Show commands:
Magma
magma: G := TransitiveGroup(45, 7);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $7$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_9\times D_5$ | ||
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $9$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,13,27,39,6,18,30,41,7,21,32,43,10,22,36,3,15,25,37,5,16,28,40,9,19,33,44,11,23,34,2,14,26,38,4,17,29,42,8,20,31,45,12,24,35), (1,13,16,28,31,45,3,15,17,29,32,43,2,14,18,30,33,44)(4,25,21,42,34,10,6,26,19,41,35,11,5,27,20,40,36,12)(7,37,23,8,39,22,9,38,24) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $9$: $C_9$ $10$: $D_{5}$ $18$: $C_{18}$ $30$: $D_5\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: $D_{5}$
Degree 9: $C_9$
Degree 15: $D_5\times C_3$
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 $ | $1$ | $1$ | $()$ |
| $ 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 $ | $5$ | $2$ | $( 4,15)( 5,13)( 6,14)( 7,25)( 8,27)( 9,26)(10,39)(11,37)(12,38)(19,30)(20,28) (21,29)(22,40)(23,42)(24,41)(34,43)(35,44)(36,45)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,45,44)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 2, 3)( 4,13, 6,15, 5,14)( 7,27, 9,25, 8,26)(10,38,11,39,12,37)(16,18,17) (19,29,20,30,21,28)(22,41,23,40,24,42)(31,33,32)(34,45,35,43,36,44)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,15,14)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,42,41) (43,44,45)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 3, 2)( 4,14, 5,15, 6,13)( 7,26, 8,25, 9,27)(10,37,12,39,11,38)(16,17,18) (19,28,21,30,20,29)(22,42,24,40,23,41)(31,32,33)(34,44,36,43,35,45)$ |
| $ 45 $ | $2$ | $45$ | $( 1, 4, 9,10,13,17,19,22,27,29,33,36,39,42,44, 3, 6, 8,11,15,18,20,23,25,30, 31,34,37,41,45, 2, 5, 7,12,14,16,21,24,26,28,32,35,38,40,43)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 4,18,20,32,35, 3, 6,16,21,33,36, 2, 5,17,19,31,34)( 7,30,22,43,37,15, 9, 28,23,44,38,14, 8,29,24,45,39,13)(10,42,25,12,40,27,11,41,26)$ |
| $ 45 $ | $2$ | $45$ | $( 1, 5, 8,10,14,18,19,24,25,29,32,34,39,40,45, 3, 4, 7,11,13,16,20,22,26,30, 33,35,37,42,43, 2, 6, 9,12,15,17,21,23,27,28,31,36,38,41,44)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 5,16,20,31,36, 3, 4,17,21,32,34, 2, 6,18,19,33,35)( 7,29,23,43,39,14, 9, 30,24,44,37,13, 8,28,22,45,38,15)(10,40,26,12,41,25,11,42,27)$ |
| $ 45 $ | $2$ | $45$ | $( 1, 6, 7,10,15,16,19,23,26,29,31,35,39,41,43, 3, 5, 9,11,14,17,20,24,27,30, 32,36,37,40,44, 2, 4, 8,12,13,18,21,22,25,28,33,34,38,42,45)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 6,17,20,33,34, 3, 5,18,21,31,35, 2, 4,16,19,32,36)( 7,28,24,43,38,13, 9, 29,22,44,39,15, 8,30,23,45,37,14)(10,41,27,12,42,26,11,40,25)$ |
| $ 45 $ | $2$ | $45$ | $( 1, 7,15,19,26,31,39,43, 5,11,17,24,30,36,40, 2, 8,13,21,25,33,38,45, 6,10, 16,23,29,35,41, 3, 9,14,20,27,32,37,44, 4,12,18,22,28,34,42)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 7,33,38,18,22, 2, 8,32,37,17,24, 3, 9,31,39,16,23)( 4,19,36, 6,20,34, 5, 21,35)(10,43,42,29,25,15,12,45,40,28,27,13,11,44,41,30,26,14)$ |
| $ 45 $ | $2$ | $45$ | $( 1, 8,14,19,25,32,39,45, 4,11,16,22,30,35,42, 2, 9,15,21,27,31,38,44, 5,10, 18,24,29,34,40, 3, 7,13,20,26,33,37,43, 6,12,17,23,28,36,41)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 8,31,38,17,23, 2, 9,33,37,16,22, 3, 7,32,39,18,24)( 4,21,34, 6,19,35, 5, 20,36)(10,45,41,29,27,14,12,44,42,28,26,15,11,43,40,30,25,13)$ |
