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Magma
magma: G := TransitiveGroup(45, 8);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^2\times D_5$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $9$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,38,2,37,3,39)(4,6,5)(7,18,8,17,9,16)(10,29,12,28,11,30)(13,41,14,42,15,40)(19,20,21)(22,32,24,31,23,33)(25,45,27,44,26,43)(34,35,36), (1,33,18)(2,32,17)(3,31,16)(4,43,21,13,34,29)(5,45,20,14,36,28)(6,44,19,15,35,30)(7,11,23,27,39,42)(8,10,22,26,38,40)(9,12,24,25,37,41) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ x 4 $6$: $C_6$ x 4 $9$: $C_3^2$ $10$: $D_{5}$ $18$: $C_6 \times C_3$ $30$: $D_5\times C_3$ x 4 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$ x 4
Degree 5: $D_{5}$
Degree 9: $C_3^2$
Degree 15: $D_5\times C_3$ 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 $ | $1$ | $1$ | $()$ |
| $ 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 $ | $5$ | $2$ | $( 4,13)( 5,14)( 6,15)( 7,27)( 8,26)( 9,25)(10,38)(11,39)(12,37)(19,30)(20,28) (21,29)(22,40)(23,42)(24,41)(34,43)(35,44)(36,45)$ |
| $ 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 $ | $5$ | $6$ | $( 1, 2, 3)( 4,14, 6,13, 5,15)( 7,26, 9,27, 8,25)(10,37,11,38,12,39)(16,18,17) (19,29,20,30,21,28)(22,41,23,40,24,42)(31,33,32)(34,45,35,43,36,44)$ |
| $ 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 $ | $5$ | $6$ | $( 1, 3, 2)( 4,15, 5,13, 6,14)( 7,25, 8,27, 9,26)(10,39,12,38,11,37)(16,17,18) (19,28,21,30,20,29)(22,42,24,40,23,41)(31,32,33)(34,44,36,43,35,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 4, 8,12,13,16,19,23,26,30,32,36,37,42,45)( 2, 5, 9,11,14,18,21,22,25,29, 31,35,39,40,44)( 3, 6, 7,10,15,17,20,24,27,28,33,34,38,41,43)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 4,16,19,32,36)( 2, 5,18,21,31,35)( 3, 6,17,20,33,34)( 7,28,24,43,38,15) ( 8,30,23,45,37,13)( 9,29,22,44,39,14)(10,41,27)(11,40,25)(12,42,26)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 5, 7,12,14,17,19,22,27,30,31,34,37,40,43)( 2, 6, 8,11,15,16,21,24,26,29, 33,36,39,41,45)( 3, 4, 9,10,13,18,20,23,25,28,32,35,38,42,44)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 5,17,19,31,34)( 2, 6,16,21,33,36)( 3, 4,18,20,32,35)( 7,30,22,43,37,14) ( 8,29,24,45,39,15)( 9,28,23,44,38,13)(10,42,25)(11,41,26)(12,40,27)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 6, 9,12,15,18,19,24,25,30,33,35,37,41,44)( 2, 4, 7,11,13,17,21,23,27,29, 32,34,39,42,43)( 3, 5, 8,10,14,16,20,22,26,28,31,36,38,40,45)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 6,18,19,33,35)( 2, 4,17,21,32,34)( 3, 5,16,20,31,36)( 7,29,23,43,39,13) ( 8,28,22,45,38,14)( 9,30,24,44,37,15)(10,40,26)(11,42,27)(12,41,25)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 7,14,19,27,31,37,43, 5,12,17,22,30,34,40)( 2, 8,15,21,26,33,39,45, 6,11, 16,24,29,36,41)( 3, 9,13,20,25,32,38,44, 4,10,18,23,28,35,42)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 7,31,37,17,22)( 2, 8,33,39,16,24)( 3, 9,32,38,18,23)( 4,20,35)( 5,19,34) ( 6,21,36)(10,44,42,28,25,13)(11,45,41,29,26,15)(12,43,40,30,27,14)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 8,13,19,26,32,37,45, 4,12,16,23,30,36,42)( 2, 9,14,21,25,31,39,44, 5,11, 18,22,29,35,40)( 3, 7,15,20,27,33,38,43, 6,10,17,24,28,34,41)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 8,32,37,16,23)( 2, 9,31,39,18,22)( 3, 7,33,38,17,24)( 4,19,36)( 5,21,35) ( 6,20,34)(10,43,41,28,27,15)(11,44,40,29,25,14)(12,45,42,30,26,13)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 9,15,19,25,33,37,44, 6,12,18,24,30,35,41)( 2, 7,13,21,27,32,39,43, 4,11, 17,23,29,34,42)( 3, 8,14,20,26,31,38,45, 5,10,16,22,28,36,40)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 9,33,37,18,24)( 2, 7,32,39,17,23)( 3, 8,31,38,16,22)( 4,21,34)( 