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Magma
magma: G := TransitiveGroup(36, 3);
Group action invariants
| Degree $n$: | $36$ | 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: | $C_3\times C_{12}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $36$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,27,16,4,25,13,2,28,15,3,26,14)(5,31,20,7,29,17,6,32,19,8,30,18)(9,35,24,12,34,21,10,36,23,11,33,22), (1,23,19,2,24,20)(3,21,17,4,22,18)(5,26,10,6,25,9)(7,27,11,8,28,12)(13,35,31,14,36,32)(15,34,29,16,33,30) | 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 $4$: $C_4$ $6$: $C_6$ x 4 $9$: $C_3^2$ $12$: $C_{12}$ x 4 $18$: $C_6 \times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$ x 4
Degree 4: $C_4$
Degree 6: $C_6$ x 4
Degree 9: $C_3^2$
Degree 12: $C_{12}$ x 4
Degree 18: $C_6 \times C_3$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,24,22,23) (25,27,26,28)(29,31,30,32)(33,36,34,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,23,22,24) (25,28,26,27)(29,32,30,31)(33,35,34,36)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5,33)( 2, 6,34)( 3, 8,36)( 4, 7,35)( 9,16,20)(10,15,19)(11,14,18) (12,13,17)(21,28,32)(22,27,31)(23,26,30)(24,25,29)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1, 6,33, 2, 5,34)( 3, 7,36, 4, 8,35)( 9,15,20,10,16,19)(11,13,18,12,14,17) (21,27,32,22,28,31)(23,25,30,24,26,29)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1, 7,34, 3, 5,35, 2, 8,33, 4, 6,36)( 9,13,19,11,16,17,10,14,20,12,15,18) (21,26,31,24,28,30,22,25,32,23,27,29)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1, 8,34, 4, 5,36, 2, 7,33, 3, 6,35)( 9,14,19,12,16,18,10,13,20,11,15,17) (21,25,31,23,28,29,22,26,32,24,27,30)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1, 9,29, 2,10,30)( 3,11,31, 4,12,32)( 5,16,24, 6,15,23)( 7,13,21, 8,14,22) (17,28,36,18,27,35)(19,26,33,20,25,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,10,29)( 2, 9,30)( 3,12,31)( 4,11,32)( 5,15,24)( 6,16,23)( 7,14,21) ( 8,13,22)(17,27,36)(18,28,35)(19,25,33)(20,26,34)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,11,30, 3,10,32, 2,12,29, 4, 9,31)( 5,14,23, 8,15,21, 6,13,24, 7,16,22) (17,25,35,20,27,33,18,26,36,19,28,34)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,12,30, 4,10,31, 2,11,29, 3, 9,32)( 5,13,23, 7,15,22, 6,14,24, 8,16,21) (17,26,35,19,27,34,18,25,36,20,28,33)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,13,26, 4,15,27, 2,14,25, 3,16,28)( 5,17,30, 7,19,31, 6,18,29, 8,20,32) ( 9,21,33,12,23,35,10,22,34,11,24,36)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,14,26, 3,15,28, 2,13,25, 4,16,27)( 5,18,30, 8,19,32, 6,17,29, 7,20,31) ( 9,22,33,11,23,36,10,21,34,12,24,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,15,25)( 2,16,26)( 3,13,27)( 4,14,28)( 5,19,29)( 6,20,30)( 7,18,32) ( 8,17,31)( 9,23,34)(10,24,33)(11,21,35)(12,22,36)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,16,25, 2,15,26)( 3,14,27, 4,13,28)( 5,20,29, 6,19,30)( 7,17,32, 8,18,31) ( 9,24,34,10,23,33)(11,22,35,12,21,36)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,17,23, 4,19,22, 2,18,24, 3,20,21)( 5,12,26, 7,10,27, 6,11,25, 8, 9,28) (13,30,35,15,31,34,14,29,36,16,32,33)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,18,23, 3,19,21, 