Properties

Label 45T50
Order \(360\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5\times C_3:S_3.C_2$

Learn more about

Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $50$
Group :  $D_5\times C_3:S_3.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,39,6,19)(2,38,7,18)(3,37,8,17)(4,36,9,16)(5,40,10,20)(11,24,26,34)(12,23,27,33)(13,22,28,32)(14,21,29,31)(15,25,30,35)(41,44)(42,43), (1,41)(2,45)(3,44)(4,43)(5,42)(7,10)(8,9)(11,31)(12,35)(13,34)(14,33)(15,32)(16,26)(17,30)(18,29)(19,28)(20,27)(21,36)(22,40)(23,39)(24,38)(25,37)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
10:  $D_{5}$
20:  $D_{10}$
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 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ $9$ $4$ $( 6,21,41,26)( 7,22,42,27)( 8,23,43,28)( 9,24,44,29)(10,25,45,30)(11,16,36,31) (12,17,37,32)(13,18,38,33)(14,19,39,34)(15,20,40,35)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ $9$ $4$ $( 6,26,41,21)( 7,27,42,22)( 8,28,43,23)( 9,29,44,24)(10,30,45,25)(11,31,36,16) (12,32,37,17)(13,33,38,18)(14,34,39,19)(15,35,40,20)$
$ 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 $ $9$ $2$ $( 6,41)( 7,42)( 8,43)( 9,44)(10,45)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31) (17,32)(18,33)(19,34)(20,35)(21,26)(22,27)(23,28)(24,29)(25,30)$
$ 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 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)$
$ 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 $ $45$ $4$ $( 2, 5)( 3, 4)( 6,26,41,21)( 7,30,42,25)( 8,29,43,24)( 9,28,44,23) (10,27,45,22)(11,31,36,16)(12,35,37,20)(13,34,38,19)(14,33,39,18)(15,32,40,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $2$ $( 2, 5)( 3, 4)( 6,41)( 7,45)( 8,44)( 9,43)(10,42)(11,36)(12,40)(13,39)(14,38) (15,37)(16,31)(17,35)(18,34)(19,33)(20,32)(21,26)(22,30)(23,29)(24,28)(25,27)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40) (41,42,43,44,45)$
$ 20, 20, 5 $ $18$ $20$ $( 1, 2, 3, 4, 5)( 6,22,43,29,10,21,42,28, 9,25,41,27, 8,24,45,26, 7,23,44,30) (11,17,38,34,15,16,37,33,14,20,36,32,13,19,40,31,12,18,39,35)$
$ 20, 20, 5 $ $18$ $20$ $( 1, 2, 3, 4, 5)( 6,27,43,24,10,26,42,23, 9,30,41,22, 8,29,45,21, 7,28,44,25) (11,32,38,19,15,31,37,18,14,35,36,17,13,34,40,16,12,33,39,20)$
$ 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 $ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)(36,38,40,37,39) (41,43,45,42,44)$
$ 20, 20, 5 $ $18$ $20$ $( 1, 3, 5, 2, 4)( 6,23,45,27, 9,21,43,30, 7,24,41,28,10,22,44,26, 8,25,42,29) (11,18,40,32,14,16,38,35,12,19,36,33,15,17,39,31,13,20,37,34)$
$ 20, 20, 5 $ $18$ $20$ $( 1, 3, 5, 2, 4)( 6,28,45,22, 9,26,43,25, 7,29,41,23,10,27,44,21, 8,30,42,24) (11,33,40,17,14,31,38,20,12,34,36,18,15,32,39,16,13,35,37,19)$
$ 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 $ $4$ $3$ $( 1, 6,41)( 2, 7,42)( 3, 8,43)( 4, 9,44)( 5,10,45)(11,16,21)(12,17,22) (13,18,23)(14,19,24)(15,20,25)(26,31,36)(27,32,37)(28,33,38)(29,34,39) (30,35,40)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $20$ $6$ $( 1, 6,41)( 2,10,42, 5, 7,45)( 3, 9,43, 4, 8,44)(11,16,21)(12,20,22,15,17,25) (13,19,23,14,18,24)(26,31,36)(27,35,37,30,32,40)(28,34,38,29,33,39)$
$ 15, 15, 15 $ $8$ $15$ $( 1, 7,43, 4,10,41, 2, 8,44, 5, 6,42, 3, 9,45)(11,17,23,14,20,21,12,18,24,15, 16,22,13,19,25)(26,32,38,29,35,36,27,33,39,30,31,37,28,34,40)$
$ 15, 15, 15 $ $8$ $15$ $( 1, 8,45, 2, 9,41, 3,10,42, 4, 6,43, 5, 7,44)(11,18,25,12,19,21,13,20,22,14, 16,23,15,17,24)(26,33,40,27,34,36,28,35,37,29,31,38,30,32,39)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,11,36)( 2,12,37)( 3,13,38)( 4,14,39)( 5,15,40)( 6,16,26)( 7,17,27) ( 8,18,28)( 9,19,29)(10,20,30)(21,31,41)(22,32,42)(23,33,43)(24,34,44) (25,35,45)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $20$ $6$ $( 1,11,36)( 2,15,37, 5,12,40)( 3,14,38, 4,13,39)( 6,16,26)( 7,20,27,10,17,30) ( 8,19,28, 9,18,29)(21,31,41)(22,35,42,25,32,45)(23,34,43,24,33,44)$
$ 15, 15, 15 $ $8$ $15$ $( 1,12,38, 4,15,36, 2,13,39, 5,11,37, 3,14,40)( 6,17,28, 9,20,26, 7,18,29,10, 16,27, 8,19,30)(21,32,43,24,35,41,22,33,44,25,31,42,23,34,45)$
$ 15, 15, 15 $ $8$ $15$ $( 1,13,40, 2,14,36, 3,15,37, 4,11,38, 5,12,39)( 6,18,30, 7,19,26, 8,20,27, 9, 16,28,10,17,29)(21,33,45,22,34,41,23,35,42,24,31,43,25,32,44)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [360, 130]
Character table: Data not available.