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Magma
magma: G := TransitiveGroup(12, 13);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $13$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $(C_6\times C_2):C_2$ | ||
CHM label: | $1/2[3:2]eD(4)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,11)(2,10)(3,9)(4,8)(5,7), (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) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $D_{4}$
Degree 6: $D_{6}$
Low degree siblings
12T15, 24T14Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 2, 2, 1, 1 $ | $6$ | $2$ | $( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)$ | |
$ 4, 4, 4 $ | $6$ | $4$ | $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$ | |
$ 6, 6 $ | $2$ | $6$ | $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$ | |
$ 6, 6 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ | |
$ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ | |
$ 6, 6 $ | $2$ | $6$ | $( 1, 6, 5,10, 9, 2)( 3, 8, 7,12,11, 4)$ | |
$ 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: | $24=2^{3} \cdot 3$ | 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: | 24.8 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 4A | 6A | 6B1 | 6B-1 | ||
Size | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 3A | 2A | 3A | 3A | 3A | |
3 P | 1A | 2A | 2B | 2C | 1A | 4A | 2B | 2A | 2B | |
Type | ||||||||||
24.8.1a | R | |||||||||
24.8.1b | R | |||||||||
24.8.1c | R | |||||||||
24.8.1d | R | |||||||||
24.8.2a | R | |||||||||
24.8.2b | R | |||||||||
24.8.2c | R | |||||||||
24.8.2d1 | C | |||||||||
24.8.2d2 | C |
magma: CharacterTable(G);