Properties

Label 45T48
Order \(360\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:S_3\times F_5$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $48$
Group :  $C_3:S_3\times F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22)(2,24)(3,23)(4,20)(5,19)(6,21)(7,18)(8,17)(9,16)(10,14)(11,13)(12,15)(25,45)(26,44)(27,43)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,36), (1,9,41,19,16,22,10,35,31,37,27,5)(2,7,40,21,17,23,12,34,32,38,26,4)(3,8,42,20,18,24,11,36,33,39,25,6)(13,29,45)(14,30,43)(15,28,44), (1,15,34,37,33,45,19,24,17,30,6,7)(2,14,36,38,31,44,21,22,18,29,5,8)(3,13,35,39,32,43,20,23,16,28,4,9)(10,42,26)(11,40,27)(12,41,25)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$ x 4
8:  $C_4\times C_2$
12:  $D_{6}$ x 4
18:  $C_3^2:C_2$
20:  $F_5$
40:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$ x 4

Degree 5: $F_5$

Degree 9: $C_3^2:C_2$

Degree 15: $F_5 \times S_3$ 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 $ $5$ $2$ $( 4,13)( 5,14)( 6,15)( 7,26)( 8,25)( 9,27)(10,37)(11,39)(12,38)(19,30)(20,28) (21,29)(22,41)(23,40)(24,42)(34,45)(35,43)(36,44)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $4$ $( 4,23,13,40)( 5,22,14,41)( 6,24,15,42)( 7,45,26,34)( 8,44,25,36)( 9,43,27,35) (10,19,37,30)(11,20,39,28)(12,21,38,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $4$ $( 4,40,13,23)( 5,41,14,22)( 6,42,15,24)( 7,34,26,45)( 8,36,25,44)( 9,35,27,43) (10,30,37,19)(11,28,39,20)(12,29,38,21)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1 $ $45$ $4$ $( 2, 3)( 4, 8,13,25)( 5, 9,14,27)( 6, 7,15,26)(10,19,37,30)(11,21,39,29) (12,20,38,28)(16,31)(17,33)(18,32)(22,43,41,35)(23,44,40,36)(24,45,42,34)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1 $ $45$ $4$ $( 2, 3)( 4,25,13, 8)( 5,27,14, 9)( 6,26,15, 7)(10,30,37,19)(11,29,39,21) (12,28,38,20)(16,31)(17,33)(18,32)(22,35,41,43)(23,36,40,44)(24,34,42,45)$
$ 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 $ $9$ $2$ $( 2, 3)( 4,36)( 5,35)( 6,34)( 7,24)( 8,23)( 9,22)(11,12)(13,44)(14,43)(15,45) (16,31)(17,33)(18,32)(20,21)(25,40)(26,42)(27,41)(28,29)(38,39)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $2$ $( 2, 3)( 4,44)( 5,43)( 6,45)( 7,42)( 8,40)( 9,41)(10,37)(11,38)(12,39)(13,36) (14,35)(15,34)(16,31)(17,33)(18,32)(19,30)(20,29)(21,28)(22,27)(23,25)(24,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,21,20) (22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,36,35)(37,38,39)(40,42,41) (43,45,44)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $10$ $6$ $( 1, 2, 3)( 4,15, 5,13, 6,14)( 7,25, 9,26, 8,27)(10,38,11,37,12,39)(16,17,18) (19,29,20,30,21,28)(22,40,24,41,23,42)(31,32,33)(34,44,35,45,36,43)$
$ 12, 12, 12, 3, 3, 3 $ $10$ $12$ $( 1, 2, 3)( 4,24,14,40, 6,22,13,42, 5,23,15,41)( 7,44,27,34, 8,43,26,36, 9,45, 25,35)(10,21,39,30,12,20,37,29,11,19,38,28)(16,17,18)(31,32,33)$
$ 12, 12, 12, 3, 3, 3 $ $10$ $12$ $( 1, 2, 3)( 4,42,14,23, 6,41,13,24, 5,40,15,22)( 7,36,27,45, 8,35,26,44, 9,34, 25,43)(10,29,39,19,12,28,37,21,11,30,38,20)(16,17,18)(31,32,33)$
$ 10, 10, 10, 10, 5 $ $36$ $10$ $( 1, 4,37,40,30,32,19,23,10,13)( 2, 5,38,41,29,31,21,22,12,14)( 3, 6,39,42,28, 33,20,24,11,15)( 7,27,45,16,34, 9,26,43,17,35)( 8,25,44,18,36)$
