Properties

Label 45T9
Order \(90\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5\times D_9$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $9$
Group :  $C_5\times D_9$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,41,20,13,37,32,11,6,28,22)(2,42,21,14,39,33,12,4,29,24)(3,40,19,15,38,31,10,5,30,23)(7,17,26,36,45,9,16,27,35,43)(8,18,25,34,44), (1,12,20,29,37,2,11,21,28,39)(3,10,19,30,38)(4,44,24,18,42,34,14,8,33,25)(5,43,23,17,40,36,15,9,31,27)(6,45,22,16,41,35,13,7,32,26)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
6:  $S_3$
10:  $C_{10}$
18:  $D_{9}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $C_5$

Degree 9: $D_{9}$

Degree 15: $S_3 \times C_5$

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

Group invariants

Order:  $90=2 \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [90, 1]
Character table: Data not available.