5,20,36) ( 6,19,35)(10,45,40,28,26,14)(11,43,42,29,27,13)(12,44,41,30,25,15)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,10,21,30,38, 2,12,20,29,37, 3,11,19,28,39)( 4,15,22,32,41, 5,13,24,31,42, 6,14,23,33,40)( 7,18,26,34,44, 8,17,25,36,43, 9,16,27,35,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,11,20,30,39, 3,12,21,28,37, 2,10,19,29,38)( 4,14,24,32,40, 6,13,22,33,42, 5,15,23,31,41)( 7,16,25,34,45, 9,17,26,35,43, 8,18,27,36,44)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,12,19,30,37)( 2,11,21,29,39)( 3,10,20,28,38)( 4,13,23,32,42) ( 5,14,22,31,40)( 6,15,24,33,41)( 7,17,27,34,43)( 8,16,26,36,45) ( 9,18,25,35,44)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,16,32)( 2,18,31)( 3,17,33)( 4,19,36)( 5,21,35)( 6,20,34)( 7,24,38) ( 8,23,37)( 9,22,39)(10,27,41)(11,25,40)(12,26,42)(13,30,45)(14,29,44) (15,28,43)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,17,31)( 2,16,33)( 3,18,32)( 4,20,35)( 5,19,34)( 6,21,36)( 7,22,37) ( 8,24,39)( 9,23,38)(10,25,42)(11,26,41)(12,27,40)(13,28,44)(14,30,43) (15,29,45)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,18,33)( 2,17,32)( 3,16,31)( 4,21,34)( 5,20,36)( 6,19,35)( 7,23,39) ( 8,22,38)( 9,24,37)(10,26,40)(11,27,42)(12,25,41)(13,29,43)(14,28,45) (15,30,44)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,19,37,12,30)( 2,21,39,11,29)( 3,20,38,10,28)( 4,23,42,13,32) ( 5,22,40,14,31)( 6,24,41,15,33)( 7,27,43,17,34)( 8,26,45,16,36) ( 9,25,44,18,35)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,20,39,12,28, 2,19,38,11,30, 3,21,37,10,29)( 4,24,40,13,33, 5,23,41,14,32, 6,22,42,15,31)( 7,25,45,17,35, 8,27,44,16,34, 9,26,43,18,36)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,21,38,12,29, 3,19,39,10,30, 2,20,37,11,28)( 4,22,41,13,31, 6,23,40,15,32, 5,24,42,14,33)( 7,26,44,17,36, 9,27,45,18,34, 8,25,43,16,35)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,22,43,19,40,17,37,14,34,12,31, 7,30, 5,27)( 2,24,45,21,41,16,39,15,36,11, 33, 8,29, 6,26)( 3,23,44,20,42,18,38,13,35,10,32, 9,28, 4,25)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,23,45,19,42,16,37,13,36,12,32, 8,30, 4,26)( 2,22,44,21,40,18,39,14,35,11, 31, 9,29, 5,25)( 3,24,43,20,41,17,38,15,34,10,33, 7,28, 6,27)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,24,44,19,41,18,37,15,35,12,33, 9,30, 6,25)( 2,23,43,21,42,17,39,13,34,11, 32, 7,29, 4,27)( 3,22,45,20,40,16,38,14,36,10,31, 8,28, 5,26)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,31,17)( 2,33,16)( 3,32,18)( 4,35,20)( 5,34,19)( 6,36,21)( 7,37,22) ( 8,39,24)( 9,38,23)(10,42,25)(11,41,26)(12,40,27)(13,44,28)(14,43,30) (15,45,29)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,32,16)( 2,31,18)( 3,33,17)( 4,36,19)( 5,35,21)( 6,34,20)( 7,38,24) ( 8,37,23)( 9,39,22)(10,41,27)(11,40,25)(12,42,26)(13,45,30)(14,44,29) (15,43,28)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,33,18)( 2,32,17)( 3,31,16)( 4,34,21)( 5,36,20)( 6,35,19)( 7,39,23) ( 8,38,22)( 9,37,24)(10,40,26)(11,42,27)(12,41,25)(13,43,29)(14,45,28) (15,44,30)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,34,22,12,43,31,19, 7,40,30,17, 5,37,27,14)( 2,36,24,11,45,33,21, 8,41,29, 16, 6,39,26,15)( 3,35,23,10,44,32,20, 9,42,28,18, 4,38,25,13)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,35,24,12,44,33,19, 9,41,30,18, 6,37,25,15)( 2,34,23,11,43,32,21, 7,42,29, 17, 4,39,27,13)( 3,36,22,10,45,31,20, 8,40,28,16, 5,38,26,14)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,36,23,12,45,32,19, 8,42,30,16, 4,37,26,13)( 2,35,22,11,44,31,21, 9,40,29, 18, 5,39,25,14)( 3,34,24,10,43,33,20, 7,41,28,17, 6,38,27,15)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $90=2 \cdot 3^{2} \cdot 5$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 90.5 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);