Properties

Label 45T34
Order \(270\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $He_3:D_5$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $34$
Group :  $He_3:D_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,43,42,39,36,33,28,27,22,21,16,15,12,8,4)(2,45,40,38,35,32,30,26,24,20,18,13,11,9,5)(3,44,41,37,34,31,29,25,23,19,17,14,10,7,6), (1,12)(2,10)(3,11)(4,24)(5,22)(6,23)(7,34)(8,35)(9,36)(13,15)(16,26)(17,25)(18,27)(19,38)(20,37)(21,39)(29,30)(31,41)(32,42)(33,40)(43,45)
$|\Aut(F/K)|$:  $1$

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:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $C_3$

Degree 5: $D_{5}$

Degree 9: $C_3^2 : S_3 $

Degree 15: $D_5\times C_3$

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

Group invariants

Order:  $270=2 \cdot 3^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [270, 14]
Character table: Data not available.