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Properties

Label 16T696
16T696 1 8 1->8 16 1->16 2 7 2->7 15 2->15 3 5 3->5 3->8 10 3->10 4 6 4->6 4->7 9 4->9 5->6 12 5->12 11 6->11 13 7->13 14 8->14 9->13 9->14 10->13 10->14 11->12 11->15 12->16 13->9 14->10 15->12 16->11
Degree $16$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4^2:C_2^2$

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Copy content magma:G := TransitiveGroup(16, 696);
 

Group invariants

Abstract group:  $D_4^2:C_2^2$
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 $t$:  $696$
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:  $(1,8)(2,7)(3,5)(4,6)(9,14,10,13)(11,15,12,16)$, $(3,8)(4,7)(5,6)(9,13)(10,14)(11,12)$, $(1,16)(2,15)(3,10)(4,9)(5,12)(6,11)(7,13)(8,14)$
Copy content magma:Generators(G);
 

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 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$
$128$:  $C_2 \wr C_2\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T659 x 2, 16T665 x 2, 16T696, 32T3217, 32T3218 x 4, 32T3219, 32T3220 x 4, 32T3221, 32T3241 x 2, 32T3242, 32T3243, 32T3344, 32T3345, 32T6289 x 4, 32T7363

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 $2^{4},1^{8}$ $2$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2C $2^{8}$ $8$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)$
2D $2^{8}$ $8$ $2$ $8$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,16)( 6,15)( 7, 9)( 8,10)$
2E $2^{4},1^{8}$ $8$ $2$ $4$ $( 9,11)(10,12)(13,15)(14,16)$
2F $2^{8}$ $8$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,16)(14,15)$
2G $2^{8}$ $8$ $2$ $8$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,15)(10,16)(11,14)(12,13)$
2H $2^{8}$ $8$ $2$ $8$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,11)(10,12)(13,15)(14,16)$
2I $2^{6},1^{4}$ $16$ $2$ $6$ $( 1, 2)( 3, 7)( 4, 8)( 9,13)(10,14)(11,12)$
4A $4^{2},2^{4}$ $4$ $4$ $10$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,10)(11,12)(13,14)(15,16)$
4B $4^{4}$ $4$ $4$ $12$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)$
4C $4^{2},1^{8}$ $4$ $4$ $6$ $(1,5,2,6)(3,8,4,7)$
4D $4^{4}$ $8$ $4$ $12$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,14, 6,13)( 7,15, 8,16)$
4E $4^{4}$ $8$ $4$ $12$ $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7, 9, 8,10)$
4F $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,15)(10,16)(11,14)(12,13)$
4G $4^{4}$ $32$ $4$ $12$ $( 1,13, 7,12)( 2,14, 8,11)( 3,15, 6, 9)( 4,16, 5,10)$
4H $4^{4}$ $32$ $4$ $12$ $( 1,11, 7, 9)( 2,12, 8,10)( 3,13, 6,15)( 4,14, 5,16)$
8A1 $8^{2}$ $4$ $8$ $14$ $( 1, 8, 5, 4, 2, 7, 6, 3)( 9,12,14,16,10,11,13,15)$
