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Magma
magma: G := TransitiveGroup(45, 3);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\times C_{15}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $15$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,26,4,28,8,33,11,34,15,38,18,40,19,44,22,3,25,5,30,7,32,12,36,13,39,17,42,20,43,23)(2,27,6,29,9,31,10,35,14,37,16,41,21,45,24), (1,30,11,39,19)(2,28,10,38,21,3,29,12,37,20)(4,32,15,42,22)(5,31,13,41,23,6,33,14,40,24)(7,35,17,45,26,9,34,16,44,27)(8,36,18,43,25) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $5$: $C_5$ $6$: $S_3$, $C_6$ $10$: $C_{10}$ $15$: $C_{15}$ $18$: $S_3\times C_3$ $30$: $S_3 \times C_5$, $C_{30}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $C_5$
Degree 9: $S_3\times C_3$
Degree 15: $C_{15}$, $S_3 \times C_5$
Low degree siblings
30T15Siblings are shown 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 5, 6)( 7, 9)(10,12)(13,14)(16,17)(20,21)(23,24)(26,27)(28,29)(31,33) (34,35)(37,38)(40,41)(44,45)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,38,39)(40,42,41) (43,45,44)$ |
| $ 30, 15 $ | $3$ | $30$ | $( 1, 4, 8,11,15,18,19,22,25,30,32,36,39,42,43)( 2, 5, 9,12,14,17,21,23,27,28, 31,34,37,40,45, 3, 6, 7,10,13,16,20,24,26,29,33,35,38,41,44)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1, 4, 8,11,15,18,19,22,25,30,32,36,39,42,43)( 2, 6, 9,10,14,16,21,24,27,29, 31,35,37,41,45)( 3, 5, 7,12,13,17,20,23,26,28,33,34,38,40,44)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 5, 9,11,13,16,19,23,27,30,33,35,39,40,45)( 2, 4, 7,10,15,17,21,22,26,29, 32,34,37,42,44)( 3, 6, 8,12,14,18,20,24,25,28,31,36,38,41,43)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 7,14,19,26,31,39,44, 6,11,17,24,30,34,41)( 2, 8,13,21,25,33,37,43, 5,10, 18,23,29,36,40)( 3, 9,15,20,27,32,38,45, 4,12,16,22,28,35,42)$ |
| $ 30, 15 $ | $3$ | $30$ | $( 1, 7,15,20,25,33,39,44, 4,12,18,23,30,34,42, 3, 8,13,19,26,32,38,43, 5,11, 17,22,28,36,40)( 2, 9,14,21,27,31,37,45, 6,10,16,24,29,35,41)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1, 8,15,19,25,32,39,43, 4,11,18,22,30,36,42)( 2, 9,14,21,27,31,37,45, 6,10, 16,24,29,35,41)( 3, 7,13,20,26,33,38,44, 5,12,17,23,28,34,40)$ |
| $ 10, 10, 10, 5, 5, 5 $ | $3$ | $10$ | $( 1,10,19,29,39, 2,11,21,30,37)( 3,12,20,28,38)( 4,14,22,31,42, 6,15,24,32,41) ( 5,13,23,33,40)( 7,17,26,34,44)( 8,16,25,35,43, 9,18,27,36,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,10,20,30,37, 3,11,21,28,39, 2,12,19,29,38)( 4,14,23,32,41, 5,15,24,33,42, 6,13,22,31,40)( 7,18,27,34,43, 9,17,25,35,44, 8,16,26,36,45)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,11,19,30,39)( 2,10,21,29,37)( 3,12,20,28,38)( 4,15,22,32,42) ( 5,13,23,33,40)( 6,14,24,31,41)( 7,17,26,34,44)( 8,18,25,36,43) ( 9,16,27,35,45)$ |
