Properties

Label 45T46
Order \(360\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_9\times F_5$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $46$
Group :  $D_9\times F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,36,15,12,16,5,28,26,32,19,44,42,2,35,14,11,17,4,30,25,33,21,43,41,3,34,13,10,18,6,29,27,31,20,45,40)(7,23,37,8,24,38,9,22,39), (1,26,19,45,37,17,10,34,30,7)(2,27,21,43,38,16,12,35,29,9)(3,25,20,44,39,18,11,36,28,8)(4,13,23,32,40)(5,15,22,33,41,6,14,24,31,42)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
18:  $D_{9}$
20:  $F_5$
40:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $F_5$

Degree 9: $D_{9}$

Degree 15: $F_5 \times S_3$

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

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [360, 39]
Character table: Data not available.