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Magma
magma: G := TransitiveGroup(36, 8);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_6:S_3$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $36$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,4)(2,3)(5,34)(6,33)(7,35)(8,36)(9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22), (1,11)(2,12)(3,10)(4,9)(5,18)(6,17)(7,20)(8,19)(13,34)(14,33)(15,35)(16,36)(21,25)(22,26)(23,28)(24,27)(29,31)(30,32), (1,34,7)(2,33,8)(3,36,6)(4,35,5)(9,18,15)(10,17,16)(11,20,13)(12,19,14)(21,29,27)(22,30,28)(23,32,26)(24,31,25) | 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 $12$: $D_{6}$ x 4 $18$: $C_3^2:C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $S_3$ x 4
Degree 4: $C_2^2$
Degree 6: $S_3$ x 4, $D_{6}$ x 8
Degree 9: $C_3^2:C_2$
Degree 12: $D_6$ x 4
Degree 18: $C_3^2 : C_2$, 18T12 x 2
Low degree siblings
18T12 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,36)( 8,35)( 9,32)(10,31)(11,29)(12,30)(13,27) (14,28)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 4)( 2, 3)( 5,34)( 6,33)( 7,35)( 8,36)( 9,31)(10,32)(11,30)(12,29)(13,28) (14,27)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,34)( 2, 8,33)( 3, 6,36)( 4, 5,35)( 9,15,18)(10,16,17)(11,13,20) (12,14,19)(21,27,29)(22,28,30)(23,26,32)(24,25,31)$ | |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 8,34, 2, 7,33)( 3, 5,36, 4, 6,35)( 9,16,18,10,15,17)(11,14,20,12,13,19) (21,28,29,22,27,30)(23,25,32,24,26,31)$ | |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 9,29, 2,10,30)( 3,12,31, 4,11,32)( 5,13,23, 6,14,24)( 7,15,21, 8,16,22) (17,28,34,18,27,33)(19,25,35,20,26,36)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,10,29)( 2, 9,30)( 3,11,31)( 4,12,32)( 5,14,23)( 6,13,24)( 7,16,21) ( 8,15,22)(17,27,34)(18,28,33)(19,26,35)(20,25,36)$ | |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,15,27, 2,16,28)( 3,14,25, 4,13,26)( 5,20,32, 6,19,31)( 7,18,29, 8,17,30) ( 9,21,33,10,22,34)(11,23,36,12,24,35)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,27)( 2,15,28)( 3,13,25)( 4,14,26)( 5,19,32)( 6,20,31)( 7,17,29) ( 8,18,30)( 9,22,33)(10,21,34)(11,24,36)(12,23,35)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,17,21)( 2,18,22)( 3,20,24)( 4,19,23)( 5,12,26)( 6,11,25)( 7,10,27) ( 8, 9,28)(13,31,36)(14,32,35)(15,30,33)(16,29,34)$ | |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,18,21, 2,17,22)( 3,19,24, 4,20,23)( 5,11,26, 6,12,25)( 7, 9,27, 8,10,28) (13,32,36,14,31,35)(15,29,33,16,30,34)$ |
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: | no | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 36.13 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | ||
Size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 3D | 3A | 3C | 3B | 3D | 3C | 3B | 3A | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | |
Type | |||||||||||||
36.13.1a | R | ||||||||||||
36.13.1b | R | ||||||||||||
36.13.1c | R | ||||||||||||
36.13.1d | R | ||||||||||||
36.13.2a | R | ||||||||||||
36.13.2b | R | ||||||||||||
36.13.2c | R | ||||||||||||
36.13.2d | R | ||||||||||||
36.13.2e | R | ||||||||||||
36.13.2f | R | ||||||||||||
36.13.2g | R | ||||||||||||
36.13.2h | R |
magma: CharacterTable(G);