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Properties

Label 16T686
16T686 1 8 1->8 14 1->14 2 7 2->7 13 2->13 3 6 3->6 16 3->16 4 5 4->5 15 4->15 5->3 11 5->11 6->4 12 6->12 7->1 10 7->10 8->2 9 8->9 9->3 9->11 10->4 10->12 11->1 11->10 12->2 12->9 13->7 13->14 14->8 15->6 16->5
Degree $16$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4.D_8$

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Copy content magma:G := TransitiveGroup(16, 686);
 

Group invariants

Abstract group:  $C_2^4.D_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $256=2^{8}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $5$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $686$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,8,2,7)(3,6,4,5)(9,11,10,12)(13,14)$, $(1,14,8,9,3,16,5,11)(2,13,7,10,4,15,6,12)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$
$32$:  $C_4\wr C_2$, $C_2^3 : C_4 $, 16T26
$64$:  $((C_8 : C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T163
$128$:  16T330

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $D_{8}$

Low degree siblings

16T682 x 2, 16T686, 16T699 x 4, 16T703 x 2, 16T705 x 2, 32T3297 x 2, 32T3298, 32T3299, 32T3317 x 2, 32T3318 x 2, 32T3349 x 2, 32T3350 x 4, 32T3351 x 4, 32T3352 x 4, 32T3353 x 2, 32T3363 x 2, 32T3364 x 2, 32T3365, 32T3366 x 2, 32T3367, 32T3368, 32T3369, 32T3370 x 2, 32T3374 x 2, 32T7424, 32T7605

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $2$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2D $2^{8}$ $2$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)$
2E $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
2F $2^{4},1^{8}$ $4$ $2$ $4$ $( 1, 2)( 3, 4)(13,14)(15,16)$
2G $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
2H $2^{2},1^{12}$ $4$ $2$ $2$ $(13,14)(15,16)$
2I $2^{8}$ $8$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)$
2J $2^{8}$ $16$ $2$ $8$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,10)( 6, 9)( 7,12)( 8,11)$
4A1 $4^{4}$ $8$ $4$ $12$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,13,12,16)(10,14,11,15)$
4A-1 $4^{4}$ $8$ $4$ $12$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,16,12,13)(10,15,11,14)$
4B $4^{4}$ $16$ $4$ $12$ $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,13,11,15)(10,14,12,16)$
4C $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,13, 2,14)( 3,16, 4,15)( 5, 9)( 6,10)( 7,11)( 8,12)$
4D $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,15, 2,16)( 3,14, 4,13)( 5,11)( 6,12)( 7, 9)( 8,10)$
4E $4^{4}$ $16$ $4$ $12$ $( 1,12, 2,11)( 3, 9, 4,10)( 5,14, 6,13)( 7,16, 8,15)$
4F1 $4^{3},2,1^{2}$ $16$ $4$ $10$ $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,11,10,12)(13,14)$
4F-1 $4^{3},2,1^{2}$ $16$ $4$ $10$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,12,10,11)(13,14)$
4G1 $4,2^{5},1^{2}$ $16$ $4$ $8$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)(11,12)(13,16,14,15)$
4G-1 $4,2^{5},1^{2}$ $16$ $4$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,10)(13,16,14,15)$
8A1 $8^{2}$ $16$ $8$ $14$ $( 1,12, 5,16, 4, 9, 7,13)( 2,11, 6,15, 3,10, 8,14)$
8A-1 $8^{2}$ $16$ $8$ $14$ $( 1,11, 5,15, 3, 9, 8,13)( 2,12, 6,16, 4,10, 7,14)$
8A3 $8^{2}$ $16$ $8$ $14$ $( 1, 9, 6,14, 3,11, 7,16)( 2,10, 5,13, 4,12, 8,15)$
8A-3 $8^{2}$ $16$ $8$ $14$ $( 1,10, 6,13, 4,11, 8,16)( 2, 9, 5,14, 3,12, 7,15)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A1 4A-1 4B 4C 4D 4E 4F1 4F-1 4G1 4G-1 8A1 8A-1 8A3 8A-3
Size 1 1 2 2 2 4 4 4 4 8 16 8 8 16 16 16 16 16 16 16 16 16 16 16 16
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2D 2D 2C 2F 2F 2A 2G 2G 2H 2H 4A1 4A-1 4A-1 4A1
Type
256.382.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.382.1b R 1 1 1 1 1 1 1 1 1 1 −1 1 1 −1 −1 −1 1 −1 −1 −1 −1 1 1 1 1
256.382.1c R 1 1 1 1 1 1 1 1 1 1 −1 1 1 −1 1 1 1 1 −1 1 −1 −1 −1 −1 −1
256.382.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 −1 −1
256.382.1e1 C 1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 −1 −i i 1 −i 1 i 1 i −i −i i
256.382.1e2 C 1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 −1 i −i 1 i 1 −i 1 −i i i −i
256.382.1f1 C 1 1 1 1 1 −1 1 −1 1 −1 1 −1 −1 1 −i i 1 −i −1 i −1 −i i i −i
256.382.1f2 C 1 1 1 1 1 −1 1 −1 1 −1 1 −1 −1 1 i −i 1 i −1 −i −1 i −i −i i
256.382.2a R 2 2 2 2 2 −2 2 −2 2 −2 0 2 2 0 0 0 −2 0 0 0 0 0 0 0 0
256.382.2b R 2 2 2 2 2 2 2 2 2 2 0 −2 −2 0 0 0 −2 0 0 0 0 0 0 0 0
256.382.2c1 R 2 2 2 −2 −2 2 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 −ζ8−1−ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8−1+ζ8
256.382.2c2 R 2 2 2 −2 −2 2 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 ζ8−1+ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−1−ζ8
256.382.2d1 C 2 2 2 −2 −2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83
256.382.2d2 C 2 2 2 −2 −2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83
256.382.2e1 C 2 2 2 −2 −2 0 −2 0 2 0 0 −2i 2i 0 −1+i −1−i 0 1−i 0 1+i 0 0 0 0 0
256.382.2e2 C 2 2 2 −2 −2 0 −2 0 2 0 0 2i −2i 0 −1−i −1+i 0 1+i 0 1−i 0 0 0 0 0
256.382.2f1 C 2 2 2 −2 −2 0 −2 0 2 0 0 −2i 2i 0 1−i 1+i 0 −1+i 0 −1−i 0 0 0 0 0
256.382.2f2 C 2 2 2 −2 −2 0 −2 0 2 0 0 2i −2i 0 1+i 1−i 0 −1−i 0 −1+i 0 0 0 0 0
256.382.4a R 4 4 4 4 4 0 −4 0 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
256.382.4b R 4 4 −4 4 −4 0 0 0 0 0 −2 0 0 2 0 0 0 0 0 0 0 0 0 0 0
256.382.4c R 4 4 −4 4 −4 0 0 0 0 0 2 0 0 −2 0 0 0 0 0 0 0 0 0 0 0
256.382.4d R 4 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 0 2 0 0 0 0
256.382.4e R 4 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 −2 0 0 0 0
256.382.8a R 8 −8 0 0 0 −4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
256.382.8b R 8 −8 0 0 0 4 0 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed