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Magma
magma: G := TransitiveGroup(16, 43);
Group invariants
Abstract group: | $C_4:D_4$ | 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 $t$: | $43$ | 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: | $(1,16)(2,15)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)$, $(1,10,15,7)(2,9,16,8)(3,11,6,14)(4,12,5,13)$, $(1,3,2,4)(5,15,6,16)(7,12,8,11)(9,14,10,13)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $D_{4}$ x 4, $C_2^3$ $16$: $D_4\times C_2$ x 2, $Q_8:C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 4
Degree 8: $D_4$, $D_4\times C_2$, $Q_8:C_2$
Low degree siblings
16T34 x 2, 16T43, 32T20Siblings 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 | $2^{8}$ | $1$ | $2$ | $8$ | $( 1,16)( 2,15)( 3, 5)( 4, 6)( 7, 9)( 8,10)(11,13)(12,14)$ |
2B | $2^{8}$ | $1$ | $2$ | $8$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
2C | $2^{8}$ | $1$ | $2$ | $8$ | $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$ |
2D | $2^{8}$ | $2$ | $2$ | $8$ | $( 1,16)( 2,15)( 3, 5)( 4, 6)( 7, 8)( 9,10)(11,12)(13,14)$ |
2E | $2^{4},1^{8}$ | $2$ | $2$ | $4$ | $( 1,15)( 2,16)( 3, 6)( 4, 5)$ |
2F | $2^{8}$ | $4$ | $2$ | $8$ | $( 1,11)( 2,12)( 3, 9)( 4,10)( 5, 7)( 6, 8)(13,16)(14,15)$ |
2G | $2^{8}$ | $4$ | $2$ | $8$ | $( 1, 8)( 2, 7)( 3,13)( 4,14)( 5,11)( 6,12)( 9,15)(10,16)$ |
4A | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 5, 2, 6)( 3,15, 4,16)( 7,11, 8,12)( 9,13,10,14)$ |
4B | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5,16, 6,15)( 7,14, 8,13)( 9,12,10,11)$ |
4C1 | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 3, 2, 4)( 5,15, 6,16)( 7,12, 8,11)( 9,14,10,13)$ |
4C-1 | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5,16, 6,15)( 7,11, 8,12)( 9,13,10,14)$ |
4D | $4^{4}$ | $4$ | $4$ | $12$ | $( 1,13,15,12)( 2,14,16,11)( 3, 7, 6,10)( 4, 8, 5, 9)$ |
4E | $4^{4}$ | $4$ | $4$ | $12$ | $( 1,10,15, 7)( 2, 9,16, 8)( 3,11, 6,14)( 4,12, 5,13)$ |
Malle's constant $a(G)$: $1/4$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C1 | 4C-1 | 4D | 4E | ||
Size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2B | 2B | 2B | 2B | 2C | 2C | |
Type | |||||||||||||||
32.28.1a | R | ||||||||||||||
32.28.1b | R | ||||||||||||||
32.28.1c | R | ||||||||||||||
32.28.1d | R | ||||||||||||||
32.28.1e | R | ||||||||||||||
32.28.1f | R | ||||||||||||||
32.28.1g | R | ||||||||||||||
32.28.1h | R | ||||||||||||||
32.28.2a | R | ||||||||||||||
32.28.2b | R | ||||||||||||||
32.28.2c | R | ||||||||||||||
32.28.2d | R | ||||||||||||||
32.28.2e1 | C | ||||||||||||||
32.28.2e2 | C |
magma: CharacterTable(G);
Regular extensions
Data not computed