# Properties

 Label 45T42 Degree $45$ Order $360$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_5\times S_3^2$

## Group action invariants

 Degree $n$: $45$ Transitive number $t$: $42$ Group: $D_5\times S_3^2$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,27,6,29,7,33,10,35,15,38,17,42,20,43,24)(2,25,5,28,8,31,11,36,14,39,16,40,21,44,22,3,26,4,30,9,32,12,34,13,37,18,41,19,45,23), (1,28)(2,30)(3,29)(4,27)(5,26)(6,25)(7,23)(8,22)(9,24)(10,19)(11,21)(12,20)(13,17)(14,16)(15,18)(31,43)(32,45)(33,44)(34,41)(35,40)(36,42)(38,39), (1,4,16,20,31,34)(2,6,18,21,33,36)(3,5,17,19,32,35)(7,28,22,43,39,14)(8,29,23,45,38,13)(9,30,24,44,37,15)(10,40,26)(11,42,25)(12,41,27)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2
$8$:  $C_2^3$
$10$:  $D_{5}$
$12$:  $D_{6}$ x 6
$20$:  $D_{10}$ x 3
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 3: $S_3$ x 2

Degree 5: $D_{5}$

Degree 9: $S_3^2$

Degree 15: $D_5\times S_3$ 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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 4,13)( 5,14)( 6,15)( 7,27)( 8,26)( 9,25)(10,38)(11,37)(12,39)(19,28)(20,29) (21,30)(22,41)(23,40)(24,42)(34,45)(35,43)(36,44)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $2$ $( 4,36)( 5,34)( 6,35)( 7,24)( 8,22)( 9,23)(13,44)(14,45)(15,43)(16,32)(17,33) (18,31)(25,40)(26,41)(27,42)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $15$ $2$ $( 4,44)( 5,45)( 6,43)( 7,42)( 8,41)( 9,40)(10,38)(11,37)(12,39)(13,36)(14,34) (15,35)(16,32)(17,33)(18,31)(19,28)(20,29)(21,30)(22,26)(23,25)(24,27)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(11,12)(13,14)(16,18)(19,21)(22,23)(25,26)(28,30)(31,32) (34,36)(37,39)(40,41)(44,45)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $15$ $2$ $( 2, 3)( 4,14)( 5,13)( 6,15)( 7,27)( 8,25)( 9,26)(10,38)(11,39)(12,37)(16,18) (19,30)(20,29)(21,28)(22,40)(23,41)(24,42)(31,32)(34,44)(35,43)(36,45)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $9$ $2$ $( 2, 3)( 4,34)( 5,36)( 6,35)( 7,24)( 8,23)( 9,22)(11,12)(13,45)(14,44)(15,43) (16,31)(17,33)(18,32)(19,21)(25,41)(26,40)(27,42)(28,30)(37,39)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $45$ $2$ $( 2, 3)( 4,45)( 5,44)( 6,43)( 7,42)( 8,40)( 9,41)(10,38)(11,39)(12,37)(13,34) (14,36)(15,35)(16,31)(17,33)(18,32)(19,30)(20,29)(21,28)(22,25)(23,26)(24,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,20,21) (22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41) (43,45,44)$ $6, 6, 6, 6, 