Properties

Label 45T20
Order \(180\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5\times D_9$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $20$
Group :  $D_5\times D_9$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,25)(2,26)(3,27)(4,24)(5,22)(6,23)(7,21)(8,20)(9,19)(10,17)(11,18)(12,16)(14,15)(28,44)(29,45)(30,43)(31,42)(32,41)(33,40)(34,37)(35,39)(36,38), (1,40,19,14,37,32,12,4,30,22)(2,41,20,15,39,33,10,5,28,24)(3,42,21,13,38,31,11,6,29,23)(7,16,27,34,45,9,18,25,36,43)(8,17,26,35,44)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
10:  $D_{5}$
12:  $D_{6}$
18:  $D_{9}$
20:  $D_{10}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 9: $D_{9}$

Degree 15: $D_5\times S_3$

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

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 7]
Character table: Data not available.