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Magma
magma: G := TransitiveGroup(12, 129);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $129$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_4\wr C_2$ | ||
CHM label: | $[1/4E(4)^{3}:3:2]3$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,10)(3,12)(4,7)(6,9), (1,7,10)(2,5,11)(3,6,9), (4,7)(6,9)(8,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 4: None
Degree 6: None
Low degree siblings
8T42, 12T126, 12T128, 16T708, 18T112, 18T113, 24T692, 24T694, 24T695, 24T702, 24T703, 24T704, 32T9306, 36T316, 36T318, 36T456, 36T457, 36T458, 36T459Siblings 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, 2, 1, 1, 1, 1, 1, 1 $ | $12$ | $2$ | $( 5, 8)( 7,10)( 9,12)$ |
$ 3, 3, 3, 1, 1, 1 $ | $32$ | $3$ | $( 4, 7,10)( 5,11, 8)( 6, 9,12)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 2, 5)( 3,12)( 6, 9)( 8,11)$ |
$ 4, 4, 2, 1, 1 $ | $36$ | $4$ | $( 2, 5,11, 8)( 3,12, 6, 9)( 7,10)$ |
$ 6, 3, 3 $ | $48$ | $6$ | $( 1, 2, 3)( 4, 5, 6,10,11,12)( 7, 8, 9)$ |
$ 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 2, 3)( 4, 5, 9)( 6,10, 8)( 7,11,12)$ |
$ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 2, 3)( 4,11, 6)( 5,12,10)( 7, 8, 9)$ |
$ 6, 6 $ | $24$ | $6$ | $( 1, 2, 6, 7,11, 9)( 3, 4, 5,12,10, 8)$ |
$ 6, 3, 3 $ | $48$ | $6$ | $( 1, 3, 2)( 4, 6,11)( 5, 7,12, 8,10, 9)$ |
$ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 3, 2)( 4, 6,11)( 5,10,12)( 7, 9, 8)$ |
$ 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3, 2)( 4, 9, 5)( 6, 8,10)( 7,12,11)$ |
$ 6, 6 $ | $24$ | $6$ | $( 1, 3,11,10, 6, 5)( 2, 4, 9, 8, 7,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 4)( 2, 5)( 3, 9)( 6,12)( 7,10)( 8,11)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $288=2^{5} \cdot 3^{2}$ | 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: | 288.1025 | magma: IdentifyGroup(G);
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Character table: |
2 5 3 . 5 3 1 2 1 2 1 1 2 2 4 3 2 1 2 . . 1 2 2 1 1 2 2 1 1 1a 2a 3a 2b 4a 6a 3b 3c 6b 6c 3d 3e 6d 2c 2P 1a 1a 3a 1a 2b 3d 3e 3d 3e 3c 3c 3b 3b 1a 3P 1a 2a 1a 2b 4a 2a 1a 1a 2c 2a 1a 1a 2c 2c 5P 1a 2a 3a 2b 4a 6c 3e 3d 6d 6a 3c 3b 6b 2c X.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 X.3 1 -1 1 1 -1 A -A -A -A /A -/A -/A -/A 1 X.4 1 -1 1 1 -1 /A -/A -/A -/A A -A -A -A 1 X.5 1 1 1 1 1 -/A -/A -/A -/A -A -A -A -A 1 X.6 1 1 1 1 1 -A -A -A -A -/A -/A -/A -/A 1 X.7 2 . -1 2 . . -1 2 -1 . 2 -1 -1 2 X.8 2 . -1 2 . . /A C /A . /C A A 2 X.9 2 . -1 2 . . A /C A . C /A /A 2 X.10 6 . . -2 . . 3 . -1 . . 3 -1 2 X.11 6 . . -2 . . B . /A . . /B A 2 X.12 6 . . -2 . . /B . A . . B /A 2 X.13 9 -3 . 1 1 . . . . . . . . -3 X.14 9 3 . 1 -1 . . . . . . . . -3 A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 C = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 |
magma: CharacterTable(G);