Show commands:
Magma
magma: G := TransitiveGroup(45, 34);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_3^2:D_{15}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,43,42,39,36,33,28,27,22,21,16,15,12,8,4)(2,45,40,38,35,32,30,26,24,20,18,13,11,9,5)(3,44,41,37,34,31,29,25,23,19,17,14,10,7,6), (1,12)(2,10)(3,11)(4,24)(5,22)(6,23)(7,34)(8,35)(9,36)(13,15)(16,26)(17,25)(18,27)(19,38)(20,37)(21,39)(29,30)(31,41)(32,42)(33,40)(43,45) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $10$: $D_{5}$ $18$: $S_3\times C_3$ $30$: $D_{15}$, $D_5\times C_3$ $54$: $C_3^2 : C_6$ $90$: 30T16 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: $D_{5}$
Degree 9: $C_3^2 : S_3 $
Degree 15: $D_5\times C_3$
Low degree siblings
45T32Siblings 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 |
| $1^{45}$ | $1$ | $1$ | $()$ | |
| $3^{10},1^{15}$ | $6$ | $3$ | $( 4, 5, 6)( 7, 9, 8)(13,14,15)(16,17,18)(22,24,23)(25,26,27)(31,33,32) (34,35,36)(40,41,42)(43,44,45)$ | |
| $2^{21},1^{3}$ | $45$ | $2$ | $( 2, 3)( 4,13)( 5,15)( 6,14)( 7,25)( 8,26)( 9,27)(10,38)(11,37)(12,39)(16,18) (19,30)(20,29)(21,28)(22,40)(23,41)(24,42)(32,33)(34,44)(35,43)(36,45)$ | |
| $3^{15}$ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,45,44)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1, 4, 7,12,15,17,21,22,25,28,33,34,39,42,44)( 2, 5, 8,11,13,16,20,24,27,30, 32,36,38,40,43)( 3, 6, 9,10,14,18,19,23,26,29,31,35,37,41,45)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1, 4, 8,12,15,16,21,22,27,28,33,36,39,42,43)( 2, 5, 9,11,13,18,20,24,26,30, 32,35,38,40,45)( 3, 6, 7,10,14,17,19,23,25,29,31,34,37,41,44)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1, 4, 9,12,15,18,21,22,26,28,33,35,39,42,45)( 2, 5, 7,11,13,17,20,24,25,30, 32,34,38,40,44)( 3, 6, 8,10,14,16,19,23,27,29,31,36,37,41,43)$ | |
| $6^{7},3$ | $45$ | $6$ | $( 1, 4,16,19,32,34)( 2, 6,18,20,31,35)( 3, 5,17,21,33,36)( 7,28,22,43,37,13) ( 8,29,24,44,39,15)( 9,30,23,45,38,14)(10,40,25,12,42,27)(11,41,26)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1, 7,13,21,25,32,39,44, 5,12,17,24,28,34,40)( 2, 8,14,20,27,31,38,43, 6,11, 16,23,30,36,41)( 3, 9,15,19,26,33,37,45, 4,10,18,22,29,35,42)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1, 7,14,21,25,31,39,44, 6,12,17,23,28,34,41)( 2, 8,15,20,27,33,38,43, 4,11, 16,22,30,36,42)( 3, 9,13,19,26,32,37,45, 5,10,18,24,29,35,40)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1, 7,15,21,25,33,39,44, 4,12,17,22,28,34,42)( 2, 8,13,20,27,32,38,43, 5,11, 16,24,30,36,40)( 3, 9,14,19,26,31,37,45, 6,10,18,23,29,35,41)$ | |
| $6^{7},3$ | $45$ | $6$ | $( 1, 7,31,39,17,23)( 2, 9,33,37,16,24)( 3, 8,32,38,18,22)( 4,19,36, 5,20,35) ( 6,21,34)(10,43,40,30,26,15)(11,45,42,29,27,13)(12,44,41,28,25,14)$ | |
| $15^{2},5^{3}$ | $6$ | $15$ | $( 1,10,20,28,37, 2,12,19,30,39, 3,11,21,29,38)( 4,13,23,33,40, 6,15,24,31,42, 5,14,22,32,41)( 7,17,25,34,44)( 8,16,27,36,43)( 9,18,26,35,45)$ | |
| $15^{3}$ | $2$ | $15$ | $( 1,10,20,28,37, 2,12,19,30,39, 3,11,21,29,38)( 4,14,24,33,41, 5,15,23,32,42, 6,13,22,31,40)( 7,18,27,34,45, 8,17,26,36,44, 9,16,25,35,43)$ | |
| $15^{2},5^{3}$ | $6$ | $15$ | $( 1,10,20,28,37, 2,12,19,30,39, 3,11,21,29,38)( 4,15,22,33,42)( 5,13,24,32,40) ( 6,14,23,31,41)( 7,16,26,34,43, 9,17,27,35,44, 8,18,25,36,45)$ | |