6, 6, 3, 3, 3$ $10$ $6$ $( 1, 2, 3)( 4,15, 5,13, 6,14)( 7,26, 9,27, 8,25)(10,37,12,38,11,39)(16,18,17) (19,29,21,28,20,30)(22,40,24,41,23,42)(31,33,32)(34,44,35,45,36,43)$ $6, 6, 6, 6, 6, 3, 3, 3, 3, 3$ $6$ $6$ $( 1, 2, 3)( 4,35, 5,36, 6,34)( 7,22, 9,24, 8,23)(10,11,12)(13,43,14,44,15,45) (16,31,17,32,18,33)(19,20,21)(25,42,26,40,27,41)(28,29,30)(37,39,38)$ $6, 6, 6, 6, 6, 6, 6, 3$ $30$ $6$ $( 1, 2, 3)( 4,43, 5,44, 6,45)( 7,41, 9,42, 8,40)(10,37,12,38,11,39) (13,35,14,36,15,34)(16,31,17,32,18,33)(19,29,21,28,20,30)(22,25,24,26,23,27)$ $30, 15$ $12$ $30$ $( 1, 4, 7,12,15,18,20,23,27,28,33,36,38,40,43, 3, 6, 9,10,13,17,19,24,25,29, 31,35,39,42,44)( 2, 5, 8,11,14,16,21,22,26,30,32,34,37,41,45)$ $6, 6, 6, 6, 6, 6, 6, 3$ $30$ $6$ $( 1, 4,17,19,33,36)( 2, 5,16,21,32,34)( 3, 6,18,20,31,35)( 7,28,24,44,38,13) ( 8,30,22,45,37,14)( 9,29,23,43,39,15)(10,40,27,12,42,25)(11,41,26)$ $10, 10, 10, 10, 5$ $18$ $10$ $( 1, 4,38,40,29,31,20,23,10,13)( 2, 5,37,41,30,32,21,22,11,14)( 3, 6,39,42,28, 33,19,24,12,15)( 7,25,43,18,35, 9,27,44,17,36)( 8,26,45,16,34)$ $15, 15, 15$ $8$ $15$ $( 1, 4, 8,10,13,16,20,23,26,29,31,34,38,40,45)( 2, 6, 9,11,15,18,21,24,25,30, 33,36,37,42,44)( 3, 5, 7,12,14,17,19,22,27,28,32,35,39,41,43)$ $6, 6, 6, 6, 6, 6, 3, 3, 3$ $20$ $6$ $( 1, 4,16,20,31,34)( 2, 6,18,21,33,36)( 3, 5,17,19,32,35)( 7,28,22,43,39,14) ( 8,29,23,45,38,13)( 9,30,24,44,37,15)(10,40,26)(11,42,25)(12,41,27)$ $30, 15$ $12$ $30$ $( 1, 4,37,42,28,32,20,23,11,15, 3, 5,38,40,30,33,19,22,10,13, 2, 6,39,41,29, 31,21,24,12,14)( 7,25,45,17,36, 8,27,44,16,35, 9,26,43,18,34)$ $15, 15, 15$ $4$ $15$ $( 1, 6, 7,10,15,17,20,24,27,29,33,35,38,42,43)( 2, 5, 8,11,14,16,21,22,26,30, 32,34,37,41,45)( 3, 4, 9,12,13,18,19,23,25,28,31,36,39,40,44)$ $6, 6, 6, 6, 6, 6, 3, 3, 3$ $10$ $6$ $( 1, 6,17,20,33,35)( 2, 5,16,21,32,34)( 3, 4,18,19,31,36)( 7,29,24,43,38,15) ( 8,30,22,45,37,14)( 9,28,23,44,39,13)(10,42,27)(11,41,26)(12,40,25)$ $10, 10, 10, 5, 5, 5$ $6$ $10$ $( 1, 6,38,42,29,33,20,24,10,15)( 2, 5,37,41,30,32,21,22,11,14)( 3, 4,39,40,28, 31,19,23,12,13)( 7,27,43,17,35)( 8,26,45,16,34)( 9,25,44,18,36)$ $15, 15, 15$ $4$ $15$ $( 1, 7,15,20,27,33,38,43, 6,10,17,24,29,35,42)( 2, 8,14,21,26,32,37,45, 5,11, 16,22,30,34,41)( 3, 9,13,19,25,31,39,44, 4,12,18,23,28,36,40)$ $10, 10, 10, 5, 5, 5$ $6$ $10$ $( 1, 7,29,35,10,17,38,43,20,27)( 2, 8,30,34,11,16,37,45,21,26)( 3, 9,28,36,12, 18,39,44,19,25)( 4,40,31,23,13)( 5,41,32,22,14)( 6,42,33,24,15)$ $30, 15$ $12$ $30$ $( 1, 