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Magma
magma: G := TransitiveGroup(45, 10);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{15}:S_3$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $5$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,13,11,22,20,32,28,41,37,6)(2,14,12,24,21,33,29,42,39,4)(3,15,10,23,19,31,30,40,38,5)(7,35,16,45,26)(8,36,18,43,25,9,34,17,44,27), (1,34,37,25,28,18,20,8,11,44)(2,35,39,26,29,16,21,7,12,45)(3,36,38,27,30,17,19,9,10,43)(4,24,42,14,33)(5,22,40,13,31,6,23,41,15,32), (1,16)(2,17)(3,18)(4,5)(7,37)(8,38)(9,39)(10,25)(11,26)(12,27)(14,15)(19,34)(20,35)(21,36)(23,24)(28,45)(29,43)(30,44)(31,33)(40,42) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $6$: $S_3$ x 4 $10$: $C_{10}$ $18$: $C_3^2:C_2$ $30$: $S_3 \times C_5$ x 4 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$ x 4
Degree 5: $C_5$
Degree 9: $C_3^2:C_2$
Degree 15: $S_3 \times C_5$ 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, 2, 2, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 2, 3)( 4,34)( 5,36)( 6,35)( 7,22)( 8,24)( 9,23)(10,12)(13,45)(14,44)(15,43) (16,32)(17,31)(18,33)(19,21)(25,42)(26,41)(27,40)(29,30)(38,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,20,21) (22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41) (43,45,44)$ |
| $ 10, 10, 10, 10, 5 $ | $9$ | $10$ | $( 1, 4,37,42,28,33,20,24,11,14)( 2, 5,39,40,29,31,21,23,12,15)( 3, 6,38,41,30, 32,19,22,10,13)( 7,27,45,17,35, 9,26,43,16,36)( 8,25,44,18,34)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 4, 8,11,14,18,20,24,25,28,33,34,37,42,44)( 2, 6, 9,12,13,17,21,22,27,29, 32,36,39,41,43)( 3, 5, 7,10,15,16,19,23,26,30,31,35,38,40,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 5, 9,11,15,17,20,23,27,28,31,36,37,40,43)( 2, 4, 7,12,14,16,21,24,26,29, 33,35,39,42,45)( 3, 6, 8,10,13,18,19,22,25,30,32,34,38,41,44)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 6, 7,11,13,16,20,22,26,28,32,35,37,41,45)( 2, 5, 8,12,15,18,21,23,25,29, 31,34,39,40,44)( 3, 4, 9,10,14,17,19,24,27,30,33,36,38,42,43)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 7,13,20,26,32,37,45, 6,11,16,22,28,35,41)( 2, 8,15,21,25,31,39,44, 5,12, 18,23,29,34,40)( 3, 9,14,19,27,33,38,43, 4,10,17,24,30,36,42)$ |
| $ 10, 10, 10, 10, 5 $ | $9$ | $10$ | $( 1, 7,28,35,11,16,37,45,20,26)( 2, 9,29,36,12,17,39,43,21,27)( 3, 8,30,34,10, 18,38,44,19,25)( 4,40,33,23,14, 5,42,31,24,15)( 6,41,32,22,13)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 8,14,20,25,33,37,44, 4,11,18,24,28,34,42)( 2, 9,13,21,27,32,39,43, 6,12, 17,22,29,36,41)( 3, 7,15,19,26,31,38,45, 5,10,16,23,30,35,40)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 9,15,20,27,31,37,43, 5,11,17,23,28,36,40)( 2, 7,14,21,26,33,39,45, 4,12, 16,24,29,35,42)( 3, 8,13,19,25,32,38,44, 6,10,18,22,30,34,41)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,10,21,28,38, 2,11,19,29,37, 3,12,20,30,39)( 4,15,22,33,40, 6,14,23,32,42, 5,13,24,31,41)( 7,17,25,35,43, 8,16,27,34,45, 9,18,26,36,44)$ |
| $ 10, 10, 10, 10, 5 $ | $9$ | $10$ | $( 1,10,20,30,37, 3,11,19,28,38)( 2,12,21,29,39)( 4,45,24,16,42,35,14, 7,33,26) ( 5,44,23,18,40,34,15, 8,31,25)( 6,43,22,17,41,36,13, 9,32,27)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,11,20,28,37)( 2,12,21,29,39)( 3,10,19,30,38)( 4,14,24,33,42) ( 5,15,23,31,40)( 6,13,22,32,41)( 7,16,26,35,45)( 8,18,25,34,44) ( 9,17,27,36,43)$ |
