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Magma
magma: G := TransitiveGroup(12, 5);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3 : C_4$ | ||
CHM label: | $1/2[3:2]4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,8,7,2)(3,6,9,12)(4,11,10,5), (1,5,9)(2,6,10)(3,7,11)(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) $2$: $C_2$ $4$: $C_4$ $6$: $S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $C_4$
Degree 6: $S_3$
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ | |
$ 4, 4, 4 $ | $3$ | $4$ | $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$ | |
$ 6, 6 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ | |
$ 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 7,10)( 2, 9, 8, 3)( 5,12,11, 6)$ | |
$ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $12=2^{2} \cdot 3$ | 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: | 12.1 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A | 4A1 | 4A-1 | 6A | ||
Size | 1 | 1 | 2 | 3 | 3 | 2 | |
2 P | 1A | 1A | 3A | 2A | 2A | 3A | |
3 P | 1A | 2A | 1A | 4A-1 | 4A1 | 2A | |
Type | |||||||
12.1.1a | R | ||||||
12.1.1b | R | ||||||
12.1.1c1 | C | ||||||
12.1.1c2 | C | ||||||
12.1.2a | R | ||||||
12.1.2b | S |
magma: CharacterTable(G);