Show commands:
Magma
magma: G := TransitiveGroup(45, 14);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_{15}:D_6$ | ||
| 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: | (1,26,32,11,17,41)(2,25,31,10,16,42)(3,27,33,12,18,40)(4,37,35,24,20,7)(5,39,34,23,19,8)(6,38,36,22,21,9)(13,29,45)(14,28,44)(15,30,43), (1,45,40,37,36,31,29,26,24,20,16,14,11,8,4,3,44,42,39,34,33,30,25,22,19,17,15,12,9,6)(2,43,41,38,35,32,28,27,23,21,18,13,10,7,5) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $10$: $D_{5}$ $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $20$: $D_{10}$ $30$: $D_5\times C_3$ $36$: $C_6\times S_3$ $60$: $D_5\times S_3$, 30T5 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 9: $S_3\times C_3$
Degree 15: $D_5\times C_3$, $D_5\times S_3$
Low degree siblings
30T44Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | Cycle Type | Size | Order | Representative |
| $1^{45}$ | $1$ | $1$ | $()$ | |
| $2^{18},1^{9}$ | $5$ | $2$ | $( 4,15)( 5,13)( 6,14)( 7,27)( 8,26)( 9,25)(10,38)(11,39)(12,37)(19,29)(20,30) (21,28)(22,42)(23,41)(24,40)(34,45)(35,43)(36,44)$ | |
| $2^{15},1^{15}$ | $3$ | $2$ | $( 2, 3)( 5, 6)( 7, 8)(10,12)(13,14)(17,18)(20,21)(22,23)(26,27)(28,30)(31,32) (34,35)(37,38)(41,42)(43,45)$ | |
| $2^{21},1^{3}$ | $15$ | $2$ | $( 2, 3)( 4,15)( 5,14)( 6,13)( 7,26)( 8,27)( 9,25)(10,37)(11,39)(12,38)(17,18) (19,29)(20,28)(21,30)(22,41)(23,42)(24,40)(31,32)(34,43)(35,45)(36,44)$ | |
| $3^{15}$ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,45,44)$ | |
| $6^{6},3^{3}$ | $10$ | $6$ | $( 1, 2, 3)( 4,13, 6,15, 5,14)( 7,26, 9,27, 8,25)(10,37,11,38,12,39)(16,18,17) (19,28,20,29,21,30)(22,40,23,42,24,41)(31,33,32)(34,44,35,45,36,43)$ | |
| $15^{3}$ | $2$ | $15$ | $( 1, 4, 9,11,15,16,19,24,25,29,33,36,39,40,44)( 2, 5, 7,10,13,18,21,23,27,28, 32,35,38,41,43)( 3, 6, 8,12,14,17,20,22,26,30,31,34,37,42,45)$ | |
| $6^{6},3^{3}$ | $5$ | $6$ | $( 1, 4,16,19,33,36)( 2, 5,18,21,32,35)( 3, 6,17,20,31,34)( 7,28,23,43,38,13) ( 8,30,22,45,37,14)( 9,29,24,44,39,15)(10,41,27)(11,40,25)(12,42,26)$ | |
| $30,15$ | $6$ | $30$ | $( 1, 4, 9,11,15,16,19,24,25,29,33,36,39,40,44)( 2, 6, 7,12,13,17,21,22,27,30, 32,34,38,42,43, 3, 5, 8,10,14,18,20,23,26,28,31,35,37,41,45)$ | |
| $6^{7},3$ | $15$ | $6$ | $( 1, 4,16,19,33,36)( 2, 6,18,20,32,34)( 3, 5,17,21,31,35)( 7,30,23,45,38,14) ( 8,28,22,43,37,13)( 9,29,24,44,39,15)(10,42,27,12,41,26)(11,40,25)$ | |
