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Magma
magma: G := TransitiveGroup(36, 41);
Group action invariants
| Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $41$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{12}:S_3$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,7)(2,8)(3,5)(4,6)(9,23)(10,24)(11,21)(12,22)(13,31)(14,32)(15,29)(16,30)(17,27)(18,28)(19,26)(20,25), (1,19,21)(2,20,22)(3,18,23)(4,17,24)(5,9,28)(6,10,27)(7,11,26)(8,12,25)(13,31,35)(14,32,36)(15,29,34)(16,30,33), (1,3,2,4)(5,35,6,36)(7,33,8,34)(9,31,10,32)(11,30,12,29)(13,27,14,28)(15,26,16,25)(17,21,18,22)(19,23,20,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ x 4 $8$: $C_4\times C_2$ $12$: $D_{6}$ x 4 $18$: $C_3^2:C_2$ $24$: $S_3 \times C_4$ x 4 $36$: 18T12 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$ x 4
Degree 4: $C_4$
Degree 6: $D_{6}$ x 4
Degree 9: $C_3^2:C_2$
Degree 12: $S_3 \times C_4$ x 4
Degree 18: 18T12
Low degree siblings
36T41Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5,33)( 6,34)( 7,36)( 8,35)( 9,30)(10,29)(11,32)(12,31)(13,25)(14,26)(15,27) (16,28)(17,24)(18,23)(19,21)(20,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,29)(10,30)(11,31)(12,32)(13,26) (14,25)(15,28)(16,27)(17,23)(18,24)(19,22)(20,21)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,32,30,31)(33,35,34,36)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 3, 2, 4)( 5,35, 6,36)( 7,33, 8,34)( 9,31,10,32)(11,30,12,29)(13,27,14,28) (15,26,16,25)(17,21,18,22)(19,23,20,24)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,31,30,32)(33,36,34,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 4, 2, 3)( 5,36, 6,35)( 7,34, 8,33)( 9,32,10,31)(11,29,12,30)(13,28,14,27) (15,25,16,26)(17,22,18,21)(19,24,20,23)$ |
| $ 12, 12, 12 $ | $2$ | $12$ | $( 1, 5,35, 4, 7,33, 2, 6,36, 3, 8,34)( 9,13,17,11,16,20,10,14,18,12,15,19) (21,28,31,24,26,30,22,27,32,23,25,29)$ |
| $ 12, 12, 12 $ | $2$ | $12$ | $( 1, 6,35, 3, 7,34, 2, 5,36, 4, 8,33)( 9,14,17,12,16,19,10,13,18,11,15,20) (21,27,31,23,26,29,22,28,32,24,25,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,36)( 2, 8,35)( 3, 5,33)( 4, 6,34)( 9,16,18)(10,15,17)(11,14,19) (12,13,20)(21,26,32)(22,25,31)(23,28,30)(24,27,29)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 8,36, 2, 7,35)( 3, 6,33, 4, 5,34)( 9,15,18,10,16,17)(11,13,19,12,14,20) (21,25,32,22,26,31)(23,27,30,24,28,29)$ |
| $ 12, 12, 12 $ | $2$ | $12$ | $( 1, 9,31, 4,11,30, 2,10,32, 3,12,29)( 5,13,24, 7,16,22, 6,14,23, 8,15,21) (17,26,33,20,27,36,18,25,34,19,28,35)$ |
| $ 12, 12, 12 $ | $2$ | $12$ | $( 1,10,31, 3,11,29, 2, 9,32, 4,12,30)( 5,14,24, 8,16,21, 6,13,23, 7,15,22) (17,25,33,19,27,35,18,26,34,20,28,36)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,11,32)( 2,12,31)( 3, 9,30)( 4,10,29)( 5,16,23)( 6,15,24)( 7,14,21) ( 8,13,22)(17,27,34)(18,28,33)(19,26,36)(20,25,35)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,12,32, 2,11,31)( 3,10,30, 4, 9,29)( 5,15,23, 6,16,24)( 7,13,21, 8,14,22) (17,28,34,18,27,33)(19,25,36,20,26,35)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,13,26, 2,14,25)( 3,15,28, 4,16,27)( 5,17,30, 6,18,29)( 7,20,32, 8,19,31) ( 9,24,33,10,23,34)(11,22,36,12,21,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,14,26)( 2,13,25)( 3,16,28)( 4,15,27)( 5,18,30)( 6,17,29)( 7,19,32) ( 8,20,31)( 9,23,33)(10,24,34)(11,21,36)(12,22,35)$ |
| $ 12, 12, 12 $ | $2$ | $12$ | $( 1,15,25, 3,14,27, 2,16,26, 4,13,28)( 5,19,29, 8,18,32, 6,20,30, 7,17,31) ( 9,21,34,12,23,36,10,22,33,11,24,35)$ |
| $ 12, 12, 12 $ | $2$ | $12$ | $( 1,16,25, 4,14,28, 2,15,26, 3,13,27)( 5,20,29, 7,18,31, 6,19,30, 8,17,32) ( 9,22,34,11,23,35,10,21,33,12,24,36)$ |
| $ 12, 12, 12 $ | $2$ | $12$ | $( 1,17,22, 3,19,24, 2,18,21, 4,20,23)( 5,11,27, 8, 9,26, 6,12,28, 7,10,25) (13,30,36,15,31,33,14,29,35,16,32,34)$ |
| $ 12, 12, 12 $ | $2$ | $12$ | $( 1,18,22, 4,19,23, 2,17,21, 3,20,24)( 5,12,27, 7, 9,25, 6,11,28, 8,10,26) (13,29,36,16,31,34,14,30,35,15,32,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,19,21)( 2,20,22)( 3,18,23)( 4,17,24)( 5, 9,28)( 6,10,27)( 7,11,26) ( 8,12,25)(13,31,35)(14,32,36)(15,29,34)(16,30,33)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,20,21, 2,19,22)( 3,17,23, 4,18,24)( 5,10,28, 6, 9,27)( 7,12,26, 8,11,25) (13,32,35,14,31,36)(15,30,34,16,29,33)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.32 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);