Properties

Label 45T35
Order \(270\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5\times He_3$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$ x 4
6:  $C_6$ x 4
9:  $C_3^2$
10:  $D_{5}$
27:  $C_3^2:C_3$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $C_3$

Degree 5: $D_{5}$

Degree 9: $C_3^2:C_3$

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

Group invariants

Order:  $270=2 \cdot 3^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [270, 6]
Character table: Data not available.