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Magma
magma: G := TransitiveGroup(16, 37);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $37$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^2.D_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,16)(2,15)(3,6)(4,5)(11,12)(13,14), (1,4,2,3)(5,16,6,15)(7,12,8,11)(9,14,10,13), (1,7,15,10)(2,8,16,9)(3,14,6,11)(4,13,5,12) | 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 2, $C_2^3$ $16$: $D_4\times C_2$, $Q_8:C_2$ x 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: $D_4\times C_2$, $Q_8:C_2$ x 2
Low degree siblings
16T54 x 2, 32T23Siblings 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, 2, 2, 1, 1, 1, 1 $ | $4$ | $2$ | $( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,14)(12,13)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$ | |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 5,15, 6,16)( 7,11, 8,12)( 9,13,10,14)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,11)( 8,12)( 9,13)(10,14)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 7, 2, 8)( 3,13, 4,14)( 5,11, 6,12)( 9,15,10,16)$ | |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7,15,10)( 2, 8,16, 9)( 3,14, 6,11)( 4,13, 5,12)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 8, 2, 7)( 3,14, 4,13)( 5,12, 6,11)( 9,16,10,15)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,15,14,16)$ | |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,11,16,13)( 2,12,15,14)( 3,10, 5, 8)( 4, 9, 6, 7)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,12, 2,11)( 3,10, 4, 9)( 5, 8, 6, 7)(13,16,14,15)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,16)( 2,15)( 3, 5)( 4, 6)( 7, 9)( 8,10)(11,13)(12,14)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $32=2^{5}$ | 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: | $2$ | ||
Label: | 32.30 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 4A1 | 4A-1 | 4B1 | 4B-1 | 4C | 4D | 4E | ||
Size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2B | 2B | 2B | 2B | 2B | 2A | 2C | |
Type | |||||||||||||||
32.30.1a | R | ||||||||||||||
32.30.1b | R | ||||||||||||||
32.30.1c | R | ||||||||||||||
32.30.1d | R | ||||||||||||||
32.30.1e | R | ||||||||||||||
32.30.1f | R | ||||||||||||||
32.30.1g | R | ||||||||||||||
32.30.1h | R | ||||||||||||||
32.30.2a | R | ||||||||||||||
32.30.2b | R | ||||||||||||||
32.30.2c1 | C | ||||||||||||||
32.30.2c2 | C | ||||||||||||||
32.30.2d1 | C | ||||||||||||||
32.30.2d2 | C |
magma: CharacterTable(G);