| $ 45 $ | $2$ | $45$ | $( 1, 9,13,19,27,33,39,44, 6,11,18,23,30,34,41, 2, 7,14,21,26,32,38,43, 4,10, 17,22,29,36,42, 3, 8,15,20,25,31,37,45, 5,12,16,24,28,35,40)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 9,32,38,16,24, 2, 7,31,37,18,23, 3, 8,33,39,17,22)( 4,20,35, 6,21,36, 5, 19,34)(10,44,40,29,26,13,12,43,41,28,25,14,11,45,42,30,27,15)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,10,19,29,39, 3,11,20,30,37, 2,12,21,28,38)( 4,13,22,33,42, 6,15,23,31,41, 5,14,24,32,40)( 7,16,26,35,43, 9,17,27,36,44, 8,18,25,34,45)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,11,21,29,37)( 2,10,20,28,39)( 3,12,19,30,38)( 4,15,24,33,41) ( 5,13,23,32,42)( 6,14,22,31,40)( 7,17,25,35,44)( 8,16,27,34,43) ( 9,18,26,36,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,12,20,29,38, 2,11,19,28,37, 3,10,21,30,39)( 4,14,23,33,40, 5,15,22,32,41, 6,13,24,31,42)( 7,18,27,35,45, 8,17,26,34,44, 9,16,25,36,43)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,16,31, 3,17,32, 2,18,33)( 4,21,34, 6,19,35, 5,20,36)( 7,23,39, 9,24,37, 8, 22,38)(10,26,41,11,27,40,12,25,42)(13,28,45,15,29,43,14,30,44)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,17,33, 3,18,31, 2,16,32)( 4,19,36, 6,20,34, 5,21,35)( 7,24,38, 9,22,39, 8, 23,37)(10,27,42,11,25,41,12,26,40)(13,29,44,15,30,45,14,28,43)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,18,32, 3,16,33, 2,17,31)( 4,20,35, 6,21,36, 5,19,34)( 7,22,37, 9,23,38, 8, 24,39)(10,25,40,11,26,42,12,27,41)(13,30,43,15,28,44,14,29,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,19,39,11,30, 2,21,38,10,29, 3,20,37,12,28)( 4,22,42,15,31, 5,24,40,13,33, 6,23,41,14,32)( 7,26,43,17,36, 8,25,45,16,35, 9,27,44,18,34)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,20,38,11,28, 3,21,39,12,29, 2,19,37,10,30)( 4,23,40,15,32, 6,24,42,14,33, 5,22,41,13,31)( 7,27,45,17,34, 9,25,43,18,35, 8,26,44,16,36)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,21,37,11,29)( 2,20,39,10,28)( 3,19,38,12,30)( 4,24,41,15,33) ( 5,23,42,13,32)( 6,22,40,14,31)( 7,25,44,17,35)( 8,27,43,16,34) ( 9,26,45,18,36)$ |
| $ 45 $ | $2$ | $45$ | $( 1,22,44,20,41,16,38,13,36,11,31, 7,28, 4,27, 3,23,45,21,40,17,39,15,34,12, 32, 9,29, 6,25, 2,24,43,19,42,18,37,14,35,10,33, 8,30, 5,26)$ |
| $ 45 $ | $2$ | $45$ | $( 1,23,43,20,40,18,38,15,35,11,32, 8,28, 6,26, 3,24,44,21,42,16,39,14,36,12, 33, 7,29, 5,27, 2,22,45,19,41,17,37,13,34,10,31, 9,30, 4,25)$ |
| $ 45 $ | $2$ | $45$ | $( 1,24,45,20,42,17,38,14,34,11,33, 9,28, 5,25, 3,22,43,21,41,18,39,13,35,12, 31, 8,29, 4,26, 2,23,44,19,40,16,37,15,36,10,32, 7,30, 6,27)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,31,17, 2,33,16, 3,32,18)( 4,34,19, 5,36,21, 6,35,20)( 7,39,24, 8,38,23, 9, 37,22)(10,41,27,12,42,26,11,40,25)(13,45,29,14,44,28,15,43,30)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,32,16, 2,31,18, 3,33,17)( 4,35,21, 5,34,20, 6,36,19)( 7,37,23, 8,39,22, 9, 38,24)(10,40,26,12,41,25,11,42,27)(13,43,28,14,45,30,15,44,29)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,33,18, 2,32,17, 3,31,16)( 4,36,20, 5,35,19, 6,34,21)( 7,38,22, 8,37,24, 9, 39,23)(10,42,25,12,40,27,11,41,26)(13,44,30,14,43,29,15,45,28)$ |
| $ 45 $ | $2$ | $45$ | $( 1,34,22,12,44,32,20, 9,41,29,16, 6,38,25,13, 2,36,24,11,43,31,19, 7,42,28, 18, 4,37,27,14, 3,35,23,10,45,33,21, 8,40,30,17, 5,39,26,15)$ |
| $ 45 $ | $2$ | $45$ | $( 1,35,24,12,45,31,20, 8,42,29,17, 4,38,26,14, 2,34,23,11,44,33,19, 9,40,28, 16, 5,37,25,15, 3,36,22,10,43,32,21, 7,41,30,18, 6,39,27,13)$ |
| $ 45 $ | $2$ | $45$ | $( 1,36,23,12,43,33,20, 7,40,29,18, 5,38,27,15, 2,35,22,11,45,32,19, 8,41,28, 17, 6,37,26,13, 3,34,24,10,44,31,21, 9,42,30,16, 4,39,25,14)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $90=2 \cdot 3^{2} \cdot 5$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 90.2 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);