8A3 $8^{2}$ $4$ $8$ $14$ $( 1, 7, 5, 3, 2, 8, 6, 4)( 9,11,14,15,10,12,13,16)$
8B $8^{2}$ $8$ $8$ $14$ $( 1, 7, 5, 3, 2, 8, 6, 4)( 9,12,14,16,10,11,13,15)$
8C $8^{2}$ $16$ $8$ $14$ $( 1,10, 5,14, 2, 9, 6,13)( 3,11, 8,16, 4,12, 7,15)$
8D $8^{2}$ $16$ $8$ $14$ $( 1,11, 5,16, 2,12, 6,15)( 3,13, 8,10, 4,14, 7, 9)$
8E1 $8,2^{3},1^{2}$ $16$ $8$ $10$ $( 1, 8, 5, 4, 2, 7, 6, 3)( 9,13)(10,14)(11,12)$
8E3 $8,2^{3},1^{2}$ $16$ $8$ $10$ $( 1, 7, 5, 3, 2, 8, 6, 4)( 9,14)(10,13)(15,16)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 8A1 8A3 8B 8C 8D 8E1 8E3
Size 1 1 2 8 8 8 8 8 8 16 4 4 4 8 8 16 32 32 4 4 8 16 16 16 16
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2B 2A 2B 2A 2A 2B 2G 2H 4B 4B 4B 4B 4B 4C 4C
Type
256.6661.1a 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
256.6661.1b 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
256.6661.1c 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
256.6661.1d 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
256.6661.1e 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
256.6661.1f 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
256.6661.1g 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
256.6661.1h 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
256.6661.2a R 2 2 2 −2 0 0 0 2 −2 0 −2 2 −2 −2 0 0 0 0 0 0 0 0 0 2 0
256.6661.2b R 2 2 2 0 −2 0 0 −2 2 0 −2 2 −2 0 −2 0 0 0 0 0 0 0 0 0 2
256.6661.2c R 2 2 2 0 0 0 0 −2 −2 −2 2 2 2 0 0 0 0 0 2 2 2 0 0 0 0
256.6661.2d R 2 2 2 0 0 0 0 −2 −2 2 2 2 2 0 0 0 0 0 −2 −2 −2 0 0 0 0
256.6661.2e R 2 2 2 0 2 0 0 −2 2 0 −2 2 −2 0 2 0 0 0 0 0 0 0 0 0 −2
256.6661.2f R 2 2 2 2 0 0 0 2 −2 0 −2 2 −2 2 0 0 0 0 0 0 0 0 0 −2 0
256.6661.4a R 4 4 4 0 0 −2 −2 0 0 0 0 −4 0 0 0 2 0 0 0 0 0 0 0 0 0
256.6661.4b R 4 4 4 0 0 2 2 0 0 0 0 −4 0 0 0 −2 0 0 0 0 0 0 0 0 0
256.6661.4c R 4 4 −4 −2 −2 0 0 0 0 0 0 0 0 2 2 0 0 0 2 2 −2 0 0 0 0
256.6661.4d R 4 4 −4 −2 2 0 0 0 0 0 0 0 0 2 −2 0 0 0 −2 −2 2 0 0 0 0
256.6661.4e R 4 4 −4 2 −2 0 0 0 0 0 0 0 0 −2 2 0 0 0 −2 −2 2 0 0 0 0
256.6661.4f R 4 4 −4 2 2 0 0 0 0 0 0 0 0 −2 −2 0 0 0 2 2 −2 0 0 0 0
256.6661.4g1 R 4 −4 0 0 0 −2 2 0 0 0 −2 0 2 0 0 0 0 0 −2ζ8−1−2ζ8 2ζ8−1+2ζ8 0 ζ8−1+ζ8 −ζ8−1−ζ8 0 0
256.6661.4g2 R 4 −4 0 0 0 −2 2 0 0 0 −2 0 2 0 0 0 0 0 2ζ8−1+2ζ8 −2ζ8−1−2ζ8 0 −ζ8−1−ζ8 ζ8−1+ζ8 0 0
256.6661.4h1 R 4 −4 0 0 0 2 −2 0 0 0 −2 0 2 0 0 0 0 0 −2ζ8−1−2ζ8 2ζ8−1+2ζ8 0 −ζ8−1−ζ8 ζ8−1+ζ8 0 0
256.6661.4h2 R 4 −4 0 0 0 2 −2 0 0 0 −2 0 2 0 0 0 0 0 2ζ8−1+2ζ8 −2ζ8−1−2ζ8 0 ζ8−1+ζ8 −ζ8−1−ζ8 0 0
256.6661.8a R 8 −8 0 0 0 0 0 0 0 0 4 0 −4 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed