Properties

Label 45T44
Order \(360\)
n \(45\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_3\times S_5$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $44$
Group :  $C_3\times S_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,17)(2,22,16)(3,24,18)(4,29,19)(5,30,20)(6,28,21)(7,26,13)(8,27,14)(9,25,15)(10,44,31)(11,43,32)(12,45,33)(34,39,41)(35,37,40)(36,38,42), (1,40,11,33)(2,41,10,32)(3,42,12,31)(4,9,23,35)(5,8,24,36)(6,7,22,34)(13,45)(14,43)(15,44)(16,39,26,20)(17,37,25,21)(18,38,27,19)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
120:  $S_5$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $C_3$

Degree 5: $S_5$

Degree 9: None

Degree 15: $S_5$, $S_5 \times C_3$

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

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [360, 119]
Character table: Data not available.