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Magma
magma: G := TransitiveGroup(16, 114);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $114$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_8:C_4$ | ||
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: | (1,16)(2,15)(3,9)(4,10)(5,12)(6,11)(7,14)(8,13), (1,14)(2,13)(3,16)(4,15)(5,9)(6,10)(7,11)(8,12), (1,10,6,13,2,9,5,14)(3,11,7,15,4,12,8,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_4$ x 4, $C_2^2$ x 7 $8$: $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$ $16$: $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$ $32$: $C_4 \times D_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 8: $Q_8:C_2$
Low degree siblings
16T114, 32T119, 32T248 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | 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 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ | |
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $4$ | $( 9,13,10,14)(11,16,12,15)$ | |
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $4$ | $( 9,14,10,13)(11,15,12,16)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ | |
$ 4, 4, 2, 2, 2, 2 $ | $2$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13,10,14)(11,16,12,15)$ | |
$ 4, 4, 2, 2, 2, 2 $ | $2$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14,10,13)(11,15,12,16)$ | |
$ 8, 8 $ | $2$ | $8$ | $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,11,13,16,10,12,14,15)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,12,13,15,10,11,14,16)$ | |
$ 8, 8 $ | $2$ | $8$ | $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,15,14,12,10,16,13,11)$ | |
$ 8, 8 $ | $2$ | $8$ | $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,16,14,11,10,15,13,12)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,11,13,16,10,12,14,15)$ | |
$ 8, 8 $ | $2$ | $8$ | $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,15,14,12,10,16,13,11)$ | |
$ 8, 8 $ | $2$ | $8$ | $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,16,14,11,10,15,13,12)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)$ | |
$ 8, 8 $ | $2$ | $8$ | $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,15,14,12,10,16,13,11)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,16,14,11,10,15,13,12)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1, 8, 6, 3, 2, 7, 5, 4)( 9,15,14,12,10,16,13,11)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$ | |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,16, 8,15)$ | |
$ 8, 8 $ | $4$ | $8$ | $( 1, 9, 5,13, 2,10, 6,14)( 3,12, 8,15, 4,11, 7,16)$ | |
$ 8, 8 $ | $4$ | $8$ | $( 1, 9, 6,14, 2,10, 5,13)( 3,12, 7,16, 4,11, 8,15)$ | |
$ 8, 8 $ | $4$ | $8$ | $( 1,11, 6,15, 2,12, 5,16)( 3,14, 7,10, 4,13, 8, 9)$ | |
$ 8, 8 $ | $4$ | $8$ | $( 1,11, 5,16, 2,12, 6,15)( 3,14, 8, 9, 4,13, 7,10)$ | |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,11, 2,12)( 3,14, 4,13)( 5,16, 6,15)( 7,10, 8, 9)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,10)( 8, 9)$ |
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: | 64.124 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 4A1 | 4A-1 | 4B | 4C1 | 4C-1 | 4D1 | 4D-1 | 4E | 4F | 8A1 | 8A-1 | 8A3 | 8A-3 | 8B1 | 8B-1 | 8C1 | 8C3 | 8D1 | 8D-1 | 8E1 | 8E-1 | 8F1 | 8F-1 | ||
Size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2B | 2B | 2A | 2B | 2B | 2A | 2A | 4A1 | 4A-1 | 4A-1 | 4A1 | 4B | 4B | 4B | 4A-1 | 4A1 | 4B | 4A1 | 4A1 | 4A-1 | 4A-1 | |
Type | |||||||||||||||||||||||||||||
64.124.1a | R | ||||||||||||||||||||||||||||
64.124.1b | R | ||||||||||||||||||||||||||||
64.124.1c | R | ||||||||||||||||||||||||||||
64.124.1d | R | ||||||||||||||||||||||||||||
64.124.1e | R | ||||||||||||||||||||||||||||
64.124.1f | R | ||||||||||||||||||||||||||||
64.124.1g | R | ||||||||||||||||||||||||||||
64.124.1h | R | ||||||||||||||||||||||||||||
64.124.1i1 | C | ||||||||||||||||||||||||||||
64.124.1i2 | C | ||||||||||||||||||||||||||||
64.124.1j1 | C | ||||||||||||||||||||||||||||
64.124.1j2 | C | ||||||||||||||||||||||||||||
64.124.1k1 | C | ||||||||||||||||||||||||||||
64.124.1k2 | C | ||||||||||||||||||||||||||||
64.124.1l1 | C | ||||||||||||||||||||||||||||
64.124.1l2 | C | ||||||||||||||||||||||||||||
64.124.2a | R | ||||||||||||||||||||||||||||
64.124.2b | R | ||||||||||||||||||||||||||||
64.124.2c1 | C | ||||||||||||||||||||||||||||
64.124.2c2 | C | ||||||||||||||||||||||||||||
64.124.2d1 | C | ||||||||||||||||||||||||||||
64.124.2d2 | C | ||||||||||||||||||||||||||||
64.124.2d3 | C | ||||||||||||||||||||||||||||
64.124.2d4 | C | ||||||||||||||||||||||||||||
64.124.2e1 | C | ||||||||||||||||||||||||||||
64.124.2e2 | C | ||||||||||||||||||||||||||||
64.124.2e3 | C | ||||||||||||||||||||||||||||
64.124.2e4 | C |
magma: CharacterTable(G);