| $15^{3}$ | $2$ | $15$ | $( 1,11,19,28,38, 3,12,20,29,39, 2,10,21,30,37)( 4,13,23,33,40, 6,15,24,31,42, 5,14,22,32,41)( 7,16,26,34,43, 9,17,27,35,44, 8,18,25,36,45)$ | |
| $5^{9}$ | $2$ | $5$ | $( 1,12,21,28,39)( 2,11,20,30,38)( 3,10,19,29,37)( 4,15,22,33,42) ( 5,13,24,32,40)( 6,14,23,31,41)( 7,17,25,34,44)( 8,16,27,36,43) ( 9,18,26,35,45)$ | |
| $3^{15}$ | $3$ | $3$ | $( 1,16,32)( 2,18,31)( 3,17,33)( 4,19,34)( 5,21,36)( 6,20,35)( 7,22,37) ( 8,24,39)( 9,23,38)(10,25,42)(11,26,41)(12,27,40)(13,28,43)(14,30,45) (15,29,44)$ | |
| $3^{15}$ | $6$ | $3$ | $( 1,16,31)( 2,18,33)( 3,17,32)( 4,20,35)( 5,19,34)( 6,21,36)( 7,24,37) ( 8,23,39)( 9,22,38)(10,25,40)(11,26,42)(12,27,41)(13,29,44)(14,28,43) (15,30,45)$ | |
| $15^{2},5^{3}$ | $6$ | $15$ | $( 1,19,38,12,29, 2,21,37,11,28, 3,20,39,10,30)( 4,22,42,15,33)( 5,24,40,13,32) ( 6,23,41,14,31)( 7,27,45,17,36, 9,25,43,18,34, 8,26,44,16,35)$ | |
| $15^{3}$ | $2$ | $15$ | $( 1,19,38,12,29, 2,21,37,11,28, 3,20,39,10,30)( 4,23,40,15,31, 5,22,41,13,33, 6,24,42,14,32)( 7,26,43,17,35, 8,25,45,16,34, 9,27,44,18,36)$ | |
| $15^{2},5^{3}$ | $6$ | $15$ | $( 1,19,38,12,29, 2,21,37,11,28, 3,20,39,10,30)( 4,24,41,15,32, 6,22,40,14,33, 5,23,42,13,31)( 7,25,44,17,34)( 8,27,43,16,36)( 9,26,45,18,35)$ | |
| $15^{3}$ | $2$ | $15$ | $( 1,20,37,12,30, 3,21,38,10,28, 2,19,39,11,29)( 4,24,41,15,32, 6,22,40,14,33, 5,23,42,13,31)( 7,27,45,17,36, 9,25,43,18,34, 8,26,44,16,35)$ | |
| $5^{9}$ | $2$ | $5$ | $( 1,21,39,12,28)( 2,20,38,11,30)( 3,19,37,10,29)( 4,22,42,15,33) ( 5,24,40,13,32)( 6,23,41,14,31)( 7,25,44,17,34)( 8,27,43,16,36) ( 9,26,45,18,35)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1,22,44,21,42,17,39,15,34,12,33, 7,28, 4,25)( 2,24,43,20,40,16,38,13,36,11, 32, 8,30, 5,27)( 3,23,45,19,41,18,37,14,35,10,31, 9,29, 6,26)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1,22,45,21,42,18,39,15,35,12,33, 9,28, 4,26)( 2,24,44,20,40,17,38,13,34,11, 32, 7,30, 5,25)( 3,23,43,19,41,16,37,14,36,10,31, 8,29, 6,27)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1,22,43,21,42,16,39,15,36,12,33, 8,28, 4,27)( 2,24,45,20,40,18,38,13,35,11, 32, 9,30, 5,26)( 3,23,44,19,41,17,37,14,34,10,31, 7,29, 6,25)$ | |
| $3^{15}$ | $6$ | $3$ | $( 1,31,18)( 2,33,17)( 3,32,16)( 4,34,20)( 5,36,19)( 6,35,21)( 7,38,22) ( 8,37,24)( 9,39,23)(10,40,27)(11,42,25)(12,41,26)(13,43,29)(14,45,28) (15,44,30)$ | |
| $3^{15}$ | $3$ | $3$ | $( 1,31,17)( 2,33,16)( 3,32,18)( 4,36,20)( 5,35,19)( 6,34,21)( 7,39,23) ( 8,38,22)( 9,37,24)(10,40,26)(11,42,27)(12,41,25)(13,45,29)(14,44,28) (15,43,30)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1,34,24,12,44,32,21, 7,40,28,17, 5,39,25,13)( 2,36,23,11,43,31,20, 8,41,30, 16, 6,38,27,14)( 3,35,22,10,45,33,19, 9,42,29,18, 4,37,26,15)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1,34,23,12,44,31,21, 7,41,28,17, 6,39,25,14)( 2,36,22,11,43,33,20, 8,42,30, 16, 4,38,27,15)( 3,35,24,10,45,32,19, 9,40,29,18, 5,37,26,13)$ | |
| $15^{3}$ | $6$ | $15$ | $( 1,34,22,12,44,33,21, 7,42,28,17, 4,39,25,15)( 2,36,24,11,43,32,20, 8,40,30, 16, 5,38,27,13)( 3,35,23,10,45,31,19, 9,41,29,18, 6,37,26,14)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $270=2 \cdot 3^{3} \cdot 5$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 270.14 | magma: IdentifyGroup(G);
| |
| Character table: | 32 x 32 character table |
magma: CharacterTable(G);