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Magma
magma: G := TransitiveGroup(16, 576);
Group action invariants
| Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $576$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $(C_2\times C_4^2).D_4$ | ||
| 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: | (1,11,15,6,2,12,16,5)(3,9,14,8,4,10,13,7), (1,11)(2,12)(3,9)(4,10)(5,13,6,14)(7,15,8,16) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 6, $C_2^2$ $8$: $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$ $16$: $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3 $32$: $C_2^3 : C_4 $ x 2, 32T41 $64$: $((C_8 : C_2):C_2):C_2$, 16T77, 16T140 $128$: 32T1313 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $C_2^3 : C_4 $
Low degree siblings
16T576 x 3, 32T2837 x 2, 32T2838 x 4, 32T6868 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 | 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 | $2^{4},1^{8}$ | $2$ | $2$ | $4$ | $( 1, 2)( 3, 4)(13,14)(15,16)$ |
| 2C | $2^{4},1^{8}$ | $4$ | $2$ | $4$ | $( 1, 2)( 3, 4)( 9,10)(11,12)$ |
| 2D | $2^{8}$ | $8$ | $2$ | $8$ | $( 1,14)( 2,13)( 3,16)( 4,15)( 5,10)( 6, 9)( 7,11)( 8,12)$ |
| 2E | $2^{8}$ | $8$ | $2$ | $8$ | $( 1,13)( 2,14)( 3,15)( 4,16)( 5,12)( 6,11)( 7,10)( 8, 9)$ |
| 4A1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$ |
| 4A-1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$ |
| 4B | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$ |
| 4C | $4^{2},2^{2},1^{4}$ | $4$ | $4$ | $8$ | $( 1, 4, 2, 3)( 9,10)(11,12)(13,16,14,15)$ |
| 4D | $4^{2},2^{4}$ | $4$ | $4$ | $10$ | $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,14)(15,16)$ |
| 4E | $4^{2},2^{2},1^{4}$ | $4$ | $4$ | $8$ | $( 1, 3, 2, 4)( 5, 6)( 7, 8)(13,15,14,16)$ |
| 4F1 | $4^{4}$ | $4$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$ |
| 4F-1 | $4^{2},1^{8}$ | $4$ | $4$ | $6$ | $( 5, 7, 6, 8)( 9,11,10,12)$ |
| 4G | $4^{4}$ | $8$ | $4$ | $12$ | $( 1,15, 2,16)( 3,14, 4,13)( 5,12, 6,11)( 7,10, 8, 9)$ |
| 4H | $4^{4}$ | $8$ | $4$ | $12$ | $( 1,15, 2,16)( 3,14, 4,13)( 5,10, 6, 9)( 7,11, 8,12)$ |
| 4I1 | $4^{2},2^{4}$ | $16$ | $4$ | $10$ | $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,15)(10,16)(11,13)(12,14)$ |
| 4I-1 | $4^{2},2^{4}$ | $16$ | $4$ | $10$ | $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,13,10,14)(11,16,12,15)$ |
| 8A1 | $8^{2}$ | $8$ | $8$ | $14$ | $( 1, 8, 4, 6, 2, 7, 3, 5)( 9,15,11,13,10,16,12,14)$ |
| 8A-1 | $8,4,2^{2}$ | $8$ | $8$ | $12$ | $( 1,13, 3,15, 2,14, 4,16)( 5, 6)( 7, 8)( 9,12,10,11)$ |
| 8B1 | $8^{2}$ | $8$ | $8$ | $14$ | $( 1, 8, 3, 5, 2, 7, 4, 6)( 9,16,12,13,10,15,11,14)$ |
| 8B-1 | $8^{2}$ | $8$ | $8$ | $14$ | $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,14,12,16,10,13,11,15)$ |
| 8C1 | $8,4,2^{2}$ | $8$ | $8$ | $12$ | $( 1,13, 3,15, 2,14, 4,16)( 5, 7, 6, 8)( 9,10)(11,12)$ |
| 8C-1 | $8,4,1^{4}$ | $8$ | $8$ | $10$ | $( 1,14, 3,16, 2,13, 4,15)( 5, 8, 6, 7)$ |
| 8D1 | $8,4,2^{2}$ | $8$ | $8$ | $12$ | $( 1,15, 4,13, 2,16, 3,14)( 5, 6)( 7, 8)( 9,11,10,12)$ |
| 8D-1 | $8,4,2^{2}$ | $8$ | $8$ | $12$ | $( 1,15, 4,13, 2,16, 3,14)( 5, 8, 6, 7)( 9,10)(11,12)$ |
| 8E1 | $8^{2}$ | $8$ | $8$ | $14$ | $( 1, 5, 4, 8, 2, 6, 3, 7)( 9,13,11,16,10,14,12,15)$ |
| 8E-1 | $8,4,1^{4}$ | $8$ | $8$ | $10$ | $( 1,16, 4,14, 2,15, 3,13)( 5, 7, 6, 8)$ |
| 8F1 | $8,4,1^{4}$ | $8$ | $8$ | $10$ | $( 1,14, 3,16, 2,13, 4,15)( 9,11,10,12)$ |
| 8F-1 | $8,4,1^{4}$ | $8$ | $8$ | $10$ | $( 1,16, 4,14, 2,15, 3,13)( 9,12,10,11)$ |
| 8G1 | $8^{2}$ | $16$ | $8$ | $14$ | $( 1, 7,16,10, 2, 8,15, 9)( 3, 6,13,11, 4, 5,14,12)$ |
| 8G-1 | $8^{2}$ | $16$ | $8$ | $14$ | $( 1, 5,15,11, 2, 6,16,12)( 3, 7,14, 9, 4, 8,13,10)$ |
| 8H1 | $8^{2}$ | $16$ | $8$ | $14$ | $( 1, 6,14, 9, 2, 5,13,10)( 3, 8,16,12, 4, 7,15,11)$ |
| 8H-1 | $8^{2}$ | $16$ | $8$ | $14$ | $( 1, 8,13,11, 2, 7,14,12)( 3, 5,15, 9, 4, 6,16,10)$ |
Malle's constant $a(G)$: $1/4$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $256=2^{8}$ | 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: | $5$ | ||
| Label: | 256.480 | magma: IdentifyGroup(G);
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| Character table: | 34 x 34 character table |
magma: CharacterTable(G);