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Magma
magma: G := TransitiveGroup(16, 18);
Group invariants
Abstract group: | $C_2 \times (C_4\times C_2):C_2$ | 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 $t$: | $18$ | 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: | $(1,16)(2,15)(3,6)(4,5)(7,10)(8,9)(11,14)(12,13)$, $(1,5,10,13)(2,6,9,14)(3,8,11,15)(4,7,12,16)$, $(1,9)(2,10)(3,4)(5,14)(6,13)(7,8)(11,12)(15,16)$, $(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)$ | 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 $16$: $Q_8:C_2$ x 2, $C_2^4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7
Low degree siblings
16T18 x 5, 32T4Siblings 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 | $2^{8}$ | $1$ | $2$ | $8$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,16)( 8,15)$ |
2C | $2^{8}$ | $1$ | $2$ | $8$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$ |
2D | $2^{8}$ | $2$ | $2$ | $8$ | $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,15)( 6,16)( 7,14)( 8,13)$ |
2E | $2^{8}$ | $2$ | $2$ | $8$ | $( 1, 8)( 2, 7)( 3,13)( 4,14)( 5,11)( 6,12)( 9,16)(10,15)$ |
2F | $2^{8}$ | $2$ | $2$ | $8$ | $( 1,16)( 2,15)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$ |
2G | $2^{8}$ | $2$ | $2$ | $8$ | $( 1, 2)( 3,12)( 4,11)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$ |
2H | $2^{4},1^{8}$ | $2$ | $2$ | $4$ | $( 3,11)( 4,12)( 7,16)( 8,15)$ |
2I | $2^{8}$ | $2$ | $2$ | $8$ | $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,16)( 6,15)( 7,13)( 8,14)$ |
4A1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1,13,10, 5)( 2,14, 9, 6)( 3,15,11, 8)( 4,16,12, 7)$ |
4A-1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 5,10,13)( 2, 6, 9,14)( 3, 8,11,15)( 4, 7,12,16)$ |
4B1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1,14,10, 6)( 2,13, 9, 5)( 3,16,11, 7)( 4,15,12, 8)$ |
4B-1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 6,10,14)( 2, 5, 9,13)( 3, 7,11,16)( 4, 8,12,15)$ |
4C | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 3,10,11)( 2, 4, 9,12)( 5, 8,13,15)( 6, 7,14,16)$ |
4D | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,12,10, 4)( 2,11, 9, 3)( 5,16,13, 7)( 6,15,14, 8)$ |
4E | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,15,10, 8)( 2,16, 9, 7)( 3,13,11, 5)( 4,14,12, 6)$ |
4F | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,16,10, 7)( 2,15, 9, 8)( 3,14,11, 6)( 4,13,12, 5)$ |
4G | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,14,10, 6)( 2,13, 9, 5)( 3, 7,11,16)( 4, 8,12,15)$ |
4H | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,13,10, 5)( 2,14, 9, 6)( 3, 8,11,15)( 4, 7,12,16)$ |
Malle's constant $a(G)$: $1/4$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A1 | 4A-1 | 4B1 | 4B-1 | 4C | 4D | 4E | 4F | 4G | 4H | ||
Size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2C | 2C | 2C | 2C | 2C | 2C | 2C | 2C | 2C | 2C | |
Type | |||||||||||||||||||||
32.48.1a | R | ||||||||||||||||||||
32.48.1b | R | ||||||||||||||||||||
32.48.1c | R | ||||||||||||||||||||
32.48.1d | R | ||||||||||||||||||||
32.48.1e | R | ||||||||||||||||||||
32.48.1f | R | ||||||||||||||||||||
32.48.1g | R | ||||||||||||||||||||
32.48.1h | R | ||||||||||||||||||||
32.48.1i | R | ||||||||||||||||||||
32.48.1j | R | ||||||||||||||||||||
32.48.1k | R | ||||||||||||||||||||
32.48.1l | R | ||||||||||||||||||||
32.48.1m | R | ||||||||||||||||||||
32.48.1n | R | ||||||||||||||||||||
32.48.1o | R | ||||||||||||||||||||
32.48.1p | R | ||||||||||||||||||||
32.48.2a1 | C | ||||||||||||||||||||
32.48.2a2 | C | ||||||||||||||||||||
32.48.2b1 | C | ||||||||||||||||||||
32.48.2b2 | C |
magma: CharacterTable(G);