Properties

Label 45T22
Order \(180\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5\times C_3:S_3$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 4
10:  $D_{5}$
12:  $D_{6}$ x 4
18:  $C_3^2:C_2$
20:  $D_{10}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$ x 4

Degree 5: $D_{5}$

Degree 9: $C_3^2:C_2$

Degree 15: $D_5\times S_3$ x 4

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

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 27]
Character table: Data not available.