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Magma
magma: G := TransitiveGroup(12, 25);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^2 \times A_4$ | ||
CHM label: | $2A_{4}(6)[x]2=[1/4.2^{6}]3$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,7)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $12$: $A_4$, $C_6\times C_2$ $24$: $A_4\times C_2$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 4: None
Degree 6: $C_6$, $A_4\times C_2$ x 2
Low degree siblings
12T25 x 2, 12T26 x 2, 16T58, 24T49 x 3, 24T50Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 4,10)( 5,11)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 8)( 3, 9)( 4,10)( 5,11)$ | |
$ 6, 6 $ | $4$ | $6$ | $( 1, 2, 5,12, 3, 4)( 6, 9,10, 7, 8,11)$ | |
$ 6, 6 $ | $4$ | $6$ | $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$ | |
$ 6, 6 $ | $4$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ | |
$ 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$ | |
$ 6, 6 $ | $4$ | $6$ | $( 1, 4, 3,12, 5, 2)( 6,11, 8, 7,10, 9)$ | |
$ 6, 6 $ | $4$ | $6$ | $( 1, 4, 9, 6,11, 2)( 3,12, 5, 8, 7,10)$ | |
$ 6, 6 $ | $4$ | $6$ | $( 1, 5, 3, 7,11, 9)( 2, 6,10, 8,12, 4)$ | |
$ 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 6)( 2, 3)( 4, 5)( 7,12)( 8, 9)(10,11)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 6)( 2, 3)( 4,11)( 5,10)( 7,12)( 8, 9)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 9)( 3, 8)( 4,11)( 5,10)( 7,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $48=2^{4} \cdot 3$ | 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: | 48.49 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A1 | 3A-1 | 6A1 | 6A-1 | 6B1 | 6B-1 | 6C1 | 6C-1 | ||
Size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3A1 | 3A-1 | 3A-1 | 3A1 | 3A1 | 3A-1 | |
3 P | 1A | 2C | 2A | 2B | 2G | 2D | 2F | 2E | 1A | 1A | 2B | 2A | 2B | 2A | 2C | 2C | |
Type | |||||||||||||||||
48.49.1a | R | ||||||||||||||||
48.49.1b | R | ||||||||||||||||
48.49.1c | R | ||||||||||||||||
48.49.1d | R | ||||||||||||||||
48.49.1e1 | C | ||||||||||||||||
48.49.1e2 | C | ||||||||||||||||
48.49.1f1 | C | ||||||||||||||||
48.49.1f2 | C | ||||||||||||||||
48.49.1g1 | C | ||||||||||||||||
48.49.1g2 | C | ||||||||||||||||
48.49.1h1 | C | ||||||||||||||||
48.49.1h2 | C | ||||||||||||||||
48.49.3a | R | ||||||||||||||||
48.49.3b | R | ||||||||||||||||
48.49.3c | R | ||||||||||||||||
48.49.3d | R |
magma: CharacterTable(G);