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Magma
magma: G := TransitiveGroup(12, 208);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $208$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_4^2:D_4$ | ||
CHM label: | $[2A_{4}^{2}]2=2A4wr2=2wrF_{18}(6)$ | ||
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: | (6,12), (2,6,10)(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 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $8$: $D_{4}$ $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $24$: $(C_6\times C_2):C_2$, $D_4 \times C_3$ $36$: $C_6\times S_3$ $72$: 12T42 $288$: $A_4\wr C_2$ $576$: 12T158 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: None
Degree 6: $S_3\times C_3$
Low degree siblings
12T208, 16T1286 x 2, 24T2815 x 2, 24T2816 x 2, 24T2817 x 2, 24T2818 x 2, 24T2819, 32T96688, 36T1607, 36T1613 x 2, 36T1635, 36T1914 x 2, 36T1915 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 6,12)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 4,10)( 6,12)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5,11)( 6,12)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 4,10)( 5,11)( 6,12)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 2, 8)( 4,10)( 6,12)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $6$ | $2$ | $( 2, 8)( 4,10)( 5,11)( 6,12)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 9)( 4,10)( 5,11)( 6,12)$ |
$ 2, 2, 2, 2, 2, 1, 1 $ | $6$ | $2$ | $( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 2, 6,10)( 4, 8,12)$ |
$ 6, 1, 1, 1, 1, 1, 1 $ | $8$ | $6$ | $( 2,12, 4, 8, 6,10)$ |
$ 3, 3, 2, 1, 1, 1, 1 $ | $24$ | $6$ | $( 2, 6,10)( 4, 8,12)( 5,11)$ |
$ 6, 2, 1, 1, 1, 1 $ | $24$ | $6$ | $( 2,12, 4, 8, 6,10)( 5,11)$ |
$ 3, 3, 2, 2, 1, 1 $ | $24$ | $6$ | $( 2, 6,10)( 3, 9)( 4, 8,12)( 5,11)$ |
$ 6, 2, 2, 1, 1 $ | $24$ | $6$ | $( 2,12, 4, 8, 6,10)( 3, 9)( 5,11)$ |
$ 3, 3, 2, 2, 2 $ | $8$ | $6$ | $( 1, 7)( 2, 6,10)( 3, 9)( 4, 8,12)( 5,11)$ |
$ 6, 2, 2, 2 $ | $8$ | $6$ | $( 1, 7)( 2,12, 4, 8, 6,10)( 3, 9)( 5,11)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 2,10, 6)( 4,12, 8)$ |
$ 6, 1, 1, 1, 1, 1, 1 $ | $8$ | $6$ | $( 2,10,12, 8, 4, 6)$ |
$ 3, 3, 2, 1, 1, 1, 1 $ | $24$ | $6$ | $( 2,10, 6)( 4,12, 8)( 5,11)$ |
$ 6, 2, 1, 1, 1, 1 $ | $24$ | $6$ | $( 2,10,12, 8, 4, 6)( 5,11)$ |
$ 3, 3, 2, 2, 1, 1 $ | $24$ | $6$ | $( 2,10, 6)( 3, 9)( 4,12, 8)( 5,11)$ |
$ 6, 2, 2, 1, 1 $ | $24$ | $6$ | $( 2,10,12, 8, 4, 6)( 3, 9)( 5,11)$ |
$ 3, 3, 2, 2, 2 $ | $8$ | $6$ | $( 1, 7)( 2,10, 6)( 3, 9)( 4,12, 8)( 5,11)$ |
$ 6, 2, 2, 2 $ | $8$ | $6$ | $( 1, 7)( 2,10,12, 8, 4, 6)( 3, 9)( 5,11)$ |
$ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
$ 6, 3, 3 $ | $32$ | $6$ | $( 1, 5, 9)( 2,12, 4, 8, 6,10)( 3, 7,11)$ |
$ 6, 6 $ | $16$ | $6$ | $( 1,11, 3, 7, 5, 9)( 2,12, 4, 8, 6,10)$ |
$ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ |
$ 6, 3, 3 $ | $32$ | $6$ | $( 1, 5, 9)( 2,10,12, 8, 4, 6)( 3, 7,11)$ |
$ 6, 3, 3 $ | $32$ | $6$ | $( 1,11, 3, 7, 5, 9)( 2,10, 6)( 4,12, 8)$ |
$ 6, 6 $ | $32$ | $6$ | $( 1,11, 3, 7, 5, 9)( 2,10,12, 8, 4, 6)$ |
$ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
$ 6, 3, 3 $ | $32$ | $6$ | $( 1, 9, 5)( 2,10,12, 8, 4, 6)( 3,11, 7)$ |
$ 6, 6 $ | $16$ | $6$ | $( 1, 9,11, 7, 3, 5)( 2,10,12, 8, 4, 6)$ |
$ 2, 2, 2, 2, 2, 2 $ | $24$ | $2$ | $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$ |
$ 4, 2, 2, 2, 2 $ | $72$ | $4$ | $( 1, 6, 7,12)( 2, 3)( 4, 5)( 8, 9)(10,11)$ |
$ 4, 4, 2, 2 $ | $72$ | $4$ | $( 1, 6, 7,12)( 2, 3)( 4, 5,10,11)( 8, 9)$ |
$ 4, 4, 4 $ | $24$ | $4$ | $( 1, 6, 7,12)( 2, 3, 8, 9)( 4, 5,10,11)$ |
$ 6, 6 $ | $96$ | $6$ | $( 1, 4, 5, 8, 9,12)( 2, 3, 6, 7,10,11)$ |
$ 12 $ | $96$ | $12$ | $( 1, 4, 5, 8, 9, 6, 7,10,11, 2, 3,12)$ |
$ 6, 6 $ | $96$ | $6$ | $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$ |
$ 12 $ | $96$ | $12$ | $( 1, 8, 9, 4, 5, 6, 7, 2, 3,10,11,12)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $1152=2^{7} \cdot 3^{2}$ | 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: | 1152.157517 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);