7,15,20,27,33,38,43, 6,10,17,24,29,35,42)( 2, 9,14,19,26,31,37,44, 5,12, 16,23,30,36,41, 3, 8,13,21,25,32,39,45, 4,11,18,22,28,34,40)$ $10, 10, 10, 10, 5$ $18$ $10$ $( 1, 7,29,35,10,17,38,43,20,27)( 2, 9,30,36,11,18,37,44,21,25)( 3, 8,28,34,12, 16,39,45,19,26)( 4,41,31,22,13, 5,40,32,23,14)( 6,42,33,24,15)$ $15, 15, 15$ $8$ $15$ $( 1, 8,13,20,26,31,38,45, 4,10,16,23,29,34,40)( 2, 9,15,21,25,33,37,44, 6,11, 18,24,30,36,42)( 3, 7,14,19,27,32,39,43, 5,12,17,22,28,35,41)$ $30, 15$ $12$ $30$ $( 1, 8,28,35,11,18,38,45,19,27, 2, 9,29,34,12,17,37,44,20,26, 3, 7,30,36,10, 16,39,43,21,25)( 4,42,32,23,15, 5,40,33,22,13, 6,41,31,24,14)$ $5, 5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,10,20,29,38)( 2,11,21,30,37)( 3,12,19,28,39)( 4,13,23,31,40) ( 5,14,22,32,41)( 6,15,24,33,42)( 7,17,27,35,43)( 8,16,26,34,45) ( 9,18,25,36,44)$ $10, 10, 10, 5, 5, 5$ $6$ $10$ $( 1,10,20,29,38)( 2,12,21,28,37, 3,11,19,30,39)( 4,14,23,32,40, 5,13,22,31,41) ( 6,15,24,33,42)( 7,17,27,35,43)( 8,18,26,36,45, 9,16,25,34,44)$ $15, 15, 15$ $4$ $15$ $( 1,11,19,29,37, 3,10,21,28,38, 2,12,20,30,39)( 4,15,22,31,42, 5,13,24,32,40, 6,14,23,33,41)( 7,16,25,35,45, 9,17,26,36,43, 8,18,27,34,44)$ $6, 6, 6, 6, 6, 3, 3, 3, 3, 3$ $6$ $6$ $( 1,16,33, 2,17,32)( 3,18,31)( 4,19,36)( 5,20,34, 6,21,35)( 7,22,38, 8,24,37) ( 9,23,39)(10,26,42,11,27,41)(12,25,40)(13,28,44)(14,29,45,15,30,43)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,16,31)( 2,18,33)( 3,17,32)( 4,20,34)( 5,19,35)( 6,21,36)( 7,22,39) ( 8,23,38)( 9,24,37)(10,26,40)(11,25,42)(12,27,41)(13,29,45)(14,28,43) (15,30,44)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,17,33)( 2,16,32)( 3,18,31)( 4,19,36)( 5,21,34)( 6,20,35)( 7,24,38) ( 8,22,37)( 9,23,39)(10,27,42)(11,26,41)(12,25,40)(13,28,44)(14,30,45) (15,29,43)$ $15, 15, 15$ $4$ $15$ $( 1,19,37,10,28, 2,20,39,11,29, 3,21,38,12,30)( 4,22,42,13,32, 6,23,41,15,31, 5,24,40,14,33)( 7,25,45,17,36, 8,27,44,16,35, 9,26,43,18,34)$ $10, 10, 10, 5, 5, 5$ $6$ $10$ $( 1,19,38,12,29, 3,20,39,10,28)( 2,21,37,11,30)( 4,24,40,15,31, 6,23,42,13,33) ( 5,22,41,14,32)( 7,25,43,18,35, 9,27,44,17,36)( 8,26,45,16,34)$ $5, 5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,20,38,10,29)( 2,21,37,11,30)( 3,19,39,12,28)( 4,23,40,13,31) ( 5,22,41,14,32)( 6,24,42,15,33)( 7,27,43,17,35)( 8,26,45,16,34) ( 9,25,44,18,36)$

## Group invariants

 Order: $360=2^{3} \cdot 3^{2} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [360, 137]
 Character table: not available.