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Magma
magma: G := TransitiveGroup(12, 87);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $87$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^4:A_4$ | ||
CHM label: | $[2^{5}]6$ | ||
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: | (1,12)(2,3), (1,3,5,7,9,11)(2,4,6,8,10,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $12$: $A_4$, $C_6\times C_2$ $24$: $A_4\times C_2$ x 3 $48$: $C_2^2 \times A_4$ $96$: $C_2^4:C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 4: None
Degree 6: $C_6$
Low degree siblings
12T87, 12T88 x 2, 16T416 x 2, 24T441 x 2, 24T442 x 2, 24T443 x 4, 24T444 x 2, 24T445 x 2, 24T446 x 2, 24T447 x 2, 24T448 x 4, 24T449, 24T450, 24T451 x 4, 24T452 x 4, 32T2183 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
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 $ | $6$ | $2$ | $( 8, 9)(10,11)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 6, 7)(10,11)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 4, 5)(10,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $6$ | $2$ | $( 4, 5)( 6, 7)( 8, 9)(10,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $6$ | $2$ | $( 2, 3)( 6, 7)( 8, 9)(10,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 4, 5)( 8, 9)(10,11)$ |
$ 6, 6 $ | $16$ | $6$ | $( 1, 2, 4, 6, 8,10)( 3, 5, 7, 9,11,12)$ |
$ 6, 6 $ | $16$ | $6$ | $( 1, 2, 4, 6, 8,11)( 3, 5, 7, 9,10,12)$ |
$ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$ |
$ 6, 6 $ | $16$ | $6$ | $( 1, 4, 8,12, 5, 9)( 2, 6,10, 3, 7,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 6)( 2, 8)( 3, 9)( 4,10)( 5,11)( 7,12)$ |
$ 4, 4, 2, 2 $ | $12$ | $4$ | $( 1, 6)( 2, 8, 3, 9)( 4,10, 5,11)( 7,12)$ |
$ 4, 4, 2, 2 $ | $12$ | $4$ | $( 1, 6,12, 7)( 2, 8)( 3, 9)( 4,10, 5,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 6)( 2, 8)( 3, 9)( 4,11)( 5,10)( 7,12)$ |
$ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 8, 4)( 2,10, 6)( 3,11, 7)( 5,12, 9)$ |
$ 6, 6 $ | $16$ | $6$ | $( 1, 8, 5,12, 9, 4)( 2,10, 7, 3,11, 6)$ |
$ 6, 6 $ | $16$ | $6$ | $( 1,10, 8, 6, 4, 2)( 3,12,11, 9, 7, 5)$ |
$ 6, 6 $ | $16$ | $6$ | $( 1,10, 9, 6, 4, 2)( 3,12,11, 8, 7, 5)$ |
$ 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: | $192=2^{6} \cdot 3$ | 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: | 192.1000 | magma: IdentifyGroup(G);
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Character table: |
2 6 5 5 6 5 5 6 2 2 2 2 4 4 4 4 2 2 2 2 6 3 1 . . . . . . 1 1 1 1 1 . . 1 1 1 1 1 1 1a 2a 2b 2c 2d 2e 2f 6a 6b 3a 6c 2g 4a 4b 2h 3b 6d 6e 6f 2i 2P 1a 1a 1a 1a 1a 1a 1a 3a 3a 3b 3b 1a 2f 2f 1a 3a 3a 3b 3b 1a 3P 1a 2a 2b 2c 2d 2e 2f 2g 2h 1a 2i 2g 4a 4b 2h 1a 2i 2g 2h 2i 5P 1a 2a 2b 2c 2d 2e 2f 6e 6f 3b 6d 2g 4a 4b 2h 3a 6c 6a 6b 2i X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 X.3 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 X.4 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 X.5 1 -1 1 -1 1 -1 1 A -A -/A /A -1 1 -1 1 -A A /A -/A -1 X.6 1 -1 1 -1 1 -1 1 /A -/A -A A -1 1 -1 1 -/A /A A -A -1 X.7 1 -1 1 -1 1 -1 1 -/A /A -A A 1 -1 1 -1 -/A /A -A A -1 X.8 1 -1 1 -1 1 -1 1 -A A -/A /A 1 -1 1 -1 -A A -/A /A -1 X.9 1 1 1 1 1 1 1 A A -/A -/A -1 -1 -1 -1 -A -A /A /A 1 X.10 1 1 1 1 1 1 1 /A /A -A -A -1 -1 -1 -1 -/A -/A A A 1 X.11 1 1 1 1 1 1 1 -/A -/A -A -A 1 1 1 1 -/A -/A -A -A 1 X.12 1 1 1 1 1 1 1 -A -A -/A -/A 1 1 1 1 -A -A -/A -/A 1 X.13 3 -1 -1 3 -1 -1 3 . . . . -3 1 1 -3 . . . . 3 X.14 3 -1 -1 3 -1 -1 3 . . . . 3 -1 -1 3 . . . . 3 X.15 3 1 -1 -3 -1 1 3 . . . . -3 -1 1 3 . . . . -3 X.16 3 1 -1 -3 -1 1 3 . . . . 3 1 -1 -3 . . . . -3 X.17 6 -2 -2 2 2 2 -2 . . . . . . . . . . . . -6 X.18 6 -2 2 -2 -2 2 -2 . . . . . . . . . . . . 6 X.19 6 2 -2 -2 2 -2 -2 . . . . . . . . . . . . 6 X.20 6 2 2 2 -2 -2 -2 . . . . . . . . . . . . -6 A = -E(3) = (1-Sqrt(-3))/2 = -b3 |
magma: CharacterTable(G);