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Magma
magma: G := TransitiveGroup(45, 36);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{45}:C_6$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,36,33,20,16,6,2,35,32,19,18,4,3,34,31,21,17,5)(7,15,39,44,22,30,8,13,38,43,24,29,9,14,37,45,23,28)(10,25,40,11,26,42,12,27,41), (1,11,20,30,39,3,12,21,28,37,2,10,19,29,38)(4,13,23,33,41)(5,14,22,32,42)(6,15,24,31,40)(7,17,27,34,44,8,16,26,36,43,9,18,25,35,45) | 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 $9$: $C_3^2$ $10$: $D_{5}$ $18$: $C_6 \times C_3$ $27$: $C_9:C_3$ $30$: $D_5\times C_3$ x 4 $54$: 18T14 $90$: 45T8 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: $D_{5}$
Degree 9: $C_9:C_3$
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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 4, 5, 6)( 7, 9, 8)(13,14,15)(16,17,18)(22,24,23)(25,26,27)(31,33,32) (34,35,36)(40,41,42)(43,44,45)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 4, 6, 5)( 7, 8, 9)(13,15,14)(16,18,17)(22,23,24)(25,27,26)(31,32,33) (34,36,35)(40,42,41)(43,45,44)$ |
| $ 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 $ | $5$ | $2$ | $( 4,13)( 5,14)( 6,15)( 7,25)( 8,27)( 9,26)(10,38)(11,39)(12,37)(19,30)(20,28) (21,29)(22,42)(23,41)(24,40)(34,43)(35,44)(36,45)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $15$ | $6$ | $( 4,14, 6,13, 5,15)( 7,26, 8,25, 9,27)(10,38)(11,39)(12,37)(16,17,18)(19,30) (20,28)(21,29)(22,40,23,42,24,41)(31,33,32)(34,44,36,43,35,45)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $15$ | $6$ | $( 4,15, 5,13, 6,14)( 7,27, 9,25, 8,26)(10,38)(11,39)(12,37)(16,18,17)(19,30) (20,28)(21,29)(22,41,24,42,23,40)(31,32,33)(34,45,35,43,36,44)$ |
| $ 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 $ | $5$ | $6$ | $( 1, 2, 3)( 4,14, 6,13, 5,15)( 7,27, 9,25, 8,26)(10,37,11,38,12,39)(16,18,17) (19,29,20,30,21,28)(22,40,23,42,24,41)(31,33,32)(34,45,35,43,36,44)$ |
| $ 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 $ | $5$ | $6$ | $( 1, 3, 2)( 4,15, 5,13, 6,14)( 7,26, 8,25, 9,27)(10,39,12,38,11,37)(16,17,18) (19,28,21,30,20,29)(22,41,24,42,23,40)(31,32,33)(34,44,36,43,35,45)$ |
| $ 45 $ | $6$ | $45$ | $( 1, 4, 7,11,14,18,20,24,26,30,33,34,39,42,45, 3, 6, 9,12,13,16,21,22,27,28, 31,35,37,41,43, 2, 5, 8,10,15,17,19,23,25,29,32,36,38,40,44)$ |
| $ 45 $ | $6$ | $45$ | $( 1, 4, 8,11,14,17,20,24,25,30,33,36,39,42,44, 3, 6, 7,12,13,18,21,22,26,28, 31,34,37,41,45, 2, 5, 9,10,15,16,19,23,27,29,32,35,38,40,43)$ |
| $ 45 $ | $6$ | $45$ | $( 1, 4, 9,11,14,16,20,24,27,30,33,35,39,42,43, 3, 6, 8,12,13,17,21,22,25,28, 31,36,37,41,44, 2, 5, 7,10,15,18,19,23,26,29,32,34,38,40,45)$ |
| $ 18, 18, 9 $ | $15$ | $18$ | $( 1, 4,16,21,32,36, 3, 6,17,19,33,34, 2, 5,18,20,31,35)( 7,29,22,45,38,15, 9, 30,23,43,39,14, 8,28,24,44,37,13)(10,40,26,12,41,25,11,42,27)$ |