| $15^{3}$ | $4$ | $15$ | $( 1, 5, 8,11,13,17,19,23,26,29,32,34,39,41,45)( 2, 6, 9,10,14,16,21,22,25,28, 31,36,38,42,44)( 3, 4, 7,12,15,18,20,24,27,30,33,35,37,40,43)$ | |
| $6^{6},3^{3}$ | $10$ | $6$ | $( 1, 5,17,19,32,34)( 2, 6,16,21,31,36)( 3, 4,18,20,33,35)( 7,30,24,43,37,15) ( 8,29,23,45,39,13)( 9,28,22,44,38,14)(10,42,25)(11,41,26)(12,40,27)$ | |
| $15^{3}$ | $4$ | $15$ | $( 1, 7,14,19,27,31,39,43, 6,11,18,22,29,35,42)( 2, 8,15,21,26,33,38,45, 4,10, 17,24,28,34,40)( 3, 9,13,20,25,32,37,44, 5,12,16,23,30,36,41)$ | |
| $6^{6},3^{3}$ | $10$ | $6$ | $( 1, 7,31,39,18,22)( 2, 8,33,38,17,24)( 3, 9,32,37,16,23)( 4,21,34)( 5,20,36) ( 6,19,35)(10,45,40,28,26,15)(11,43,42,29,27,14)(12,44,41,30,25,13)$ | |
| $30,15$ | $6$ | $30$ | $( 1, 7,15,21,25,32,39,43, 4,10,16,23,29,35,40, 2, 9,13,19,27,33,38,44, 5,11, 18,24,28,36,41)( 3, 8,14,20,26,31,37,45, 6,12,17,22,30,34,42)$ | |
| $6^{7},3$ | $15$ | $6$ | $( 1, 7,33,38,16,23)( 2, 9,32,39,18,24)( 3, 8,31,37,17,22)( 4,21,36, 5,19,35) ( 6,20,34)(10,44,41,29,27,15)(11,43,40,28,25,13)(12,45,42,30,26,14)$ | |
| $15^{3}$ | $2$ | $15$ | $( 1, 9,15,19,25,33,39,44, 4,11,16,24,29,36,40)( 2, 7,13,21,27,32,38,43, 5,10, 18,23,28,35,41)( 3, 8,14,20,26,31,37,45, 6,12,17,22,30,34,42)$ | |
| $6^{6},3^{3}$ | $5$ | $6$ | $( 1, 9,33,39,16,24)( 2, 7,32,38,18,23)( 3, 8,31,37,17,22)( 4,19,36)( 5,21,35) ( 6,20,34)(10,43,41,28,27,13)(11,44,40,29,25,15)(12,45,42,30,26,14)$ | |
| $10^{3},5^{3}$ | $6$ | $10$ | $( 1,10,19,28,39, 2,11,21,29,38)( 3,12,20,30,37)( 4,13,24,32,40, 5,15,23,33,41) ( 6,14,22,31,42)( 7,16,27,36,43, 9,18,25,35,44)( 8,17,26,34,45)$ | |
| $15^{3}$ | $4$ | $15$ | $( 1,10,20,29,38, 3,11,21,30,39, 2,12,19,28,37)( 4,13,22,33,41, 6,15,23,31,40, 5,14,24,32,42)( 7,17,25,35,45, 9,18,26,36,43, 8,16,27,34,44)$ | |
| $5^{9}$ | $2$ | $5$ | $( 1,11,19,29,39)( 2,10,21,28,38)( 3,12,20,30,37)( 4,15,24,33,40) ( 5,13,23,32,41)( 6,14,22,31,42)( 7,18,27,35,43)( 8,17,26,34,45) ( 9,16,25,36,44)$ | |
| $6^{5},3^{5}$ | $3$ | $6$ | $( 1,16,33)( 2,17,32, 3,18,31)( 4,19,36)( 5,20,35, 6,21,34)( 7,22,38, 8,23,37) ( 9,24,39)(10,26,41,12,27,42)(11,25,40)(13,30,43,14,28,45)(15,29,44)$ | |
| $3^{15}$ | $1$ | $3$ | $( 1,16,33)( 2,18,32)( 3,17,31)( 4,19,36)( 5,21,35)( 6,20,34)( 7,23,38) ( 8,22,37)( 9,24,39)(10,27,41)(11,25,40)(12,26,42)(13,28,43)(14,30,45) (15,29,44)$ | |
| $3^{15}$ | $2$ | $3$ | $( 1,17,32)( 2,16,31)( 3,18,33)( 4,20,35)( 5,19,34)( 6,21,36)( 7,24,37) ( 8,23,39)( 9,22,38)(10,25,42)(11,26,41)(12,27,40)(13,29,45)(14,28,44) (15,30,43)$ | |
| $10^{3},5^{3}$ | $6$ | $10$ | $( 1,19,39,11,29)( 2,20,38,12,28, 3,21,37,10,30)( 4,24,40,15,33) ( 5,22,41,14,32, 6,23,42,13,31)( 7,26,43,17,35, 8,27,45,18,34)( 9,25,44,16,36)$ | |
| $5^{9}$ | $2$ | $5$ | $( 1,19,39,11,29)( 2,21,38,10,28)( 3,20,37,12,30)( 4,24,40,15,33) ( 5,23,41,13,32)( 6,22,42,14,31)( 7,27,43,18,35)( 8,26,45,17,34) ( 9,25,44,16,36)$ | |
| $15^{3}$ | $4$ | $15$ | $( 1,20,38,11,30, 2,19,37,10,29, 3,21,39,12,28)( 4,22,41,15,31, 5,24,42,13,33, 6,23,40,14,32)( 7,25,45,18,36, 8,27,44,17,35, 9,26,43,16,34)$ | |
| $30,15$ | $6$ | $30$ | $( 1,22,44,20,40,17,39,14,36,12,33, 8,29, 6,25, 3,24,45,19,42,16,37,15,34,11, 31, 9,30, 4,26)( 2,23,43,21,41,18,38,13,35,10,32, 7,28, 5,27)$ | |
| $15^{3}$ | $4$ | $15$ | $( 1,22,43,19,42,18,39,14,35,11,31, 7,29, 6,27)( 2,24,45,21,40,17,38,15,34,10, 33, 8,28, 4,26)( 3,23,44,20,41,16,37,13,36,12,32, 9,30, 5,25)$ | |
| $15^{3}$ | $2$ | $15$ | $( 1,24,44,19,40,16,39,15,36,11,33, 9,29, 4,25)( 2,23,43,21,41,18,38,13,35,10, 32, 7,28, 5,27)( 3,22,45,20,42,17,37,14,34,12,31, 8,30, 6,26)$ | |
| $6^{5},3^{5}$ | $3$ | $6$ | $( 1,31,16, 3,33,17)( 2,32,18)( 4,34,19, 6,36,20)( 5,35,21)( 7,38,23) ( 8,39,22, 9,37,24)(10,41,27)(11,42,25,12,40,26)(13,43,28)(14,44,30,15,45,29)$ | |
| $3^{15}$ | $2$ | $3$ | $( 1,31,18)( 2,33,17)( 3,32,16)( 4,34,21)( 5,36,20)( 6,35,19)( 7,39,22) ( 8,38,24)( 9,37,23)(10,40,26)(11,42,27)(12,41,25)(13,44,30)(14,43,29) (15,45,28)$ | |
| $3^{15}$ | $1$ | $3$ | $( 1,33,16)( 2,32,18)( 3,31,17)( 4,36,19)( 5,35,21)( 6,34,20)( 7,38,23) ( 8,37,22)( 9,39,24)(10,41,27)(11,40,25)(12,42,26)(13,43,28)(14,45,30) (15,44,29)$ | |
| $30,15$ | $6$ | $30$ | $( 1,34,24,12,44,31,19, 8,40,30,16, 6,39,26,15, 3,36,22,11,45,33,20, 9,42,29, 17, 4,37,25,14)( 2,35,23,10,43,32,21, 7,41,28,18, 5,38,27,13)$ | |
| $15^{3}$ | $4$ | $15$ | $( 1,34,23,11,45,32,19, 8,41,29,17, 5,39,26,13)( 2,36,22,10,44,31,21, 9,42,28, 16, 6,38,25,14)( 3,35,24,12,43,33,20, 7,40,30,18, 4,37,27,15)$ | |
| $15^{3}$ | $2$ | $15$ | $( 1,36,24,11,44,33,19, 9,40,29,16, 4,39,25,15)( 2,35,23,10,43,32,21, 7,41,28, 18, 5,38,27,13)( 3,34,22,12,45,31,20, 8,42,30,17, 6,37,26,14)$ |
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);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 180.26 | magma: IdentifyGroup(G);
| |
| Character table: | 36 x 36 character table |
magma: CharacterTable(G);