Properties

Label 45T45
Order \(360\)
n \(45\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_3:S_5$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $45$
Group :  $C_3:S_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,40,11,33)(2,42,10,31)(3,41,12,32)(4,8,23,36)(5,9,24,35)(6,7,22,34)(13,43)(14,45)(15,44)(16,38,27,19)(17,37,26,21)(18,39,25,20)(28,30), (1,43,26,35)(2,45,27,36)(3,44,25,34)(4,41,17,28)(5,42,18,30)(6,40,16,29)(7,19)(8,20)(9,21)(10,37,22,14)(11,39,23,15)(12,38,24,13)(32,33)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $S_5$

Degree 9: None

Degree 15: $S_5$, $\GL(2,4):C_2$ x 2, $\GL(2,4):C_2$

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 $ $15$ $2$ $( 4,14)( 5,13)( 6,15)( 7,38)( 8,37)( 9,39)(10,28)(11,29)(12,30)(19,42)(20,41) (21,40)(25,44)(26,45)(27,43)(31,34)(32,35)(33,36)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1 $ $90$ $4$ $( 2, 3)( 4,21,14,40)( 5,20,13,41)( 6,19,15,42)( 7,28,38,10)( 8,29,37,11) ( 9,30,39,12)(16,24)(17,23)(18,22)(25,32,44,35)(26,33,45,36)(27,31,43,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $30$ $2$ $( 2, 3)( 4,26)( 5,27)( 6,25)( 7,39)( 8,37)( 9,38)(10,12)(13,43)(14,45)(15,44) (16,24)(17,23)(18,22)(19,35)(20,34)(21,36)(28,30)(31,41)(32,42)(33,40)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,44,45)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $30$ $6$ $( 1, 2, 3)( 4,15, 5,14, 6,13)( 7,37, 9,38, 8,39)(10,30,11,28,12,29)(16,18,17) (19,40,20,42,21,41)(22,24,23)(25,45,27,44,26,43)(31,36,32,34,33,35)$
$ 6, 6, 6, 6, 6, 6, 6, 3 $ $60$ $6$ $( 1, 4,23,11,17,26)( 2, 5,22,12,16,25)( 3, 6,24,10,18,27)( 7,39,44,20,34,15) ( 8,37,45,21,36,14)( 9,38,43,19,35,13)(28,31,41,30,32,42)(29,33,40)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 4,21,40,14)( 2, 6,20,41,15)( 3, 5,19,42,13)( 7,38,44,24,25) ( 8,37,45,23,26)( 9,39,43,22,27)(10,32,35,28,16)(11,33,36,29,17) (12,31,34,30,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 4,36)( 2, 6,35)( 3, 5,34)( 7,31,19)( 8,33,21)( 9,32,20)(10,39,16) (11,37,17)(12,38,18)(13,30,44)(14,29,45)(15,28,43)(22,27,41)(23,26,40) (24,25,42)$
$ 15, 15, 15 $ $24$ $15$ $( 1, 5,20,40,13, 2, 4,19,41,14, 3, 6,21,42,15)( 7,39,45,24,27, 8,38,43,23,25, 9,37,44,22,26)(10,33,34,28,17,12,32,36,30,16,11,31,35,29,18)$
$ 15, 15, 15 $ $24$ $15$ $( 1, 5,28,37,31, 2, 4,30,39,33, 3, 6,29,38,32)( 7,16,11,44,41, 8,18,10,45,42, 9,17,12,43,40)(13,35,21,24,27,14,34,20,23,25,15,36,19,22,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 5,35)( 2, 4,34)( 3, 6,36)( 7,32,21)( 8,31,20)( 9,33,19)(10,37,18) (11,38,16)(12,39,17)(13,28,45)(14,30,43)(15,29,44)(22,26,42)(23,25,41) (24,27,40)$

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  3  2  2  2  2  1  .  1   .   .  .
      3  2  1  .  1  2  1  1  1  2   1   1  2
      5  1  .  .  .  1  .  .  1  .   1   1  .

        1a 2a 4a 2b 3a 6a 6b 5a 3b 15a 15b 3c
     2P 1a 1a 2a 1a 3a 3a 3b 5a 3b 15a 15b 3c
     3P 1a 2a 4a 2b 1a 2a 2b 5a 1a  5a  5a 1a
     5P 1a 2a 4a 2b 3a 6a 6b 1a 3b  3a  3a 3c
     7P 1a 2a 4a 2b 3a 6a 6b 5a 3b 15b 15a 3c
    11P 1a 2a 4a 2b 3a 6a 6b 5a 3b 15b 15a 3c
    13P 1a 2a 4a 2b 3a 6a 6b 5a 3b 15b 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  2  .  . -1 -1  .  2  2  -1  -1 -1
X.4      4  .  . -2  4  .  1 -1  1  -1  -1  1
X.5      4  .  .  2  4  . -1 -1  1  -1  -1  1
X.6      5  1 -1  1  5  1  1  . -1   .   . -1
X.7      5  1  1 -1  5  1 -1  . -1   .   . -1
X.8      6 -2  .  .  6 -2  .  1  .   1   1  .
X.9      6 -2  .  . -3  1  .  1  .   A  /A  .
X.10     6 -2  .  . -3  1  .  1  .  /A   A  .
X.11     8  .  .  . -4  .  . -2  2   1   1 -1
X.12    10  2  .  . -5 -1  .  . -2   .   .  1

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