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Magma
magma: G := TransitiveGroup(45, 19);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $19$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^2:F_5$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,15,39,42)(2,13,38,41)(3,14,37,40)(4,21)(5,20)(6,19)(7,26,18,44)(8,27,17,45)(9,25,16,43)(10,33,30,24)(11,31,29,22)(12,32,28,23)(35,36), (1,21,11,38)(2,19,12,39)(3,20,10,37)(4,26,23,17)(5,25,24,16)(6,27,22,18)(7,31,36,42)(8,32,35,41)(9,33,34,40)(13,44)(14,43)(15,45)(28,29), (1,25,6,29,9,31,11,34,15,39,16,42,19,43,22)(2,27,5,28,7,33,12,36,14,38,18,40,21,45,24)(3,26,4,30,8,32,10,35,13,37,17,41,20,44,23) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $6$: $S_3$ x 4 $12$: $C_3 : C_4$ x 4 $18$: $C_3^2:C_2$ $20$: $F_5$ $36$: 36T7 $60$: $C_{15} : C_4$ x 4 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$ x 4
Degree 5: $F_5$
Degree 9: $C_3^2:C_2$
Degree 15: $C_{15} : C_4$ 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 $ | $5$ | $2$ | $( 4,13)( 5,14)( 6,15)( 7,27)( 8,26)( 9,25)(10,37)(11,39)(12,38)(19,29)(20,30) (21,28)(22,42)(23,41)(24,40)(34,43)(35,44)(36,45)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1 $ | $45$ | $4$ | $( 2, 3)( 4, 7,13,27)( 5, 8,14,26)( 6, 9,15,25)(10,21,37,28)(11,19,39,29) (12,20,38,30)(16,31)(17,33)(18,32)(22,43,42,34)(23,45,41,36)(24,44,40,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1 $ | $45$ | $4$ | $( 2, 3)( 4,27,13, 7)( 5,26,14, 8)( 6,25,15, 9)(10,28,37,21)(11,29,39,19) (12,30,38,20)(16,31)(17,33)(18,32)(22,34,42,43)(23,36,41,45)(24,35,40,44)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,45,44)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $10$ | $6$ | $( 1, 2, 3)( 4,15, 5,13, 6,14)( 7,26, 9,27, 8,25)(10,39,12,37,11,38)(16,18,17) (19,28,20,29,21,30)(22,40,23,42,24,41)(31,33,32)(34,45,35,43,36,44)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 4, 7,11,13,18,19,23,27,29,32,36,39,41,45)( 2, 6, 8,12,15,17,21,22,26,28, 31,35,38,42,44)( 3, 5, 9,10,14,16,20,24,25,30,33,34,37,40,43)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $10$ | $6$ | $( 1, 4,18,19,32,36)( 2, 6,17,21,31,35)( 3, 5,16,20,33,34)( 7,29,23,45,39,13) ( 8,28,22,44,38,15)( 9,30,24,43,37,14)(10,40,25)(11,41,27)(12,42,26)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 5, 8,11,14,17,19,24,26,29,33,35,39,40,44)( 2, 4, 9,12,13,16,21,23,25,28, 32,34,38,41,43)( 3, 6, 7,10,15,18,20,22,27,30,31,36,37,42,45)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $10$ | $6$ | $( 1, 5,17,19,33,35)( 2, 4,16,21,32,34)( 3, 6,18,20,31,36)( 7,30,22,45,37,15) ( 8,29,24,44,39,14)( 9,28,23,43,38,13)(10,42,27)(11,40,26)(12,41,25)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 6, 9,11,15,16,19,22,25,29,31,34,39,42,43)( 2, 5, 7,12,14,18,21,24,27,28, 33,36,38,40,45)( 3, 4, 8,10,13,17,20,23,26,30,32,35,37,41,44)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $10$ | $6$ | $( 1, 6,16,19,31,34)( 2, 5,18,21,33,36)( 3, 4,17,20,32,35)( 7,28,24,45,38,14) ( 8,30,23,44,37,13)( 9,29,22,43,39,15)(10,41,26)(11,42,25)(12,40,27)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,10,21,29,37, 2,11,20,28,39, 3,12,19,30,38)( 4,14,22,32,40, 6,13,24,31,41, 5,15,23,33,42)( 7,16,26,36,43, 8,18,25,35,45, 9,17,27,34,44)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,11,19,29,39)( 2,12,21,28,38)( 3,10,20,30,37)( 4,13,23,32,41) ( 5,14,24,33,40)( 6,15,22,31,42)( 7,18,27,36,45)( 8,17,26,35,44) ( 9,16,25,34,43)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,12,20,29,38, 3,11,21,30,39, 2,10,19,28,37)( 4,15,24,32,42, 5,13,22,33,41, 6,14,23,31,40)( 7,17,25,36,44, 9,18,26,34,45, 8,16,27,35,43)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,31)( 2,18,33)( 3,17,32)( 4,20,35)( 5,21,36)( 6,19,34)( 7,24,38) ( 8,23,37)( 9,22,39)(10,26,41)(11,25,42)(12,27,40)(13,30,44)(14,28,45) (15,29,43)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,17,33)( 2,16,32)( 3,18,31)( 4,21,34)( 5,19,35)( 6,20,36)( 7,22,37) ( 8,24,39)( 9,23,38)(10,27,42)(11,26,40)(12,25,41)(13,28,43)(14,29,44) (15,30,45)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,18,32)( 2,17,31)( 3,16,33)( 4,19,36)( 5,20,34)( 6,21,35)( 7,23,39) ( 8,22,38)( 9,24,37)(10,25,40)(11,27,41)(12,26,42)(13,29,45)(14,30,43) (15,28,44)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,22,43,19,42,16,39,15,34,11,31, 9,29, 6,25)( 2,24,45,21,40,18,38,14,36,12, 33, 7,28, 5,27)( 3,23,44,20,41,17,37,13,35,10,32, 8,30, 4,26)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,23,45,19,41,18,39,13,36,11,32, 7,29, 4,27)( 2,22,44,21,42,17,38,15,35,12, 31, 8,28, 6,26)( 3,24,43,20,40,16,37,14,34,10,33, 9,30, 5,25)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,24,44,19,40,17,39,14,35,11,33, 8,29, 5,26)( 2,23,43,21,41,16,38,13,34,12, 32, 9,28, 4,25)( 3,22,45,20,42,18,37,15,36,10,31, 7,30, 6,27)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $180=2^{2} \cdot 3^{2} \cdot 5$ | 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: | 180.22 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);