2,17,24, 4,20,22)( 5,11,26, 8,10,28, 6,12,25, 7, 9,27) (13,29,35,16,31,33,14,30,36,15,32,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,19,24)( 2,20,23)( 3,17,22)( 4,18,21)( 5,10,25)( 6, 9,26)( 7,11,28) ( 8,12,27)(13,31,36)(14,32,35)(15,29,33)(16,30,34)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,20,24, 2,19,23)( 3,18,22, 4,17,21)( 5, 9,25, 6,10,26)( 7,12,28, 8,11,27) (13,32,36,14,31,35)(15,30,33,16,29,34)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,21,20, 3,24,18, 2,22,19, 4,23,17)( 5,28, 9, 8,25,11, 6,27,10, 7,26,12) (13,33,32,16,36,29,14,34,31,15,35,30)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,22,20, 4,24,17, 2,21,19, 3,23,18)( 5,27, 9, 7,25,12, 6,28,10, 8,26,11) (13,34,32,15,36,30,14,33,31,16,35,29)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,23,19, 2,24,20)( 3,21,17, 4,22,18)( 5,26,10, 6,25, 9)( 7,27,11, 8,28,12) (13,35,31,14,36,32)(15,34,29,16,33,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,24,19)( 2,23,20)( 3,22,17)( 4,21,18)( 5,25,10)( 6,26, 9)( 7,28,11) ( 8,27,12)(13,36,31)(14,35,32)(15,33,29)(16,34,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,25,15)( 2,26,16)( 3,27,13)( 4,28,14)( 5,29,19)( 6,30,20)( 7,32,18) ( 8,31,17)( 9,34,23)(10,33,24)(11,35,21)(12,36,22)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,26,15, 2,25,16)( 3,28,13, 4,27,14)( 5,30,19, 6,29,20)( 7,31,18, 8,32,17) ( 9,33,23,10,34,24)(11,36,21,12,35,22)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,27,16, 4,25,13, 2,28,15, 3,26,14)( 5,31,20, 7,29,17, 6,32,19, 8,30,18) ( 9,35,24,12,34,21,10,36,23,11,33,22)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,28,16, 3,25,14, 2,27,15, 4,26,13)( 5,32,20, 8,29,18, 6,31,19, 7,30,17) ( 9,36,24,11,34,22,10,35,23,12,33,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,29,10)( 2,30, 9)( 3,31,12)( 4,32,11)( 5,24,15)( 6,23,16)( 7,21,14) ( 8,22,13)(17,36,27)(18,35,28)(19,33,25)(20,34,26)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,30,10, 2,29, 9)( 3,32,12, 4,31,11)( 5,23,15, 6,24,16)( 7,22,14, 8,21,13) (17,35,27,18,36,28)(19,34,25,20,33,26)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,31, 9, 4,29,12, 2,32,10, 3,30,11)( 5,22,16, 7,24,13, 6,21,15, 8,23,14) (17,34,28,19,36,26,18,33,27,20,35,25)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,32, 9, 3,29,11, 2,31,10, 4,30,12)( 5,21,16, 8,24,14, 6,22,15, 7,23,13) (17,33,28,20,36,25,18,34,27,19,35,26)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,33, 5)( 2,34, 6)( 3,36, 8)( 4,35, 7)( 9,20,16)(10,19,15)(11,18,14) (12,17,13)(21,32,28)(22,31,27)(23,30,26)(24,29,25)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,34, 5, 2,33, 6)( 3,35, 8, 4,36, 7)( 9,19,16,10,20,15)(11,17,14,12,18,13) (21,31,28,22,32,27)(23,29,26,24,30,25)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,35, 6, 3,33, 7, 2,36, 5, 4,34, 8)( 9,17,15,11,20,13,10,18,16,12,19,14) (21,30,27,24,32,26,22,29,28,23,31,25)$ |
| $ 12, 12, 12 $ | $1$ | $12$ | $( 1,36, 6, 4,33, 8, 2,35, 5, 3,34, 7)( 9,18,15,12,20,14,10,17,16,11,19,13) (21,29,27,23,32,25,22,30,28,24,31,26)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $36=2^{2} \cdot 3^{2}$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 36.8 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);