Properties

Label 45T33
Order \(270\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{45}:C_3$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
10:  $D_{5}$
18:  $S_3\times C_3$
54:  $(C_9:C_3):C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 9: $(C_9:C_3):C_2$

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

Group invariants

Order:  $270=2 \cdot 3^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [270, 15]
Character table: Data not available.