| $ 18, 18, 9 $ | $15$ | $18$ | $( 1, 4,17,21,32,34, 3, 6,18,19,33,35, 2, 5,16,20,31,36)( 7,28,24,45,37,13, 9, 29,22,43,38,15, 8,30,23,44,39,14)(10,40,27,12,41,26,11,42,25)$ |
| $ 18, 18, 9 $ | $15$ | $18$ | $( 1, 4,18,21,32,35, 3, 6,16,19,33,36, 2, 5,17,20,31,34)( 7,30,23,45,39,14, 9, 28,24,43,37,13, 8,29,22,44,38,15)(10,40,25,12,41,27,11,42,26)$ |
| $ 45 $ | $6$ | $45$ | $( 1, 7,13,20,26,31,39,45, 5,12,16,23,28,35,40, 2, 8,14,19,25,33,38,44, 6,11, 18,22,30,34,41, 3, 9,15,21,27,32,37,43, 4,10,17,24,29,36,42)$ |
| $ 45 $ | $6$ | $45$ | $( 1, 7,14,20,26,33,39,45, 6,12,16,22,28,35,41, 2, 8,15,19,25,32,38,44, 4,11, 18,24,30,34,42, 3, 9,13,21,27,31,37,43, 5,10,17,23,29,36,40)$ |
| $ 45 $ | $6$ | $45$ | $( 1, 7,15,20,26,32,39,45, 4,12,16,24,28,35,42, 2, 8,13,19,25,31,38,44, 5,11, 18,23,30,34,40, 3, 9,14,21,27,33,37,43, 6,10,17,22,29,36,41)$ |
| $ 18, 18, 9 $ | $15$ | $18$ | $( 1, 7,31,38,17,22, 2, 8,33,37,16,24, 3, 9,32,39,18,23)( 4,19,34, 6,20,35, 5, 21,36)(10,44,42,29,27,13,12,43,40,28,26,14,11,45,41,30,25,15)$ |
| $ 18, 18, 9 $ | $15$ | $18$ | $( 1, 7,33,38,17,24, 2, 8,32,37,16,23, 3, 9,31,39,18,22)( 4,20,35, 6,21,36, 5, 19,34)(10,44,40,29,27,14,12,43,41,28,26,15,11,45,42,30,25,13)$ |
| $ 18, 18, 9 $ | $15$ | $18$ | $( 1, 7,32,38,17,23, 2, 8,31,37,16,22, 3, 9,33,39,18,24)( 4,21,36, 6,19,34, 5, 20,35)(10,44,41,29,27,15,12,43,42,28,26,13,11,45,40,30,25,14)$ |
| $ 15, 15, 5, 5, 5 $ | $6$ | $15$ | $( 1,10,21,30,38, 2,12,20,29,37, 3,11,19,28,39)( 4,13,23,33,41)( 5,14,22,32,42) ( 6,15,24,31,40)( 7,18,26,34,45, 9,16,27,35,43, 8,17,25,36,44)$ |
| $ 15, 15, 5, 5, 5 $ | $6$ | $15$ | $( 1,10,21,30,38, 2,12,20,29,37, 3,11,19,28,39)( 4,14,24,33,42, 6,13,22,31,41, 5,15,23,32,40)( 7,16,25,34,43)( 8,18,27,36,45)( 9,17,26,35,44)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,10,21,30,38, 2,12,20,29,37, 3,11,19,28,39)( 4,15,22,33,40, 5,13,24,32,41, 6,14,23,31,42)( 7,17,27,34,44, 8,16,26,36,43, 9,18,25,35,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,11,20,30,39, 3,12,21,28,37, 2,10,19,29,38)( 4,14,24,33,42, 6,13,22,31,41, 5,15,23,32,40)( 7,18,26,34,45, 9,16,27,35,43, 8,17,25,36,44)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,12,19,30,37)( 2,11,21,29,39)( 3,10,20,28,38)( 4,13,23,33,41) ( 5,14,22,32,42)( 6,15,24,31,40)( 7,16,25,34,43)( 8,18,27,36,45) ( 9,17,26,35,44)$ |
| $ 9, 9, 9, 9, 9 $ | $3$ | $9$ | $( 1,16,31, 3,17,32, 2,18,33)( 4,19,34, 6,20,35, 5,21,36)( 7,24,38, 9,22,39, 8, 23,37)(10,26,42,11,27,41,12,25,40)(13,30,43,15,28,44,14,29,45)$ |
| $ 9, 9, 9, 9, 9 $ | $3$ | $9$ | $( 1,16,33, 3,17,31, 2,18,32)( 4,20,35, 6,21,36, 5,19,34)( 7,23,38, 9,24,39, 8, 22,37)(10,26,40,11,27,42,12,25,41)(13,28,44,15,29,45,14,30,43)$ |
