Show commands:
Magma
magma: G := TransitiveGroup(45, 2);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $2$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_3\times C_{15}$ | ||
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $45$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,7,13,21,27,32,39,44,4,10,18,22,30,36,40)(2,8,15,20,26,31,37,43,6,12,16,24,29,35,42)(3,9,14,19,25,33,38,45,5,11,17,23,28,34,41), (1,35,23,10,43,33,21,8,41,30,16,5,39,26,14)(2,34,22,12,45,32,20,9,40,29,17,4,37,25,13)(3,36,24,11,44,31,19,7,42,28,18,6,38,27,15) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $5$: $C_5$ $9$: $C_3^2$ $15$: $C_{15}$ x 4 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$ x 4
Degree 5: $C_5$
Degree 9: $C_3^2$
Degree 15: $C_{15}$ x 4
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,38,39)(40,42,41) (43,45,44)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,39,38)(40,41,42) (43,44,45)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1, 4, 7,10,13,18,21,22,27,30,32,36,39,40,44)( 2, 6, 8,12,15,16,20,24,26,29, 31,35,37,42,43)( 3, 5, 9,11,14,17,19,23,25,28,33,34,38,41,45)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1, 5, 8,10,14,16,21,23,26,30,33,35,39,41,43)( 2, 4, 9,12,13,17,20,22,25,29, 32,34,37,40,45)( 3, 6, 7,11,15,18,19,24,27,28,31,36,38,42,44)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1, 6, 9,10,15,17,21,24,25,30,31,34,39,42,45)( 2, 5, 7,12,14,18,20,23,27,29, 33,36,37,41,44)( 3, 4, 8,11,13,16,19,22,26,28,32,35,38,40,43)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1, 7,13,21,27,32,39,44, 4,10,18,22,30,36,40)( 2, 8,15,20,26,31,37,43, 6,12, 16,24,29,35,42)( 3, 9,14,19,25,33,38,45, 5,11,17,23,28,34,41)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1, 8,14,21,26,33,39,43, 5,10,16,23,30,35,41)( 2, 9,13,20,25,32,37,45, 4,12, 17,22,29,34,40)( 3, 7,15,19,27,31,38,44, 6,11,18,24,28,36,42)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1, 9,15,21,25,31,39,45, 6,10,17,24,30,34,42)( 2, 7,14,20,27,33,37,44, 5,12, 18,23,29,36,41)( 3, 8,13,19,26,32,38,43, 4,11,16,22,28,35,40)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,10,21,30,39)( 2,12,20,29,37)( 3,11,19,28,38)( 4,13,22,32,40) ( 5,14,23,33,41)( 6,15,24,31,42)( 7,18,27,36,44)( 8,16,26,35,43) ( 9,17,25,34,45)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,11,20,30,38, 2,10,19,29,39, 3,12,21,28,37)( 4,14,24,32,41, 6,13,23,31,40, 5,15,22,33,42)( 7,17,26,36,45, 8,18,25,35,44, 9,16,27,34,43)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,12,19,30,37, 3,10,20,28,39, 2,11,21,29,38)( 4,15,23,32,42, 5,13,24,33,40, 6,14,22,31,41)( 7,16,25,36,43, 9,18,26,34,44, 8,17,27,35,45)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,13,27,39, 4,18,30,40, 7,21,32,44,10,22,36)( 2,15,26,37, 6,16,29,42, 8,20, 31,43,12,24,35)( 3,14,25,38, 5,17,28,41, 9,19,33,45,11,23,34)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,14,26,39, 5,16,30,41, 8,21,33,43,10,23,35)( 2,13,25,37, 4,17,29,40, 9,20, 32,45,12,22,34)( 3,15,27,38, 6,18,28,42, 7,19,31,44,11,24,36)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,15,25,39, 6,17,30,42, 9,21,31,45,10,24,34)( 2,14,27,37, 5,18,29,41, 7,20, 33,44,12,23,36)( 3,13,26,38, 4,16,28,40, 8,19,32,43,11,22,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,16,33)( 2,17,32)( 3,18,31)( 4,20,34)( 5,21,35)( 6,19,36)( 7,24,38) ( 8,23,39)( 9,22,37)(10,26,41)(11,27,42)(12,25,40)(13,29,45)(14,30,43) (15,28,44)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,17,31)( 2,18,33)( 3,16,32)( 4,19,35)( 5,20,36)( 6,21,34)( 7,23,37) ( 8,22,38)( 9,24,39)(10,25,42)(11,26,40)(12,27,41)(13,28,43)(14,29,44) (15,30,45)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,18,32)( 2,16,31)( 3,17,33)( 4,21,36)( 5,19,34)( 6,20,35)( 7,22,39) ( 8,24,37)( 9,23,38)(10,27,40)(11,25,41)(12,26,42)(13,30,44)(14,28,45) (15,29,43)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,19,37,10,28, 2,21,38,12,30, 3,20,39,11,29)( 4,23,42,13,33, 6,22,41,15,32, 5,24,40,14,31)( 7,25,43,18,34, 8,27,45,16,36, 9,26,44,17,35)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,20,38,10,29, 3,21,37,11,30, 2,19,39,12,28)( 4,24,41,13,31, 5,22,42,14,32, 6,23,40,15,33)( 7,26,45,18,35, 9,27,43,17,36, 8,25,44,16,34)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,21,39,10,30)( 2,20,37,12,29)( 3,19,38,11,28)( 4,22,40,13,32) ( 5,23,41,14,33)( 6,24,42,15,31)( 7,27,44,18,36)( 8,26,43,16,35) ( 9,25,45,17,34)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,22,44,21,40,18,39,13,36,10,32, 7,30, 4,27)( 2,24,43,20,42,16,37,15,35,12, 31, 8,29, 6,26)( 3,23,45,19,41,17,38,14,34,11,33, 9,28, 5,25)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,23,43,21,41,16,39,14,35,10,33, 8,30, 5,26)( 2,22,45,20,40,17,37,13,34,12, 32, 9,29, 4,25)( 3,24,44,19,42,18,38,15,36,11,31, 7,28, 6,27)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,24,45,21,42,17,39,15,34,10,31, 9,30, 6,25)( 2,23,44,20,41,18,37,14,36,12, 33, 7,29, 5,27)( 3,22,43,19,40,16,38,13,35,11,32, 8,28, 4,26)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,25, 6,30, 9,31,10,34,15,39,17,42,21,45,24)( 2,27, 5,29, 7,33,12,36,14,37, 18,41,20,44,23)( 3,26, 4,28, 8,32,11,35,13,38,16,40,19,43,22)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,26, 5,30, 8,33,10,35,14,39,16,41,21,43,23)( 2,25, 4,29, 9,32,12,34,13,37, 17,40,20,45,22)( 3,27, 6,28, 7,31,11,36,15,38,18,42,19,44,24)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,27, 4,30, 7,32,10,36,13,39,18,40,21,44,22)( 2,26, 6,29, 8,31,12,35,15,37, 16,42,20,43,24)( 3,25, 5,28, 9,33,11,34,14,38,17,41,19,45,23)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,28,12,39,19, 2,30,11,37,21, 3,29,10,38,20)( 4,33,15,40,23, 6,32,14,42,22, 5,31,13,41,24)( 7,34,16,44,25, 8,36,17,43,27, 9,35,18,45,26)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,29,11,39,20, 3,30,12,38,21, 2,28,10,37,19)( 4,31,14,40,24, 5,32,15,41,22, 6,33,13,42,23)( 7,35,17,44,26, 9,36,16,45,27, 8,34,18,43,25)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,30,10,39,21)( 2,29,12,37,20)( 3,28,11,38,19)( 4,32,13,40,22) ( 5,33,14,41,23)( 6,31,15,42,24)( 7,36,18,44,27)( 8,35,16,43,26) ( 9,34,17,45,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,31,17)( 