Properties

Label 45T31
Order \(270\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_3\times C_{15}):S_3$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $31$
Group :  $(C_3\times C_{15}):S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,45,20,41,16,37,14,36,11,33,8,30,6,25)(2,24,44,21,42,18,39,15,34,12,31,7,29,5,27)(3,22,43,19,40,17,38,13,35,10,32,9,28,4,26), (1,43)(2,45)(3,44)(4,40)(5,42)(6,41)(7,38)(8,39)(9,37)(10,34)(11,35)(12,36)(13,32)(14,33)(15,31)(16,29)(17,30)(18,28)(19,27)(20,26)(21,25), (1,23)(2,24)(3,22)(4,19)(5,21)(6,20)(7,18)(8,16)(9,17)(10,13)(11,14)(12,15)(25,45)(26,43)(27,44)(28,40)(29,42)(30,41)(31,39)(32,38)(33,37)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
10:  $D_{5}$
18:  $C_3^2:C_2$
54:  $(C_3^2:C_3):C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 9: $(C_3^2:C_3):C_2$

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

Group invariants

Order:  $270=2 \cdot 3^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [270, 19]
Character table: Data not available.