# Properties

 Label 45T45 Degree $45$ Order $360$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $C_3:S_5$

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magma: G := TransitiveGroup(45, 45);

## Group action invariants

 Degree $n$: $45$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $45$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_3:S_5$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,40,11,33)(2,42,10,31)(3,41,12,32)(4,8,23,36)(5,9,24,35)(6,7,22,34)(13,43)(14,45)(15,44)(16,38,27,19)(17,37,26,21)(18,39,25,20)(28,30), (1,43,26,35)(2,45,27,36)(3,44,25,34)(4,41,17,28)(5,42,18,30)(6,40,16,29)(7,19)(8,20)(9,21)(10,37,22,14)(11,39,23,15)(12,38,24,13)(32,33) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$120$:  $S_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 5: $S_5$

Degree 9: None

Degree 15: $S_5$, $\GL(2,4):C_2$ x 2, $\GL(2,4):C_2$

## Low degree siblings

15T21 x 2, 15T22, 18T146, 30T89, 30T93 x 2, 30T101, 36T554

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $15$ $2$ $( 4,14)( 5,13)( 6,15)( 7,38)( 8,37)( 9,39)(10,28)(11,29)(12,30)(19,42)(20,41) (21,40)(25,44)(26,45)(27,43)(31,34)(32,35)(33,36)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1$ $90$ $4$ $( 2, 3)( 4,21,14,40)( 5,20,13,41)( 6,19,15,42)( 7,28,38,10)( 8,29,37,11) ( 9,30,39,12)(16,24)(17,23)(18,22)(25,32,44,35)(26,33,45,36)(27,31,43,34)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $30$ $2$ $( 2, 3)( 4,26)( 5,27)( 6,25)( 7,39)( 8,37)( 9,38)(10,12)(13,43)(14,45)(15,44) (16,24)(17,23)(18,22)(19,35)(20,34)(21,36)(28,30)(31,41)(32,42)(33,40)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,44,45)$ $6, 6, 6, 6, 6, 6, 3, 3, 3$ $30$ $6$ $( 1, 2, 3)( 4,15, 5,14, 6,13)( 7,37, 9,38, 8,39)(10,30,11,28,12,29)(16,18,17) (19,40,20,42,21,41)(22,24,23)(25,45,27,44,26,43)(31,36,32,34,33,35)$ $6, 6, 6, 6, 6, 6, 6, 3$ $60$ $6$ $( 1, 4,23,11,17,26)( 2, 5,22,12,16,25)( 3, 6,24,10,18,27)( 7,39,44,20,34,15) ( 8,37,45,21,36,14)( 9,38,43,19,35,13)(28,31,41,30,32,42)(29,33,40)$ $5, 5, 5, 5, 5, 5, 5, 5, 5$ $24$ $5$ $( 1, 4,21,40,14)( 2, 6,20,41,15)( 3, 5,19,42,13)( 7,38,44,24,25) ( 8,37,45,23,26)( 9,39,43,22,27)(10,32,35,28,16)(11,33,36,29,17) (12,31,34,30,18)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $20$ $3$ $( 1, 4,36)( 2, 6,35)( 3, 5,34)( 7,31,19)( 8,33,21)( 9,32,20)(10,39,16) (11,37,17)(12,38,18)(13,30,44)(14,29,45)(15,28,43)(22,27,41)(23,26,40) (24,25,42)$ $15, 15, 15$ $24$ $15$ $( 1, 5,20,40,13, 2, 4,19,41,14, 3, 6,21,42,15)( 7,39,45,24,27, 8,38,43,23,25, 9,37,44,22,26)(10,33,34,28,17,12,32,36,30,16,11,31,35,29,18)$ $15, 15, 15$ $24$ $15$ $( 1, 5,28,37,31, 2, 4,30,39,33, 3, 6,29,38,32)( 7,16,11,44,41, 8,18,10,45,42, 9,17,12,43,40)(13,35,21,24,27,14,34,20,23,25,15,36,19,22,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $40$ $3$ $( 1, 5,35)( 2, 4,34)( 3, 6,36)( 7,32,21)( 8,31,20)( 9,33,19)(10,37,18) (11,38,16)(12,39,17)(13,28,45)(14,30,43)(15,29,44)(22,26,42)(23,25,41) (24,27,40)$

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); Label: 360.120 magma: IdentifyGroup(G);
 Character table:  2 3 3 2 2 2 2 1 . 1 . . . 3 2 1 . 1 2 1 1 1 2 1 1 2 5 1 . . . 1 . . 1 . 1 1 . 1a 2a 4a 2b 3a 6a 6b 5a 3b 15a 15b 3c 2P 1a 1a 2a 1a 3a 3a 3b 5a 3b 15a 15b 3c 3P 1a 2a 4a 2b 1a 2a 2b 5a 1a 5a 5a 1a 5P 1a 2a 4a 2b 3a 6a 6b 1a 3b 3a 3a 3c 7P 1a 2a 4a 2b 3a 6a 6b 5a 3b 15b 15a 3c 11P 1a 2a 4a 2b 3a 6a 6b 5a 3b 15b 15a 3c 13P 1a 2a 4a 2b 3a 6a 6b 5a 3b 15b 15a 3c X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 -1 1 1 1 1 1 X.3 2 2 . . -1 -1 . 2 2 -1 -1 -1 X.4 4 . . -2 4 . 1 -1 1 -1 -1 1 X.5 4 . . 2 4 . -1 -1 1 -1 -1 1 X.6 5 1 -1 1 5 1 1 . -1 . . -1 X.7 5 1 1 -1 5 1 -1 . -1 . . -1 X.8 6 -2 . . 6 -2 . 1 . 1 1 . X.9 6 -2 . . -3 1 . 1 . A /A . X.10 6 -2 . . -3 1 . 1 . /A A . X.11 8 . . . -4 . . -2 2 1 1 -1 X.12 10 2 . . -5 -1 . . -2 . . 1 A = -E(15)-E(15)^2-E(15)^4-E(15)^8 = (-1-Sqrt(-15))/2 = -1-b15 

magma: CharacterTable(G);