2,33,18)( 3,32,16)( 4,35,19)( 5,36,20)( 6,34,21)( 7,37,23) ( 8,38,22)( 9,39,24)(10,42,25)(11,40,26)(12,41,27)(13,43,28)(14,44,29) (15,45,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,32,18)( 2,31,16)( 3,33,17)( 4,36,21)( 5,34,19)( 6,35,20)( 7,39,22) ( 8,37,24)( 9,38,23)(10,40,27)(11,41,25)(12,42,26)(13,44,30)(14,45,28) (15,43,29)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,33,16)( 2,32,17)( 3,31,18)( 4,34,20)( 5,35,21)( 6,36,19)( 7,38,24) ( 8,39,23)( 9,37,22)(10,41,26)(11,42,27)(12,40,25)(13,45,29)(14,43,30) (15,44,28)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,34,24,10,45,31,21, 9,42,30,17, 6,39,25,15)( 2,36,23,12,44,33,20, 7,41,29, 18, 5,37,27,14)( 3,35,22,11,43,32,19, 8,40,28,16, 4,38,26,13)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,35,23,10,43,33,21, 8,41,30,16, 5,39,26,14)( 2,34,22,12,45,32,20, 9,40,29, 17, 4,37,25,13)( 3,36,24,11,44,31,19, 7,42,28,18, 6,38,27,15)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,36,22,10,44,32,21, 7,40,30,18, 4,39,27,13)( 2,35,24,12,43,31,20, 8,42,29, 16, 6,37,26,15)( 3,34,23,11,45,33,19, 9,41,28,17, 5,38,25,14)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,37,28,21,12, 3,39,29,19,10, 2,38,30,20,11)( 4,42,33,22,15, 5,40,31,23,13, 6,41,32,24,14)( 7,43,34,27,16, 9,44,35,25,18, 8,45,36,26,17)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,38,29,21,11, 2,39,28,20,10, 3,37,30,19,12)( 4,41,31,22,14, 6,40,33,24,13, 5,42,32,23,15)( 7,45,35,27,17, 8,44,34,26,18, 9,43,36,25,16)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,39,30,21,10)( 2,37,29,20,12)( 3,38,28,19,11)( 4,40,32,22,13) ( 5,41,33,23,14)( 6,42,31,24,15)( 7,44,36,27,18)( 8,43,35,26,16) ( 9,45,34,25,17)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,40,36,30,22,18,10, 4,44,39,32,27,21,13, 7)( 2,42,35,29,24,16,12, 6,43,37, 31,26,20,15, 8)( 3,41,34,28,23,17,11, 5,45,38,33,25,19,14, 9)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,41,35,30,23,16,10, 5,43,39,33,26,21,14, 8)( 2,40,34,29,22,17,12, 4,45,37, 32,25,20,13, 9)( 3,42,36,28,24,18,11, 6,44,38,31,27,19,15, 7)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,42,34,30,24,17,10, 6,45,39,31,25,21,15, 9)( 2,41,36,29,23,18,12, 5,44,37, 33,27,20,14, 7)( 3,40,35,28,22,16,11, 4,43,38,32,26,19,13, 8)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,43,41,39,35,33,30,26,23,21,16,14,10, 8, 5)( 2,45,40,37,34,32,29,25,22,20, 17,13,12, 9, 4)( 3,44,42,38,36,31,28,27,24,19,18,15,11, 7, 6)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,44,40,39,36,32,30,27,22,21,18,13,10, 7, 4)( 2,43,42,37,35,31,29,26,24,20, 16,15,12, 8, 6)( 3,45,41,38,34,33,28,25,23,19,17,14,11, 9, 5)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,45,42,39,34,31,30,25,24,21,17,15,10, 9, 6)( 2,44,41,37,36,33,29,27,23,20, 18,14,12, 7, 5)( 3,43,40,38,35,32,28,26,22,19,16,13,11, 8, 4)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $45=3^{2} \cdot 5$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | yes | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | $1$ | ||
| Label: | 45.2 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);