,22,30,19,42,12,23,28,21,40)
$|\Aut(F/K)|$:  $9$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
60:  $A_5$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $C_3$

Degree 5: $A_5$

Degree 9: None

Degree 15: $A_5$, $\GL(2,4)$ x 2, $\GL(2,4)$

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

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [180, 19]
Character table:   
      2  2  2  2   2  2   2  .  .  .   .   .   .   .   .   .
      3  2  1  2   1  2   1  1  1  2   1   1   2   1   1   2
      5  1  .  1   .  1   .  1  1  .   1   1   .   1   1   .

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

X.1      1  1  1   1  1   1  1  1  1   1   1   1   1   1   1
X.2      1  1  A   A /A  /A  1  1  1   A   A   A  /A  /A  /A
X.3      1  1 /A  /A  A   A  1  1  1  /A  /A  /A   A   A   A
X.4      3 -1  3  -1  3  -1  E *E  .   E  *E   .   E  *E   .
X.5      3 -1  3  -1  3  -1 *E  E  .  *E   E   .  *E   E   .
X.6      3 -1  B  -A /B -/A  E *E  .   F   G   .  /F  /G   .
X.7      3 -1  B  -A /B -/A *E  E  .   G   F   .  /G  /F   .
X.8      3 -1 /B -/A  B  -A  E *E  .  /F  /G   .   F   G   .
X.9      3 -1 /B -/A  B  -A *E  E  .  /G  /F   .   G   F   .
X.10     4  .  4   .  4   . -1 -1  1  -1  -1   1  -1  -1   1
X.11     4  .  C   . /C   . -1 -1  1  -A  -A   A -/A -/A  /A
X.12     4  . /C   .  C   . -1 -1  1 -/A -/A  /A  -A  -A   A
X.13     5  1  5   1  5   1  .  . -1   .   .  -1   .   .  -1
X.14     5  1  D   A /D  /A  .  . -1   .   .  -A   .   . -/A
X.15     5  1 /D  /A  D   A  .  . -1   .   . -/A   .   .  -A

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3
D = 5*E(3)^2
  = (-5-5*Sqrt(-3))/2 = -5-5b3
E = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
F = -E(15)^7-E(15)^13
G = -E(15)-E(15)^4