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Magma
magma: G := TransitiveGroup(36, 47);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $47$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_{36}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,7)(2,8)(3,6)(4,5)(9,36)(10,35)(11,33)(12,34)(13,31)(14,32)(15,30)(16,29)(17,25)(18,26)(19,27)(20,28)(21,23)(22,24), (1,36,31,28,24,18,16,12,5,4,34,29,26,22,20,13,9,7,2,35,32,27,23,17,15,11,6,3,33,30,25,21,19,14,10,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ $18$: $D_{9}$ $24$: $D_{12}$ $36$: $D_{18}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $D_{4}$
Degree 6: $D_{6}$
Degree 9: $D_{9}$
Degree 12: $D_{12}$
Degree 18: $D_{18}$
Low degree siblings
36T47Siblings 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, 2, 1, 1 $ | $18$ | $2$ | $( 3, 4)( 5,33)( 6,34)( 7,35)( 8,36)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28) (15,26)(16,25)(17,22)(18,21)(19,24)(20,23)$ |
$ 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 $ | $18$ | $2$ | $( 1, 3)( 2, 4)( 5,35)( 6,36)( 7,34)( 8,33)( 9,29)(10,30)(11,31)(12,32)(13,26) (14,25)(15,28)(16,27)(17,24)(18,23)(19,21)(20,22)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)(17,20,18,19)(21,23,22,24) (25,27,26,28)(29,31,30,32)(33,35,34,36)$ |
$ 9, 9, 9, 9 $ | $2$ | $9$ | $( 1, 5, 9,15,19,24,26,32,33)( 2, 6,10,16,20,23,25,31,34)( 3, 8,12,13,17,21,28, 29,35)( 4, 7,11,14,18,22,27,30,36)$ |
$ 18, 18 $ | $2$ | $18$ | $( 1, 6, 9,16,19,23,26,31,33, 2, 5,10,15,20,24,25,32,34)( 3, 7,12,14,17,22,28, 30,35, 4, 8,11,13,18,21,27,29,36)$ |
$ 36 $ | $2$ | $36$ | $( 1, 7,10,13,19,22,25,29,33, 4, 6,12,15,18,23,28,32,36, 2, 8, 9,14,20,21,26, 30,34, 3, 5,11,16,17,24,27,31,35)$ |
$ 36 $ | $2$ | $36$ | $( 1, 8,10,14,19,21,25,30,33, 3, 6,11,15,17,23,27,32,35, 2, 7, 9,13,20,22,26, 29,34, 4, 5,12,16,18,24,28,31,36)$ |
$ 9, 9, 9, 9 $ | $2$ | $9$ | $( 1, 9,19,26,33, 5,15,24,32)( 2,10,20,25,34, 6,16,23,31)( 3,12,17,28,35, 8,13, 21,29)( 4,11,18,27,36, 7,14,22,30)$ |
$ 18, 18 $ | $2$ | $18$ | $( 1,10,19,25,33, 6,15,23,32, 2, 9,20,26,34, 5,16,24,31)( 3,11,17,27,35, 7,13, 22,29, 4,12,18,28,36, 8,14,21,30)$ |
$ 36 $ | $2$ | $36$ | $( 1,11,20,28,33, 7,16,21,32, 4,10,17,26,36, 6,13,24,30, 2,12,19,27,34, 8,15, 22,31, 3, 9,18,25,35, 5,14,23,29)$ |
$ 36 $ | $2$ | $36$ | $( 1,12,20,27,33, 8,16,22,32, 3,10,18,26,35, 6,14,24,29, 2,11,19,28,34, 7,15, 21,31, 4, 9,17,25,36, 5,13,23,30)$ |
$ 12, 12, 12 $ | $2$ | $12$ | $( 1,13,25, 4,15,28, 2,14,26, 3,16,27)( 5,17,31, 7,19,29, 6,18,32, 8,20,30) ( 9,21,34,11,24,35,10,22,33,12,23,36)$ |
$ 12, 12, 12 $ | $2$ | $12$ | $( 1,14,25, 3,15,27, 2,13,26, 4,16,28)( 5,18,31, 8,19,30, 6,17,32, 7,20,29) ( 9,22,34,12,24,36,10,21,33,11,23,35)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,15,26)( 2,16,25)( 3,13,28)( 4,14,27)( 5,19,32)( 6,20,31)( 7,18,30) ( 8,17,29)( 9,24,33)(10,23,34)(11,22,36)(12,21,35)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,16,26, 2,15,25)( 3,14,28, 4,13,27)( 5,20,32, 6,19,31)( 7,17,30, 8,18,29) ( 9,23,33,10,24,34)(11,21,36,12,22,35)$ |
$ 36 $ | $2$ | $36$ | $( 1,17,34,14,32,12,25, 7,24, 3,20,36,15,29,10,27, 5,21, 2,18,33,13,31,11,26, 8,23, 4,19,35,16,30, 9,28, 6,22)$ |
$ 36 $ | $2$ | $36$ | $( 1,18,34,13,32,11,25, 8,24, 4,20,35,15,30,10,28, 5,22, 2,17,33,14,31,12,26, 7,23, 3,19,36,16,29, 9,27, 6,21)$ |
$ 9, 9, 9, 9 $ | $2$ | $9$ | $( 1,19,33,15,32, 9,26, 5,24)( 2,20,34,16,31,10,25, 6,23)( 3,17,35,13,29,12,28, 8,21)( 4,18,36,14,30,11,27, 7,22)$ |
$ 18, 18 $ | $2$ | $18$ | $( 1,20,33,16,32,10,26, 6,24, 2,19,34,15,31, 9,25, 5,23)( 3,18,35,14,29,11,28, 7,21, 4,17,36,13,30,12,27, 8,22)$ |
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: | 72.6 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);