# Properties

 Label 45T25 Degree $45$ Order $180$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^2:C_{20}$

Show commands: Magma

magma: G := TransitiveGroup(45, 25);

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $C_{10}$
$20$:  20T1
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: None

Degree 5: $C_5$

Degree 9: $C_3^2:C_4$

Degree 15: None

## Low degree siblings

30T49 x 2

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $180=2^{2} \cdot 3^{2} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 180.23 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);