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Magma
magma: G := TransitiveGroup(36, 22);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_6^2:C_2$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $18$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,6,2,5)(3,8,4,7)(9,24,10,23)(11,22,12,21)(13,30,14,29)(15,32,16,31)(17,26,18,25)(19,27,20,28)(33,36,34,35), (1,28,16,2,27,15)(3,26,13)(4,25,14)(5,32,19)(6,31,20)(7,30,17,8,29,18)(9,34,22,10,33,21)(11,35,23)(12,36,24), (1,10,29)(2,9,30)(3,12,31,4,11,32)(5,13,24,6,14,23)(7,16,21)(8,15,22)(17,27,34)(18,28,33)(19,26,36,20,25,35) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ x 4 $8$: $D_{4}$ $12$: $D_{6}$ x 4 $18$: $C_3^2:C_2$ $24$: $(C_6\times C_2):C_2$ x 4 $36$: 18T12 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$ x 4
Degree 4: $D_{4}$
Degree 6: $S_3$ x 4
Degree 9: $C_3^2:C_2$
Degree 12: $(C_6\times C_2):C_2$ x 4
Degree 18: $C_3^2 : C_2$
Low degree siblings
36T42Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 3, 4)( 5, 6)(11,12)(13,14)(19,20)(23,24)(25,26)(31,32)(35,36)$ |
$ 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 $ | $18$ | $2$ | $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,35)( 8,36)( 9,32)(10,31)(11,29)(12,30)(13,27) (14,28)(15,25)(16,26)(17,23)(18,24)(19,22)(20,21)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $18$ | $4$ | $( 1, 3, 2, 4)( 5,34, 6,33)( 7,35, 8,36)( 9,32,10,31)(11,30,12,29)(13,28,14,27) (15,25,16,26)(17,23,18,24)(19,21,20,22)$ |
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1, 7,34)( 2, 8,33)( 3, 5,35, 4, 6,36)( 9,15,18)(10,16,17)(11,14,20,12,13,19) (21,27,29)(22,28,30)(23,25,31,24,26,32)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,34)( 2, 8,33)( 3, 6,35)( 4, 5,36)( 9,15,18)(10,16,17)(11,13,20) (12,14,19)(21,27,29)(22,28,30)(23,26,31)(24,25,32)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 8,34, 2, 7,33)( 3, 5,35, 4, 6,36)( 9,16,18,10,15,17)(11,14,20,12,13,19) (21,28,29,22,27,30)(23,25,31,24,26,32)$ |
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1, 8,34, 2, 7,33)( 3, 6,35)( 4, 5,36)( 9,16,18,10,15,17)(11,13,20)(12,14,19) (21,28,29,22,27,30)(23,26,31)(24,25,32)$ |
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1, 9,29, 2,10,30)( 3,11,31)( 4,12,32)( 5,14,24)( 6,13,23)( 7,15,21, 8,16,22) (17,28,34,18,27,33)(19,25,36)(20,26,35)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 9,29, 2,10,30)( 3,12,31, 4,11,32)( 5,13,24, 6,14,23)( 7,15,21, 8,16,22) (17,28,34,18,27,33)(19,26,36,20,25,35)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,10,29)( 2, 9,30)( 3,11,31)( 4,12,32)( 5,14,24)( 6,13,23)( 7,16,21) ( 8,15,22)(17,27,34)(18,28,33)(19,25,36)(20,26,35)$ |
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1,10,29)( 2, 9,30)( 3,12,31, 4,11,32)( 5,13,24, 6,14,23)( 7,16,21)( 8,15,22) (17,27,34)(18,28,33)(19,26,36,20,25,35)$ |
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1,15,27, 2,16,28)( 3,13,26)( 4,14,25)( 5,19,32)( 6,20,31)( 7,18,29, 8,17,30) ( 9,21,33,10,22,34)(11,23,35)(12,24,36)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,15,27, 2,16,28)( 3,14,26, 4,13,25)( 5,20,32, 6,19,31)( 7,18,29, 8,17,30) ( 9,21,33,10,22,34)(11,24,35,12,23,36)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,27)( 2,15,28)( 3,13,26)( 4,14,25)( 5,19,32)( 6,20,31)( 7,17,29) ( 8,18,30)( 9,22,33)(10,21,34)(11,23,35)(12,24,36)$ |
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1,16,27)( 2,15,28)( 3,14,26, 4,13,25)( 5,20,32, 6,19,31)( 7,17,29)( 8,18,30) ( 9,22,33)(10,21,34)(11,24,35,12,23,36)$ |
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1,17,21)( 2,18,22)( 3,19,23, 4,20,24)( 5,11,25, 6,12,26)( 7,10,27)( 8, 9,28) (13,32,35,14,31,36)(15,30,33)(16,29,34)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,17,21)( 2,18,22)( 3,20,23)( 4,19,24)( 5,12,25)( 6,11,26)( 7,10,27) ( 8, 9,28)(13,31,35)(14,32,36)(15,30,33)(16,29,34)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,18,21, 2,17,22)( 3,19,23, 4,20,24)( 5,11,25, 6,12,26)( 7, 9,27, 8,10,28) (13,32,35,14,31,36)(15,29,33,16,30,34)$ |
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1,18,21, 2,17,22)( 3,20,23)( 4,19,24)( 5,12,25)( 6,11,26)( 7, 9,27, 8,10,28) (13,31,35)(14,32,36)(15,29,33,16,30,34)$ |
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: | 72.35 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);