| $ 10, 10, 10, 10, 5 $ | $9$ | $10$ | $( 1,13,11,22,20,32,28,41,37, 6)( 2,14,12,24,21,33,29,42,39, 4)( 3,15,10,23,19, 31,30,40,38, 5)( 7,35,16,45,26)( 8,36,18,43,25, 9,34,17,44,27)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,13,26,37, 6,16,28,41, 7,20,32,45,11,22,35)( 2,15,25,39, 5,18,29,40, 8,21, 31,44,12,23,34)( 3,14,27,38, 4,17,30,42, 9,19,33,43,10,24,36)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,14,25,37, 4,18,28,42, 8,20,33,44,11,24,34)( 2,13,27,39, 6,17,29,41, 9,21, 32,43,12,22,36)( 3,15,26,38, 5,16,30,40, 7,19,31,45,10,23,35)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,15,27,37, 5,17,28,40, 9,20,31,43,11,23,36)( 2,14,26,39, 4,16,29,42, 7,21, 33,45,12,24,35)( 3,13,25,38, 6,18,30,41, 8,19,32,44,10,22,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,32)( 2,18,31)( 3,17,33)( 4,19,36)( 5,21,34)( 6,20,35)( 7,22,37) ( 8,23,39)( 9,24,38)(10,27,42)(11,26,41)(12,25,40)(13,28,45)(14,30,43) (15,29,44)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,17,31)( 2,16,33)( 3,18,32)( 4,21,35)( 5,20,36)( 6,19,34)( 7,24,39) ( 8,22,38)( 9,23,37)(10,25,41)(11,27,40)(12,26,42)(13,30,44)(14,29,45) (15,28,43)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,18,33)( 2,17,32)( 3,16,31)( 4,20,34)( 5,19,35)( 6,21,36)( 7,23,38) ( 8,24,37)( 9,22,39)(10,26,40)(11,25,42)(12,27,41)(13,29,43)(14,28,44) (15,30,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,19,39,11,30, 2,20,38,12,28, 3,21,37,10,29)( 4,23,41,14,31, 6,24,40,13,33, 5,22,42,15,32)( 7,27,44,16,36, 8,26,43,18,35, 9,25,45,17,34)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,20,37,11,28)( 2,21,39,12,29)( 3,19,38,10,30)( 4,24,42,14,33) ( 5,23,40,15,31)( 6,22,41,13,32)( 7,26,45,16,35)( 8,25,44,18,34) ( 9,27,43,17,36)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,25, 4,28, 8,33,11,34,14,37,18,42,20,44,24)( 2,27, 6,29, 9,32,12,36,13,39, 17,41,21,43,22)( 3,26, 5,30, 7,31,10,35,15,38,16,40,19,45,23)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,26, 6,28, 7,32,11,35,13,37,16,41,20,45,22)( 2,25, 5,29, 8,31,12,34,15,39, 18,40,21,44,23)( 3,27, 4,30, 9,33,10,36,14,38,17,42,19,43,24)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,27, 5,28, 9,31,11,36,15,37,17,40,20,43,23)( 2,26, 4,29, 7,33,12,35,14,39, 16,42,21,45,24)( 3,25, 6,30, 8,32,10,34,13,38,18,41,19,44,22)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,28,11,37,20)( 2,29,12,39,21)( 3,30,10,38,19)( 4,33,14,42,24) ( 5,31,15,40,23)( 6,32,13,41,22)( 7,35,16,45,26)( 8,34,18,44,25) ( 9,36,17,43,27)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,29,10,37,21, 3,28,12,38,20, 2,30,11,39,19)( 4,32,15,42,22, 5,33,13,40,24, 6,31,14,41,23)( 7,34,17,45,25, 9,35,18,43,26, 8,36,16,44,27)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,37,28,20,11)( 2,39,29,21,12)( 3,38,30,19,10)( 4,42,33,24,14) ( 5,40,31,23,15)( 6,41,32,22,13)( 7,45,35,26,16)( 8,44,34,25,18) ( 9,43,36,27,17)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,38,29,20,10, 2,37,30,21,11, 3,39,28,19,12)( 4,40,32,24,15, 6,42,31,22,14, 5,41,33,23,13)( 7,43,34,26,17, 8,45,36,25,16, 9,44,35,27,18)$ |
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: | 90.8 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);