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Properties

Label 45T40
Degree $45$
Order $360$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_3\times A_5$

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Show commands: Magma

magma: G := TransitiveGroup(45, 40);
 

Group action invariants

Degree $n$:  $45$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times A_5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,5,30,38,33,2,4,28,37,32)(3,6,29,39,31)(7,17,11,45,40,9,16,10,43,42)(8,18,12,44,41)(13,35,19,22,26,14,36,21,24,25)(15,34,20,23,27), (1,18,24,3,16,23)(2,17,22)(4,20,30,6,19,29)(5,21,28)(7,15,26,8,13,27)(9,14,25)(10,32,45)(11,31,43,12,33,44)(34,40,39,36,41,37)(35,42,38)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$60$:  $A_5$
$120$:  $A_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $A_5$

Degree 9: None

Degree 15: $A_5$, $A_5 \times S_3$

Low degree siblings

15T23, 18T145, 30T85, 30T94, 30T102, 36T551, 36T552, 36T553

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle 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 $ $15$ $2$ $( 4,13)( 5,14)( 6,15)( 7,37)( 8,39)( 9,38)(10,28)(11,30)(12,29)(19,40)(20,41) (21,42)(25,45)(26,43)(27,44)(31,34)(32,35)(33,36)$
$ 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 $ $3$ $2$ $( 2, 3)( 5, 6)( 8, 9)(10,12)(14,15)(17,18)(20,21)(22,23)(25,27)(28,29)(31,32) (34,35)(38,39)(41,42)(44,45)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2, 3)( 4,13)( 5,15)( 6,14)( 7,37)( 8,38)( 9,39)(10,29)(11,30)(12,28)(17,18) (19,40)(20,42)(21,41)(22,23)(25,44)(26,43)(27,45)(31,35)(32,34)(33,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,14,15)(16,17,18)(19,21,20) (22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,38,39)(40,42,41) (43,45,44)$
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ $30$ $6$ $( 1, 2, 3)( 4,14, 6,13, 5,15)( 7,38, 8,37, 9,39)(10,29,11,28,12,30)(16,17,18) (19,42,20,40,21,41)(22,23,24)(25,44,26,45,27,43)(31,36,32,34,33,35)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 4,19,40,13)( 2, 5,21,42,14)( 3, 6,20,41,15)( 7,37,43,24,26) ( 8,39,44,23,27)( 9,38,45,22,25)(10,32,35,28,17)(11,33,36,30,16) (12,31,34,29,18)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 4,30,37,33)( 2, 5,28,38,32)( 3, 6,29,39,31)( 7,16,11,43,40) ( 8,18,12,44,41)( 9,17,10,45,42)(13,36,19,24,26)(14,35,21,22,25) (15,34,20,23,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 4,36)( 2, 5,35)( 3, 6,34)( 7,33,19)( 8,31,20)( 9,32,21)(10,38,17) (11,37,16)(12,39,18)(13,30,43)(14,28,45)(15,29,44)(22,25,42)(23,27,41) (24,26,40)$
$ 10, 10, 10, 5, 5, 5 $ $36$ $10$ $( 1, 4,19,40,13)( 2, 6,21,41,14, 3, 5,20,42,15)( 7,37,43,24,26) ( 8,38,44,22,27, 9,39,45,23,25)(10,31,35,29,17,12,32,34,28,18)(11,33,36,30,16)$
$ 10, 10, 10, 5, 5, 5 $ $36$ $10$ $( 1, 4,30,37,33)( 2, 6,28,39,32, 3, 5,29,38,31)( 7,16,11,43,40) ( 8,17,12,45,41, 9,18,10,44,42)(13,36,19,24,26)(14,34,21,23,25,15,35,20,22,27)$
$ 6, 6, 6, 6, 6, 3, 3, 3, 3, 3 $ $60$ $6$ $( 1, 4,36)( 2, 6,35, 3, 5,34)( 7,33,19)( 8,32,20, 9,31,21)(10,39,17,12,38,18) (11,37,16)(13,30,43)(14,29,45,15,28,44)(22,27,42,23,25,41)(24,26,40)$
$ 15, 15, 15 $ $24$ $15$ $( 1, 5,20,40,14, 3, 4,21,41,13, 2, 6,19,42,15)( 7,38,44,24,25, 8,37,45,23,26, 9,39,43,22,27)(10,31,36,28,18,11,32,34,30,17,12,33,35,29,16)$
$ 15, 15, 15 $ $24$ $15$ $( 1, 5,29,37,32, 3, 4,28,39,33, 2, 6,30,38,31)( 7,17,12,43,42, 8,16,10,44,40, 9,18,11,45,41)(13,35,20,24,25,15,36,21,23,26,14,34,19,22,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 5,34)( 2, 6,36)( 3, 4,35)( 7,32,20)( 8,33,21)( 9,31,19)(10,39,16) (11,38,18)(12,37,17)(13,28,44)(14,29,43)(15,30,45)(22,27,40)(23,26,42) (24,25,41)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  360.121
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 5A1 5A2 6A 6B 10A1 10A3 15A1 15A2
Size 1 3 15 45 2 20 40 12 12 30 60 36 36 24 24
2 P 1A 1A 1A 1A 3A 3B 3C 5A2 5A1 3A 3B 5A1 5A2 15A2 15A1
3 P 1A 2A 2B 2C 1A 1A 1A 5A2 5A1 2B 2A 10A3 10A1 5A1 5A2
5 P 1A 2A 2B 2C 3A 3B 3C 1A 1A 6A 6B 2A 2A 3A 3A
Type
360.121.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
360.121.1b R 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 1 1
360.121.2a R 2 0 2 0 −1 2 −1 2 2 −1 0 0 0 −1 −1
360.121.3a1 R 3 3 −1 −1 3 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
360.121.3a2 R 3 3 −1 −1 3 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
360.121.3b1 R 3 −3 −1 1 3 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −1 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
360.121.3b2 R 3 −3 −1 1 3 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −1 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
360.121.4a R 4 4 0 0 4 1 1 −1 −1 0 1 −1 −1 −1 −1
360.121.4b R 4 −4 0 0 4 1 1 −1 −1 0 −1 1 1 −1 −1
360.121.5a R 5 5 1 1 5 −1 −1 0 0 1 −1 0 0 0 0
360.121.5b R 5 −5 1 −1 5 −1 −1 0 0 1 1 0 0 0 0
360.121.6a1 R 6 0 −2 0 −3 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 1 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
360.121.6a2 R 6 0 −2 0 −3 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 1 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
360.121.8a R 8 0 0 0 −4 2 −1 −2 −2 0 0 0 0 1 1
360.121.10a R 10 0 2 0 −5 −2 1 0 0 −1 0 0 0 0 0

magma: CharacterTable(G);