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Magma
magma: G := TransitiveGroup(16, 149);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $149$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\wr C_2^2$ | ||
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: | (1,2)(3,15)(4,16)(5,6)(7,8)(9,13)(10,14)(11,12), (1,15,6,4)(2,16,5,3)(7,11)(8,12)(9,10)(13,14), (1,8,16,10)(2,7,15,9)(3,13,6,11)(4,14,5,12) | 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: $D_4\times C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$
Low degree siblings
8T29 x 6, 8T31 x 2, 16T127, 16T128 x 3, 16T129 x 3, 16T147, 16T149 x 5, 16T150 x 3, 32T136 x 3, 32T137 x 2, 32T163 x 3Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 7, 9)( 8,10)(11,13)(12,14)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 7,12)( 8,11)( 9,14)(10,13)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3,15)( 4,16)( 5, 6)( 7, 8)( 9,13)(10,14)(11,12)$ |
$ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3,15)( 4,16)( 5, 6)( 7,10,12,13)( 8, 9,11,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7, 9)( 8,10)(11,13)(12,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,12)( 8,11)( 9,14)(10,13)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,14)( 8,13)( 9,12)(10,11)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 6,15)( 2, 3, 5,16)( 7,10,12,13)( 8, 9,11,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,12)( 8,11)( 9,14)(10,13)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,15)(14,16)$ |
$ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 7, 3, 9)( 2, 8, 4,10)( 5,11,15,13)( 6,12,16,14)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7, 6,12)( 2, 8, 5,11)( 3, 9,16,14)( 4,10,15,13)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 8)( 2, 7)( 3,13)( 4,14)( 5,12)( 6,11)( 9,15)(10,16)$ |
$ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 8, 3,13)( 2, 7, 4,14)( 5,12,15, 9)( 6,11,16,10)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 8, 6,11)( 2, 7, 5,12)( 3,13,16,10)( 4,14,15, 9)$ |
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: | 64.138 | magma: IdentifyGroup(G);
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Character table: |
2 6 4 5 4 3 5 4 5 4 6 4 3 4 4 3 4 1a 2a 2b 2c 4a 2d 2e 2f 4b 2g 2h 4c 4d 2i 4e 4f 2P 1a 1a 1a 1a 2b 1a 1a 1a 2g 1a 1a 2d 2g 1a 2f 2g 3P 1a 2a 2b 2c 4a 2d 2e 2f 4b 2g 2h 4c 4d 2i 4e 4f 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);