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Magma
magma: G := TransitiveGroup(16, 141);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $141$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_4:D_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,14)(2,13)(3,4)(5,10)(6,9)(7,15)(8,16)(11,12), (1,3,6,8,10,12,13,15)(2,4,5,7,9,11,14,16), (1,9)(2,10)(3,4)(5,6)(7,15)(8,16)(11,12)(13,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $D_{4}$ x 6, $C_2^3$ $16$: $D_4\times C_2$ x 3 $32$: $C_2^2 \wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4\times C_2$, $(C_4^2 : C_2):C_2$ x 2
Low degree siblings
8T26 x 4, 16T135 x 2, 16T141, 16T142 x 2, 16T152 x 2, 32T147 x 2, 32T148 x 2, 32T155, 32T156Siblings 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, 1, 1 $ | $1$ | $1$ | $()$ |
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 3, 8,12,15)( 4, 7,11,16)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 3,12)( 4,11)( 7,16)( 8,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,10)(11,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 2)( 3, 7)( 4, 8)( 5,13)( 6,14)( 9,10)(11,15)(12,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$ |
$ 8, 8 $ | $8$ | $8$ | $( 1, 3, 6, 8,10,12,13,15)( 2, 4, 5, 7, 9,11,14,16)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3,10,12)( 2, 4, 9,11)( 5, 7,14,16)( 6, 8,13,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$ |
$ 8, 8 $ | $8$ | $8$ | $( 1, 4, 6,16,10,11,13, 7)( 2, 3, 5,15, 9,12,14, 8)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4,10,11)( 2, 3, 9,12)( 5,15,14, 8)( 6,16,13, 7)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 6,10,13)( 2, 5, 9,14)( 3, 8,12,15)( 4, 7,11,16)$ |
$ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 6,10,13)( 2, 5, 9,14)( 3,12)( 4,11)( 7,16)( 8,15)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 6,10,13)( 2, 5, 9,14)( 3,15,12, 8)( 4,16,11, 7)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $64=2^{6}$ | 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: | $3$ | ||
Label: | 64.134 | magma: IdentifyGroup(G);
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Character table: |
2 6 4 5 4 3 4 3 4 4 3 4 4 5 4 5 6 1a 4a 2a 2b 2c 2d 8a 4b 2e 8b 4c 2f 4d 4e 4f 2g 2P 1a 2a 1a 1a 1a 1a 4d 2g 1a 4f 2g 1a 2g 2a 2g 1a 3P 1a 4a 2a 2b 2c 2d 8a 4b 2e 8b 4c 2f 4d 4e 4f 2g 5P 1a 4a 2a 2b 2c 2d 8a 4b 2e 8b 4c 2f 4d 4e 4f 2g 7P 1a 4a 2a 2b 2c 2d 8a 4b 2e 8b 4c 2f 4d 4e 4f 2g X.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 X.3 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 X.5 1 -1 1 1 -1 1 -1 1 1 -1 1 1 1 -1 1 1 X.6 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 1 1 X.7 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 X.8 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 X.9 2 . 2 -2 . . . . . . . 2 -2 . -2 2 X.10 2 . 2 2 . . . . . . . -2 -2 . -2 2 X.11 2 . -2 . . -2 . 2 . . . . -2 . 2 2 X.12 2 . -2 . . . . . -2 . 2 . 2 . -2 2 X.13 2 . -2 . . . . . 2 . -2 . 2 . -2 2 X.14 2 . -2 . . 2 . -2 . . . . -2 . 2 2 X.15 4 -2 . . . . . . . . . . . 2 . -4 X.16 4 2 . . . . . . . . . . . -2 . -4 |
magma: CharacterTable(G);