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Magma
magma: G := TransitiveGroup(12, 18);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_6\times S_3$ | ||
CHM label: | $[3^{2}]E(4)$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (2,6,10)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: None
Degree 4: $C_2^2$
Degree 6: $S_3\times C_3$
Low degree siblings
18T6 x 2, 36T6Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 2, 6,10)( 4, 8,12)$ | |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 2,10, 6)( 4,12, 8)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ | |
$ 6, 6 $ | $3$ | $6$ | $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$ | |
$ 6, 6 $ | $3$ | $6$ | $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$ | |
$ 6, 6 $ | $1$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ | |
$ 6, 2, 2, 2 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$ | |
$ 6, 6 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$ | |
$ 6, 6 $ | $3$ | $6$ | $( 1, 4, 5, 8, 9,12)( 2, 3, 6, 7,10,11)$ | |
$ 6, 6 $ | $3$ | $6$ | $( 1, 4, 9,12, 5, 8)( 2, 7,10, 3, 6,11)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ | |
$ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ | |
$ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ | |
$ 6, 2, 2, 2 $ | $2$ | $6$ | $( 1, 7)( 2,12,10, 8, 6, 4)( 3, 9)( 5,11)$ | |
$ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ | |
$ 6, 6 $ | $1$ | $6$ | $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$ |
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: | 36.12 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A1 | 3A-1 | 3B | 3C1 | 3C-1 | 6A1 | 6A-1 | 6B | 6C1 | 6C-1 | 6D1 | 6D-1 | 6E1 | 6E-1 | ||
Size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | |
2 P | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3B | 3C-1 | 3C1 | 3A-1 | 3A1 | 3C-1 | 3B | 3C1 | 3A1 | 3A-1 | 3A1 | 3A-1 | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 2B | 2B | 2C | 2C | |
Type | |||||||||||||||||||
36.12.1a | R | ||||||||||||||||||
36.12.1b | R | ||||||||||||||||||
36.12.1c | R | ||||||||||||||||||
36.12.1d | R | ||||||||||||||||||
36.12.1e1 | C | ||||||||||||||||||
36.12.1e2 | C | ||||||||||||||||||
36.12.1f1 | C | ||||||||||||||||||
36.12.1f2 | C | ||||||||||||||||||
36.12.1g1 | C | ||||||||||||||||||
36.12.1g2 | C | ||||||||||||||||||
36.12.1h1 | C | ||||||||||||||||||
36.12.1h2 | C | ||||||||||||||||||
36.12.2a | R | ||||||||||||||||||
36.12.2b | R | ||||||||||||||||||
36.12.2c1 | C | ||||||||||||||||||
36.12.2c2 | C | ||||||||||||||||||
36.12.2d1 | C | ||||||||||||||||||
36.12.2d2 | C |
magma: CharacterTable(G);