Properties

Label 45T47
Order \(360\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:(S_3\times F_5)$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $47$
Group :  $C_3:(S_3\times F_5)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,43,10,9,19,18,28,25,37,34)(2,44,12,8,21,17,30,26,38,35)(3,45,11,7,20,16,29,27,39,36)(4,33,13,40,23,5,31,14,42,22)(6,32,15,41,24), (1,15,7,12,31,44,37,41,16,30,23,26)(2,13,8,10,32,45,38,42,17,28,24,27)(3,14,9,11,33,43,39,40,18,29,22,25)(4,35,19,6,36,21)(5,34,20)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$ x 2
8:  $C_4\times C_2$
12:  $D_{6}$ x 2
20:  $F_5$
36:  $S_3^2$
40:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$ x 2

Degree 5: $F_5$

Degree 9: $S_3^2$

Degree 15: $F_5 \times S_3$ x 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 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $15$ $4$ $( 4, 7,13,27)( 5, 9,14,25)( 6, 8,15,26)(10,19,37,28)(11,20,39,29)(12,21,38,30) (16,31)(17,32)(18,33)(22,43,40,34)(23,45,42,36)(24,44,41,35)$
$ 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 $ $5$ $2$ $( 4,13)( 5,14)( 6,15)( 7,27)( 8,26)( 9,25)(10,37)(11,39)(12,38)(19,28)(20,29) (21,30)(22,40)(23,42)(24,41)(34,43)(35,44)(36,45)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $15$ $4$ $( 4,27,13, 7)( 5,25,14, 9)( 6,26,15, 8)(10,28,37,19)(11,29,39,20)(12,30,38,21) (16,31)(17,32)(18,33)(22,34,40,43)(23,36,42,45)(24,35,41,44)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $15$ $4$ $( 2, 3)( 4,23,13,42)( 5,24,14,41)( 6,22,15,40)( 7,45,27,36)( 8,43,26,34) ( 9,44,25,35)(10,19,37,28)(11,21,39,30)(12,20,38,29)(17,18)(32,33)$
$ 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 $ $9$ $2$ $( 2, 3)( 4,36)( 5,35)( 6,34)( 7,23)( 8,22)( 9,24)(11,12)(13,45)(14,44)(15,43) (16,31)(17,33)(18,32)(20,21)(25,41)(26,40)(27,42)(29,30)(38,39)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $15$ $4$ $( 2, 3)( 4,42,13,23)( 5,41,14,24)( 6,40,15,22)( 7,36,27,45)( 8,34,26,43) ( 9,35,25,44)(10,28,37,19)(11,30,39,21)(12,29,38,20)(17,18)(32,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $2$ $( 2, 3)( 4,45)( 5,44)( 6,43)( 7,42)( 8,40)( 9,41)(10,37)(11,38)(12,39)(13,36) (14,35)(15,34)(16,31)(17,33)(18,32)(19,28)(20,30)(21,29)(22,26)(23,27)(24,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,21,20) (22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,36,35)(37,38,39)(40,42,41) (43,45,44)$
$ 12, 12, 12, 6, 3 $ $30$ $12$ $( 1, 2, 3)( 4, 8,14,27, 6, 9,13,26, 5, 7,15,25)(10,21,39,28,12,20,37,30,11,19, 38,29)(16,32,18,31,17,33)(22,45,41,34,23,44,40,36,24,43,42,35)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $10$ $6$ $( 1, 2, 3)( 4,15, 5,13, 6,14)( 7,26, 9,27, 8,25)(10,38,11,37,12,39)(16,17,18) (19,30,20,28,21,29)(22,42,24,40,23,41)(31,32,33)(34,45,35,43,36,44)$