| $ 30, 15 $ | $3$ | $30$ | $( 1,13,25,38, 4,17,30,40, 8,20,32,44,11,23,36, 3,15,26,39, 5,18,28,42, 7,19, 33,43,12,22,34)( 2,14,27,37, 6,16,29,41, 9,21,31,45,10,24,35)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,13,27,39, 5,16,30,40, 9,19,33,45,11,23,35)( 2,15,26,37, 4,17,29,42, 7,21, 32,44,10,22,34)( 3,14,25,38, 6,18,28,41, 8,20,31,43,12,24,36)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,15,25,39, 4,18,30,42, 8,19,32,43,11,22,36)( 2,14,27,37, 6,16,29,41, 9,21, 31,45,10,24,35)( 3,13,26,38, 5,17,28,40, 7,20,33,44,12,23,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,33)( 2,17,32)( 3,18,31)( 4,21,34)( 5,19,35)( 6,20,36)( 7,22,37) ( 8,24,38)( 9,23,39)(10,26,42)(11,27,40)(12,25,41)(13,30,45)(14,28,43) (15,29,44)$ |
| $ 6, 6, 6, 6, 6, 3, 3, 3, 3, 3 $ | $3$ | $6$ | $( 1,16,32, 2,18,31)( 3,17,33)( 4,21,36, 6,19,35)( 5,20,34)( 7,23,38) ( 8,24,39, 9,22,37)(10,25,41,11,27,42)(12,26,40)(13,28,44)(14,30,45,15,29,43)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,18,32)( 2,16,31)( 3,17,33)( 4,19,36)( 5,20,34)( 6,21,35)( 7,23,38) ( 8,22,39)( 9,24,37)(10,27,41)(11,25,42)(12,26,40)(13,28,44)(14,29,45) (15,30,43)$ |
| $ 10, 10, 10, 5, 5, 5 $ | $3$ | $10$ | $( 1,19,39,11,30)( 2,20,37,12,29, 3,21,38,10,28)( 4,22,42,15,32) ( 5,24,40,14,33, 6,23,41,13,31)( 7,27,44,16,34, 9,26,45,17,35)( 8,25,43,18,36)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,19,39,11,30)( 2,21,37,10,29)( 3,20,38,12,28)( 4,22,42,15,32) ( 5,23,40,13,33)( 6,24,41,14,31)( 7,26,44,17,34)( 8,25,43,18,36) ( 9,27,45,16,35)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,20,37,11,28, 2,19,38,10,30, 3,21,39,12,29)( 4,23,41,15,33, 6,22,40,14,32, 5,24,42,13,31)( 7,27,43,17,35, 8,26,45,18,34, 9,25,44,16,36)$ |
| $ 30, 15 $ | $3$ | $30$ | $( 1,22,43,19,42,18,39,15,36,11,32, 8,30, 4,25)( 2,23,45,20,41,17,37,13,35,12, 31, 7,29, 5,27, 3,24,44,21,40,16,38,14,34,10,33, 9,28, 6,26)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,22,43,19,42,18,39,15,36,11,32, 8,30, 4,25)( 2,24,45,21,41,16,37,14,35,10, 31, 9,29, 6,27)( 3,23,44,20,40,17,38,13,34,12,33, 7,28, 5,26)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,23,45,19,40,16,39,13,35,11,33, 9,30, 5,27)( 2,22,44,21,42,17,37,15,34,10, 32, 7,29, 4,26)( 3,24,43,20,41,18,38,14,36,12,31, 8,28, 6,25)$ |
| $ 30, 15 $ | $3$ | $30$ | $( 1,25, 4,30, 8,32,11,36,15,39,18,42,19,43,22)( 2,26, 6,28, 9,33,10,34,14,38, 16,40,21,44,24, 3,27, 5,29, 7,31,12,35,13,37,17,41,20,45,23)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,25, 4,30, 8,32,11,36,15,39,18,42,19,43,22)( 2,27, 6,29, 9,31,10,35,14,37, 16,41,21,45,24)( 3,26, 5,28, 7,33,12,34,13,38,17,40,20,44,23)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,26, 6,30, 7,31,11,34,14,39,17,41,19,44,24)( 2,25, 5,29, 8,33,10,36,13,37, 18,40,21,43,23)( 3,27, 4,28, 9,32,12,35,15,38,16,42,20,45,22)$ |
| $ 10, 10, 10, 5, 5, 5 $ | $3$ | $10$ | $( 1,28,11,38,19, 3,30,12,39,20)( 2,29,10,37,21)( 4,33,15,40,22, 5,32,13,42,23) ( 6,31,14,41,24)( 7,36,17,43,26, 8,34,18,44,25)( 9,35,16,45,27)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,28,10,39,20, 2,30,12,37,19, 3,29,11,38,21)( 4,33,14,42,23, 6,32,13,41,22, 5,31,15,40,24)( 7,35,18,44,27, 8,34,16,43,26, 9,36,17,45,25)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,30,11,39,19)( 2,29,10,37,21)( 3,28,12,38,20)( 4,32,15,42,22) ( 5,33,13,40,23)( 6,31,14,41,24)( 7,34,17,44,26)( 8,36,18,43,25) ( 9,35,16,45,27)$ |
