Properties

Label 45T4
Order \(90\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{45}$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 9: $D_{9}$

Degree 15: $D_{15}$

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

Group invariants

Order:  $90=2 \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [90, 3]
Character table: Data not available.