Properties

Label 45T15
Order \(180\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5\times S_3^2$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $15$
Group :  $C_5\times S_3^2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,3,16,2,17)(4,6,5)(7,38,9,39,8,37)(10,25,12,26,11,27)(13,15,14)(19,34,21,35,20,36)(22,23,24)(28,43,30,44,29,45)(31,33,32)(40,42,41), (1,19,38,12,30)(2,21,37,11,28,3,20,39,10,29)(4,8,42,45,33,35,23,27,15,16)(5,9,40,44,31,34,22,26,13,18)(6,7,41,43,32,36,24,25,14,17)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
5:  $C_5$
6:  $S_3$ x 2
10:  $C_{10}$ x 3
12:  $D_{6}$ x 2
36:  $S_3^2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$ x 2

Degree 5: $C_5$

Degree 9: $S_3^2$

Degree 15: $S_3 \times C_5$ x 2

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

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 28]
Character table: Data not available.