Properties

Label 45T13
Order \(180\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times D_{15}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
10:  $D_{5}$
12:  $D_{6}$ x 2
20:  $D_{10}$
36:  $S_3^2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$ x 2

Degree 5: $D_{5}$

Degree 9: $S_3^2$

Degree 15: $D_{15}$, $D_5\times S_3$

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

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 29]
Character table: Data not available.