| $ 6, 6, 6, 6, 6, 3, 3, 3, 3, 3 $ | $3$ | $6$ | $( 1,31,18, 2,32,16)( 3,33,17)( 4,35,19, 6,36,21)( 5,34,20)( 7,38,23) ( 8,37,22, 9,39,24)(10,42,27,11,41,25)(12,40,26)(13,44,28)(14,43,29,15,45,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,31,17)( 2,33,18)( 3,32,16)( 4,35,20)( 5,36,21)( 6,34,19)( 7,39,24) ( 8,37,23)( 9,38,22)(10,40,25)(11,41,26)(12,42,27)(13,43,29)(14,44,30) (15,45,28)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,32,18)( 2,31,16)( 3,33,17)( 4,36,19)( 5,34,20)( 6,35,21)( 7,38,23) ( 8,39,22)( 9,37,24)(10,41,27)(11,42,25)(12,40,26)(13,44,28)(14,45,29) (15,43,30)$ |
| $ 30, 15 $ | $3$ | $30$ | $( 1,34,22,12,43,33,19, 7,42,28,18, 5,39,26,15, 3,36,23,11,44,32,20, 8,40,30, 17, 4,38,25,13)( 2,35,24,10,45,31,21, 9,41,29,16, 6,37,27,14)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,34,24,11,44,31,19, 7,41,30,17, 6,39,26,14)( 2,36,23,10,43,33,21, 8,40,29, 18, 5,37,25,13)( 3,35,22,12,45,32,20, 9,42,28,16, 4,38,27,15)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,36,22,11,43,32,19, 8,42,30,18, 4,39,25,15)( 2,35,24,10,45,31,21, 9,41,29, 16, 6,37,27,14)( 3,34,23,12,44,33,20, 7,40,28,17, 5,38,26,13)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,37,28,19,10, 3,39,29,20,11, 2,38,30,21,12)( 4,41,33,22,14, 5,42,31,23,15, 6,40,32,24,13)( 7,43,35,26,18, 9,44,36,27,17, 8,45,34,25,16)$ |
| $ 10, 10, 10, 5, 5, 5 $ | $3$ | $10$ | $( 1,37,30,21,11, 2,39,29,19,10)( 3,38,28,20,12)( 4,41,32,24,15, 6,42,31,22,14) ( 5,40,33,23,13)( 7,44,34,26,17)( 8,45,36,27,18, 9,43,35,25,16)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,39,30,19,11)( 2,37,29,21,10)( 3,38,28,20,12)( 4,42,32,22,15) ( 5,40,33,23,13)( 6,41,31,24,14)( 7,44,34,26,17)( 8,43,36,25,18) ( 9,45,35,27,16)$ |
| $ 30, 15 $ | $3$ | $30$ | $( 1,40,36,28,22,17,11, 5,43,38,32,26,19,13, 8, 3,42,34,30,23,18,12, 4,44,39, 33,25,20,15, 7)( 2,41,35,29,24,16,10, 6,45,37,31,27,21,14, 9)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,40,35,30,23,16,11, 5,45,39,33,27,19,13, 9)( 2,42,34,29,22,17,10, 4,44,37, 32,26,21,15, 7)( 3,41,36,28,24,18,12, 6,43,38,31,25,20,14, 8)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,42,36,30,22,18,11, 4,43,39,32,25,19,15, 8)( 2,41,35,29,24,16,10, 6,45,37, 31,27,21,14, 9)( 3,40,34,28,23,17,12, 5,44,38,33,26,20,13, 7)$ |
| $ 30, 15 $ | $3$ | $30$ | $( 1,43,42,39,36,32,30,25,22,19,18,15,11, 8, 4)( 2,44,41,38,35,33,29,26,24,20, 16,13,10, 7, 6, 3,45,40,37,34,31,28,27,23,21,17,14,12, 9, 5)$ |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,43,42,39,36,32,30,25,22,19,18,15,11, 8, 4)( 2,45,41,37,35,31,29,27,24,21, 16,14,10, 9, 6)( 3,44,40,38,34,33,28,26,23,20,17,13,12, 7, 5)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,44,41,39,34,31,30,26,24,19,17,14,11, 7, 6)( 2,43,40,37,36,33,29,25,23,21, 18,13,10, 8, 5)( 3,45,42,38,35,32,28,27,22,20,16,15,12, 9, 4)$ |
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: | 90.6 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);