$ 15, 15, 15 $ $8$ $15$ $( 1, 4, 8,10,13,18,19,23,25,30,32,36,37,40,44)( 2, 6, 9,12,15,16,21,24,27,29, 33,35,38,42,43)( 3, 5, 7,11,14,17,20,22,26,28,31,34,39,41,45)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $10$ $6$ $( 1, 4,18,19,32,36)( 2, 6,16,21,33,35)( 3, 5,17,20,31,34)( 7,28,22,45,39,14) ( 8,30,23,44,37,13)( 9,29,24,43,38,15)(10,40,25)(11,41,26)(12,42,27)$
$ 12, 12, 12, 3, 3, 3 $ $10$ $12$ $( 1, 4,25,37,32,36,10,23,18,19,40, 8)( 2, 6,27,38,33,35,12,24,16,21,42, 9) ( 3, 5,26,39,31,34,11,22,17,20,41, 7)(13,44,30)(14,45,28)(15,43,29)$
$ 12, 12, 12, 3, 3, 3 $ $10$ $12$ $( 1, 4,44,10,32,36,30,40,18,19,13,25)( 2, 6,43,12,33,35,29,42,16,21,15,27) ( 3, 5,45,11,31,34,28,41,17,20,14,26)( 7,39,22)( 8,37,23)( 9,38,24)$
$ 15, 15, 15 $ $8$ $15$ $( 1, 5, 9,10,14,16,19,22,27,30,31,35,37,41,43)( 2, 4, 7,12,13,17,21,23,26,29, 32,34,38,40,45)( 3, 6, 8,11,15,18,20,24,25,28,33,36,39,42,44)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $10$ $6$ $( 1, 5,16,19,31,35)( 2, 4,17,21,32,34)( 3, 6,18,20,33,36)( 7,29,23,45,38,13) ( 8,28,24,44,39,15)( 9,30,22,43,37,14)(10,41,27)(11,42,25)(12,40,26)$
$ 12, 12, 12, 3, 3, 3 $ $10$ $12$ $( 1, 5,27,37,31,35,10,22,16,19,41, 9)( 2, 4,26,38,32,34,12,23,17,21,40, 7) ( 3, 6,25,39,33,36,11,24,18,20,42, 8)(13,45,29)(14,43,30)(15,44,28)$
$ 12, 12, 12, 3, 3, 3 $ $10$ $12$ $( 1, 5,43,10,31,35,30,41,16,19,14,27)( 2, 4,45,12,32,34,29,40,17,21,13,26) ( 3, 6,44,11,33,36,28,42,18,20,15,25)( 7,38,23)( 8,39,24)( 9,37,22)$
$ 15, 15, 15 $ $8$ $15$ $( 1, 6, 7,10,15,17,19,24,26,30,33,34,37,42,45)( 2, 5, 8,12,14,18,21,22,25,29, 31,36,38,41,44)( 3, 4, 9,11,13,16,20,23,27,28,32,35,39,40,43)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $10$ $6$ $( 1, 6,17,19,33,34)( 2, 5,18,21,31,36)( 3, 4,16,20,32,35)( 7,30,24,45,37,15) ( 8,29,22,44,38,14)( 9,28,23,43,39,13)(10,42,26)(11,40,27)(12,41,25)$
$ 12, 12, 12, 3, 3, 3 $ $10$ $12$ $( 1, 6,26,37,33,34,10,24,17,19,42, 7)( 2, 5,25,38,31,36,12,22,18,21,41, 8) ( 3, 4,27,39,32,35,11,23,16,20,40, 9)(13,43,28)(14,44,29)(15,45,30)$
$ 12, 12, 12, 3, 3, 3 $ $10$ $12$ $( 1, 6,45,10,33,34,30,42,17,19,15,26)( 2, 5,44,12,31,36,29,41,18,21,14,25) ( 3, 4,43,11,32,35,28,40,16,20,13,27)( 7,37,24)( 8,38,22)( 9,39,23)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $4$ $5$ $( 1,10,19,30,37)( 2,12,21,29,38)( 3,11,20,28,39)( 4,13,23,32,40) ( 5,14,22,31,41)( 6,15,24,33,42)( 7,17,26,34,45)( 8,18,25,36,44) ( 9,16,27,35,43)$
$ 15, 15, 15 $ $8$ $15$ $( 1,11,21,30,39, 2,10,20,29,37, 3,12,19,28,38)( 4,14,24,32,41, 6,13,22,33,40, 5,15,23,31,42)( 7,16,25,34,43, 8,17,27,36,45, 9,18,26,35,44)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,16,31)( 2,17,32)( 3,18,33)( 4,21,34)( 5,19,35)( 6,20,36)( 7,23,38) ( 8,24,39)( 9,22,37)(10,27,41)(11,25,42)(12,26,40)(13,29,45)(14,30,43) (15,28,44)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,17,33)( 2,18,31)( 3,16,32)( 4,20,35)( 5,21,36)( 6,19,34)( 7,24,37) ( 8,22,38)( 9,23,39)(10,26,42)(11,27,40)(12,25,41)(13,28,43)(14,29,44) (15,30,45)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,18,32)( 2,16,33)( 3,17,31)( 4,19,36)( 5,20,34)( 6,21,35)( 7,22,39) ( 8,23,37)( 9,24,38)(10,25,40)(11,26,41)(12,27,42)(13,30,44)(14,28,45) (15,29,43)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [360, 127]
Character table: Data not available.