Properties

Label 45T26
Order \(180\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:(C_5:C_4)$

Learn more about

Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $26$
Group :  $C_3^2:(C_5:C_4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,39,12,5,38,11,4,37,15,3,36,14,2,40,13)(6,29,17,10,28,16,9,27,20,8,26,19,7,30,18)(21,44,32,25,43,31,24,42,35,23,41,34,22,45,33), (1,36,6,16)(2,40,7,20)(3,39,8,19)(4,38,9,18)(5,37,10,17)(11,21,26,31)(12,25,27,35)(13,24,28,34)(14,23,29,33)(15,22,30,32)(42,45)(43,44)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: None

Degree 5: $D_{5}$

Degree 9: $C_3^2:C_4$

Degree 15: None

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

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 24]
Character table:   
      2  2  2  2  2  1   1  1   1  .   .   .   .   .  .   .   .   .   .
      3  2  .  .  .  2   .  2   .  2   2   2   2   2  2   2   2   2   2
      5  1  1  .  .  1   1  1   1  1   1   1   1   1  1   1   1   1   1

        1a 2a 4a 4b 5a 10a 5b 10b 3a 15a 15b 15c 15d 3b 15e 15f 15g 15h
     2P 1a 1a 2a 2a 5b  5b 5a  5a 3a 15b 15d 15a 15c 3b 15f 15h 15e 15g
     3P 1a 2a 4b 4a 5b 10b 5a 10a 1a  5b  5a  5a  5b 1a  5b  5a  5a  5b
     5P 1a 2a 4a 4b 1a  2a 1a  2a 3a  3a  3a  3a  3a 3b  3b  3b  3b  3b
     7P 1a 2a 4b 4a 5b 10b 5a 10a 3a 15b 15d 15a 15c 3b 15f 15h 15e 15g
    11P 1a 2a 4b 4a 5a 10a 5b 10b 3a 15a 15b 15c 15d 3b 15e 15f 15g 15h
    13P 1a 2a 4a 4b 5b 10b 5a 10a 3a 15c 15a 15d 15b 3b 15g 15e 15h 15f

X.1      1  1  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   1   1   1
X.3      1 -1  A -A  1  -1  1  -1  1   1   1   1   1  1   1   1   1   1
X.4      1 -1 -A  A  1  -1  1  -1  1   1   1   1   1  1   1   1   1   1
X.5      2 -2  .  .  B  -B *B -*B  2   B  *B  *B   B  2   B  *B  *B   B
X.6      2 -2  .  . *B -*B  B  -B  2  *B   B   B  *B  2  *B   B   B  *B
X.7      2  2  .  .  B   B *B  *B  2   B  *B  *B   B  2   B  *B  *B   B
X.8      2  2  .  . *B  *B  B   B  2  *B   B   B  *B  2  *B   B   B  *B
X.9      4  .  .  .  4   .  4   . -2  -2  -2  -2  -2  1   1   1   1   1
X.10     4  .  .  .  4   .  4   .  1   1   1   1   1 -2  -2  -2  -2  -2
X.11     4  .  .  .  C   . *C   . -2  -B -*B -*B  -B  1   D  /E   E  /D
X.12     4  .  .  .  C   . *C   . -2  -B -*B -*B  -B  1  /D   E  /E   D
X.13     4  .  .  . *C   .  C   . -2 -*B  -B  -B -*B  1   E   D  /D  /E
X.14     4  .  .  . *C   .  C   . -2 -*B  -B  -B -*B  1  /E  /D   D   E
X.15     4  .  .  .  C   . *C   .  1   D  /E   E  /D -2  -B -*B -*B  -B
X.16     4  .  .  .  C   . *C   .  1  /D   E  /E   D -2  -B -*B -*B  -B
X.17     4  .  .  . *C   .  C   .  1   E   D  /D  /E -2 -*B  -B  -B -*B
X.18     4  .  .  . *C   .  C   .  1  /E  /D   D   E -2 -*B  -B  -B -*B

A = -E(4)
  = -Sqrt(-1) = -i
B = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
C = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5
D = -E(5)^2+2*E(5)^3
E = -E(5)+2*E(5)^4