Properties

Label 45T24
Order \(180\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2\times F_5$

Learn more about

Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $24$
Group :  $C_3^2\times F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,44,22,10,16,15,37,25,31,28,8,42)(2,43,23,11,18,13,38,27,32,29,7,40)(3,45,24,12,17,14,39,26,33,30,9,41)(4,21,34)(5,19,36)(6,20,35), (1,32,17)(2,33,16)(3,31,18)(4,44,19,14,35,29)(5,45,20,13,34,28)(6,43,21,15,36,30)(7,12,22,27,39,42)(8,11,24,25,38,41)(9,10,23,26,37,40)
$|\Aut(F/K)|$:  $9$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$ x 4
4:  $C_4$
6:  $C_6$ x 4
9:  $C_3^2$
20:  $F_5$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $C_3$ x 4

Degree 5: $F_5$

Degree 9: $C_3^2$

Degree 15: $F_5\times C_3$ x 4

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

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 20]
Character table: Data not available.