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Magma
magma: G := TransitiveGroup(16, 20);
Group invariants
Abstract group: | $(C_2 \times Q_8):C_2$ | magma: IdentifyGroup(G);
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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: | $2$ | magma: NilpotencyClass(G);
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Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,7,2,8)(3,6,4,5)(9,12,10,11)(13,15,14,16)$, $(1,2)(3,4)(5,6)(7,8)$, $(1,4,2,3)(5,7,6,8)(9,16,10,15)(11,13,12,14)$, $(1,16)(2,15)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 15 $4$: $C_2^2$ x 35 $8$: $C_2^3$ x 15 $16$: $C_2^4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7
Degree 8: $C_2^3$
Low degree siblings
16T20 x 4, 32T6Siblings 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 | Index | Representative |
1A | $1^{16}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{8}$ | $1$ | $2$ | $8$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
2B | $2^{8}$ | $2$ | $2$ | $8$ | $( 1,14)( 2,13)( 3,11)( 4,12)( 5,10)( 6, 9)( 7,16)( 8,15)$ |
2C | $2^{8}$ | $2$ | $2$ | $8$ | $( 1, 9)( 2,10)( 3,15)( 4,16)( 5,14)( 6,13)( 7,11)( 8,12)$ |
2D | $2^{8}$ | $2$ | $2$ | $8$ | $( 1,12)( 2,11)( 3,14)( 4,13)( 5,16)( 6,15)( 7, 9)( 8,10)$ |
2E | $2^{8}$ | $2$ | $2$ | $8$ | $( 1,16)( 2,15)( 3, 9)( 4,10)( 5,11)( 6,12)( 7,13)( 8,14)$ |
2F | $2^{4},1^{8}$ | $2$ | $2$ | $4$ | $(1,2)(3,4)(5,6)(7,8)$ |
4A | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,13, 2,14)( 3,12, 4,11)( 5, 9, 6,10)( 7,15, 8,16)$ |
4B | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$ |
4C | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 9, 2,10)( 3,15, 4,16)( 5,14, 6,13)( 7,11, 8,12)$ |
4D | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$ |
4E | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,12,10,11)(13,15,14,16)$ |
4F | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,16, 2,15)( 3, 9, 4,10)( 5,11, 6,12)( 7,13, 8,14)$ |
4G | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,16,10,15)(11,13,12,14)$ |
4H | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,11, 2,12)( 3,13, 4,14)( 5,15, 6,16)( 7,10, 8, 9)$ |
4I | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,11,10,12)(13,16,14,15)$ |
4J | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,16,10,15)(11,13,12,14)$ |
Malle's constant $a(G)$: $1/4$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ||
Size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | |
Type | ||||||||||||||||||
32.50.1a | R | |||||||||||||||||
32.50.1b | R | |||||||||||||||||
32.50.1c | R | |||||||||||||||||
32.50.1d | R | |||||||||||||||||
32.50.1e | R | |||||||||||||||||
32.50.1f | R | |||||||||||||||||
32.50.1g | R | |||||||||||||||||
32.50.1h | R | |||||||||||||||||
32.50.1i | R | |||||||||||||||||
32.50.1j | R | |||||||||||||||||
32.50.1k | R | |||||||||||||||||
32.50.1l | R | |||||||||||||||||
32.50.1m | R | |||||||||||||||||
32.50.1n | R | |||||||||||||||||
32.50.1o | R | |||||||||||||||||
32.50.1p | R | |||||||||||||||||
32.50.4a | S |
magma: CharacterTable(G);
Regular extensions
Data not computed