$ 12, 12, 12, 6, 3 $ $30$ $12$ $( 1, 2, 3)( 4,26,14, 7, 6,25,13, 8, 5,27,15, 9)(10,30,39,19,12,29,37,21,11,28, 38,20)(16,32,18,31,17,33)(22,36,41,43,23,35,40,45,24,34,42,44)$
$ 12, 12, 12, 6, 3 $ $30$ $12$ $( 1, 4,27,37,31,36,10,23,16,19,42, 7)( 2, 5,26,39,32,34,12,22,17,20,41, 9) ( 3, 6,25,38,33,35,11,24,18,21,40, 8)(13,45,28)(14,44,29,15,43,30)$
$ 10, 10, 10, 10, 5 $ $36$ $10$ $( 1, 4,37,42,28,31,19,23,10,13)( 2, 5,38,40,30,33,21,22,12,14)( 3, 6,39,41,29, 32,20,24,11,15)( 7,27,45,16,36)( 8,25,44,18,35, 9,26,43,17,34)$
$ 12, 12, 12, 6, 3 $ $30$ $12$ $( 1, 4,45,10,31,36,28,42,16,19,13,27)( 2, 5,44,11,32,34,30,40,17,20,15,25) ( 3, 6,43,12,33,35,29,41,18,21,14,26)( 7,37,23)( 8,39,24, 9,38,22)$
$ 15, 15, 15 $ $8$ $15$ $( 1, 4, 7,10,13,16,19,23,27,28,31,36,37,42,45)( 2, 6, 8,12,15,17,21,24,26,30, 32,35,38,41,44)( 3, 5, 9,11,14,18,20,22,25,29,33,34,39,40,43)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $10$ $6$ $( 1, 4,16,19,31,36)( 2, 6,17,21,32,35)( 3, 5,18,20,33,34)( 7,28,23,45,37,13) ( 8,30,24,44,38,15)( 9,29,22,43,39,14)(10,42,27)(11,40,25)(12,41,26)$
$ 15, 15, 15 $ $8$ $15$ $( 1, 5, 8,10,14,17,19,22,26,28,33,35,37,40,44)( 2, 4, 9,12,13,18,21,23,25,30, 31,34,38,42,43)( 3, 6, 7,11,15,16,20,24,27,29,32,36,39,41,45)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $20$ $6$ $( 1, 5,17,19,33,35)( 2, 4,18,21,31,34)( 3, 6,16,20,32,36)( 7,29,24,45,39,15) ( 8,28,22,44,37,14)( 9,30,23,43,38,13)(10,40,26)(11,41,27)(12,42,25)$
$ 15, 15, 15 $ $8$ $15$ $( 1, 6, 9,10,15,18,19,24,25,28,32,34,37,41,43)( 2, 5, 7,12,14,16,21,22,27,30, 33,36,38,40,45)( 3, 4, 8,11,13,17,20,23,26,29,31,35,39,42,44)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $4$ $5$ $( 1,10,19,28,37)( 2,12,21,30,38)( 3,11,20,29,39)( 4,13,23,31,42) ( 5,14,22,33,40)( 6,15,24,32,41)( 7,16,27,36,45)( 8,17,26,35,44) ( 9,18,25,34,43)$
$ 15, 15, 15 $ $8$ $15$ $( 1,11,21,28,39, 2,10,20,30,37, 3,12,19,29,38)( 4,14,24,31,40, 6,13,22,32,42, 5,15,23,33,41)( 7,18,26,36,43, 8,16,25,35,45, 9,17,27,34,44)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,16,31)( 2,17,32)( 3,18,33)( 4,19,36)( 5,20,34)( 6,21,35)( 7,23,37) ( 8,24,38)( 9,22,39)(10,27,42)(11,25,40)(12,26,41)(13,28,45)(14,29,43) (15,30,44)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,17,33)( 2,18,31)( 3,16,32)( 4,21,34)( 5,19,35)( 6,20,36)( 7,24,39) ( 8,22,37)( 9,23,38)(10,26,40)(11,27,41)(12,25,42)(13,30,43)(14,28,44) (15,29,45)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [360, 129]
Character table: Data not available.