| $ 9, 9, 9, 9, 9 $ | $3$ | $9$ | $( 1,16,32, 3,17,33, 2,18,31)( 4,21,36, 6,19,34, 5,20,35)( 7,22,38, 9,23,39, 8, 24,37)(10,26,41,11,27,40,12,25,42)(13,29,45,15,30,43,14,28,44)$ |
| $ 15, 15, 5, 5, 5 $ | $6$ | $15$ | $( 1,19,37,12,30)( 2,21,39,11,29)( 3,20,38,10,28)( 4,22,40,13,32, 6,23,42,15, 33, 5,24,41,14,31)( 7,26,45,16,35, 8,25,44,18,34, 9,27,43,17,36)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,19,37,12,30)( 2,21,39,11,29)( 3,20,38,10,28)( 4,23,41,13,33) ( 5,22,42,14,32)( 6,24,40,15,31)( 7,25,43,16,34)( 8,27,45,18,36) ( 9,26,44,17,35)$ |
| $ 15, 15, 5, 5, 5 $ | $6$ | $15$ | $( 1,19,37,12,30)( 2,21,39,11,29)( 3,20,38,10,28)( 4,24,42,13,31, 5,23,40,14, 33, 6,22,41,15,32)( 7,27,44,16,36, 9,25,45,17,34, 8,26,43,18,35)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,20,39,12,28, 2,19,38,11,30, 3,21,37,10,29)( 4,24,42,13,31, 5,23,40,14,33, 6,22,41,15,32)( 7,26,45,16,35, 8,25,44,18,34, 9,27,43,17,36)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,21,38,12,29, 3,19,39,10,30, 2,20,37,11,28)( 4,22,40,13,32, 6,23,42,15,33, 5,24,41,14,31)( 7,27,44,16,36, 9,25,45,17,34, 8,26,43,18,35)$ |
| $ 45 $ | $6$ | $45$ | $( 1,22,45,21,40,17,38,13,34,12,32, 8,29, 6,26, 3,23,43,19,42,18,39,15,35,10, 33, 7,30, 5,27, 2,24,44,20,41,16,37,14,36,11,31, 9,28, 4,25)$ |
| $ 45 $ | $6$ | $45$ | $( 1,22,43,21,40,18,38,13,35,12,32, 7,29, 6,27, 3,23,44,19,42,16,39,15,36,10, 33, 9,30, 5,25, 2,24,45,20,41,17,37,14,34,11,31, 8,28, 4,26)$ |
| $ 45 $ | $6$ | $45$ | $( 1,22,44,21,40,16,38,13,36,12,32, 9,29, 6,25, 3,23,45,19,42,17,39,15,34,10, 33, 8,30, 5,26, 2,24,43,20,41,18,37,14,35,11,31, 7,28, 4,27)$ |
| $ 9, 9, 9, 9, 9 $ | $3$ | $9$ | $( 1,31,17, 2,33,16, 3,32,18)( 4,34,20, 5,36,19, 6,35,21)( 7,38,22, 8,37,24, 9, 39,23)(10,42,27,12,40,26,11,41,25)(13,43,28,14,45,30,15,44,29)$ |
| $ 9, 9, 9, 9, 9 $ | $3$ | $9$ | $( 1,31,18, 2,33,17, 3,32,16)( 4,35,20, 5,34,19, 6,36,21)( 7,37,24, 8,39,23, 9, 38,22)(10,42,25,12,40,27,11,41,26)(13,44,28,14,43,30,15,45,29)$ |
| $ 9, 9, 9, 9, 9 $ | $3$ | $9$ | $( 1,31,16, 2,33,18, 3,32,17)( 4,36,20, 5,35,19, 6,34,21)( 7,39,23, 8,38,22, 9, 37,24)(10,42,26,12,40,25,11,41,27)(13,45,28,14,44,30,15,43,29)$ |
| $ 45 $ | $6$ | $45$ | $( 1,34,24,10,44,32,21, 8,41,30,16, 6,38,26,14, 2,36,23,12,43,31,20, 9,42,29, 18, 4,37,25,15, 3,35,22,11,45,33,19, 7,40,28,17, 5,39,27,13)$ |
| $ 45 $ | $6$ | $45$ | $( 1,34,23,10,44,31,21, 8,42,30,16, 4,38,26,15, 2,36,22,12,43,33,20, 9,40,29, 18, 5,37,25,13, 3,35,24,11,45,32,19, 7,41,28,17, 6,39,27,14)$ |
| $ 45 $ | $6$ | $45$ | $( 1,34,22,10,44,33,21, 8,40,30,16, 5,38,26,13, 2,36,24,12,43,32,20, 9,41,29, 18, 6,37,25,14, 3,35,23,11,45,31,19, 7,42,28,17, 4,39,27,15)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $270=2 \cdot 3^{3} \cdot 5$ | 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: | 270.7 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);