Properties

Label 45T43
Order \(360\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times S_3\times F_5$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$, $C_6$ x 3
8:  $C_4\times C_2$
12:  $D_{6}$
18:  $S_3\times C_3$
20:  $F_5$
40:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $C_3$, $S_3$

Degree 5: $F_5$

Degree 9: $S_3\times C_3$

Degree 15: $F_5\times C_3$, $F_5 \times S_3$

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

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [360, 126]
Character table: Data not available.