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Properties

Label 16T1069
16T1069 1 3 1->3 15 1->15 2 4 2->4 16 2->16 14 3->14 13 4->13 5 9 5->9 5->14 6 10 6->10 6->13 7 12 7->12 7->15 8 11 8->11 8->16 9->4 9->11 10->3 10->12 11->2 11->10 12->1 12->9 13->6 14->5 15->7 16->8
Degree $16$
Order $768$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4.\GL(2,3)$

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Copy content magma:G := TransitiveGroup(16, 1069);
 

Group invariants

Abstract group:  $C_2^4.\GL(2,3)$
Copy content magma:IdentifyGroup(G);
 
Order:  $768=2^{8} \cdot 3$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1069$
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:  $(1,3)(2,4)(5,14)(6,13)(7,15)(8,16)(9,11,10,12)$, $(1,15,7,12)(2,16,8,11)(3,14,5,9,4,13,6,10)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$
$24$:  $S_4$
$48$:  $\textrm{GL(2,3)}$, 12T27, 16T65
$96$:  32T415
$192$:  16T432
$384$:  24T811

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $S_4$

Degree 8: $S_4$

Low degree siblings

16T1061 x 2, 16T1069, 32T34707 x 2, 32T34715, 32T34716, 32T35038

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 2)( 7, 8)( 9,10)(13,14)$
2C $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$
2D $2^{2},1^{12}$ $4$ $2$ $2$ $(5,6)(7,8)$
2E $2^{4},1^{8}$ $6$ $2$ $4$ $( 5, 6)( 7, 8)(13,14)(15,16)$
2F $2^{4},1^{8}$ $6$ $2$ $4$ $( 1, 2)( 5, 6)(11,12)(13,14)$
2G $2^{4},1^{8}$ $8$ $2$ $4$ $( 3, 4)( 5, 6)(11,12)(13,14)$
3A $3^{4},1^{4}$ $32$ $3$ $8$ $( 1,14, 9)( 2,13,10)( 3,15,12)( 4,16,11)$
4A1 $4^{2},2^{4}$ $24$ $4$ $10$ $( 1,14, 2,13)( 3,16)( 4,15)( 5,11, 6,12)( 7,10)( 8, 9)$
4A-1 $4^{2},2^{4}$ $24$ $4$ $10$ $( 1,13, 2,14)( 3,16)( 4,15)( 5,12, 6,11)( 7,10)( 8, 9)$
4B $4^{2},2^{4}$ $48$ $4$ $10$ $( 1, 8, 2, 7)( 3, 6)( 4, 5)( 9,13,10,14)(11,16)(12,15)$
4C1 $4^{3},2^{2}$ $48$ $4$ $11$ $( 1,16, 2,15)( 3,14, 4,13)( 5, 8)( 6, 7)( 9,12,10,11)$
4C-1 $4^{3},2^{2}$ $48$ $4$ $11$ $( 1,15, 2,16)( 3,13, 4,14)( 5, 8)( 6, 7)( 9,11,10,12)$
4D1 $4,2^{6}$ $48$ $4$ $9$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5, 7, 6, 8)( 9,12)(10,11)$
4D-1 $4,2^{6}$ $48$ $4$ $9$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5, 8, 6, 7)( 9,12)(10,11)$
6A $6^{2},1^{4}$ $32$ $6$ $10$ $( 1,10,14, 2, 9,13)( 3,11,15, 4,12,16)$
6B $3^{4},2^{2}$ $32$ $6$ $10$ $( 1, 9,13)( 2,10,14)( 3,12,16)( 4,11,15)( 5, 6)( 7, 8)$
6C $6^{2},2^{2}$ $32$ $6$ $12$ $( 1,14, 7, 2,13, 8)( 3,16, 6, 4,15, 5)( 9,10)(11,12)$
6D1 $6,3^{2},2,1^{2}$ $32$ $6$ $10$ $( 1, 7,10)( 2, 8, 9)( 3, 6,12, 4, 5,11)(13,14)$
6D-1 $6,3^{2},2,1^{2}$ $32$ $6$ $10$ $( 1,10, 7)( 2, 9, 8)( 3,11, 5, 4,12, 6)(13,14)$
6E1 $6,3^{2},2,1^{2}$ $32$ $6$ $10$ $( 1, 8,14)( 2, 7,13)( 3, 6,16, 4, 5,15)(11,12)$
6E-1 $6,3^{2},2,1^{2}$ $32$ $6$ $10$ $( 1,14, 8)( 2,13, 7)( 3,15, 5, 4,16, 6)(11,12)$
8A1 $8,4^{2}$ $48$ $8$ $13$ $( 1,12,14, 5, 2,11,13, 6)( 3,10,16, 7)( 4, 9,15, 8)$
8A-1 $8,4^{2}$ $48$ $8$ $13$ $( 1, 6,13,11, 2, 5,14,12)( 3, 7,16,10)( 4, 8,15, 9)$
8A3 $8,4^{2}$ $48$ $8$ $13$ $( 1, 5,13,12, 2, 6,14,11)( 3, 7,16,10)( 4, 8,15, 9)$
8A-3 $8,4^{2}$ $48$ $8$ $13$ $( 1,11,14, 6, 2,12,13, 5)( 3,10,16, 7)( 4, 9,15, 8)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A 4A1 4A-1 4B 4C1 4C-1 4D1 4D-1 6A 6B 6C 6D1 6D-1 6E1 6E-1 8A1 8A-1 8A3 8A-3
Size 1 1 2 4 4 6 6 8 32 24 24 48 48 48 48 48 32 32 32 32 32 32 32 48 48 48 48
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 2F 2F 2B 2C 2C 2D 2D 3A 3A 3A 3A 3A 3A 3A 4A1 4A-1 4A-1 4A1
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 4A-1 4A1 4B 4C-1 4C1 4D-1 4D1 2C 2D 2A 2G 2G 2B 2B 8A3 8A-3 8A1 8A-1
Type
768.1085346.1a 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 1 1
768.1085346.1b 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 −1 −1
768.1085346.1c1 C 1 1 1 −1 −1 1 1 −1 1 −1 −1 −i i −i i 1 1 −1 1 −1 −1 −1 1 −i −i i i
768.1085346.1c2 C 1 1 1 −1 −1 1 1 −1 1 −1 −1 i −i i −i 1 1 −1 1 −1 −1 −1 1 i i −i −i
768.1085346.2a R 2 2 2 2 2 2 2 2 −1 2 2 0 0 0 0 2 −1 −1 −1 −1 −1 −1 −1 0 0 0 0
768.1085346.2b S 2 2 2 −2 −2 2 2 −2 −1 −2 −2 0 0 0 0 2 −1 1 −1 1 1 1 −1 0 0 0 0
768.1085346.2c1 C 2 2 −2 2 2 2 −2 −2 −1 0 0 0 0 0 0 0 1 −1 −1 1 1 −1 1 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83
768.1085346.2c2 C 2 2 −2 2 2 2 −2 −2 −1 0 0 0 0 0 0 0 1 −1 −1 1 1 −1 1 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83
768.1085346.2d1 S 2 2 −2 −2 −2 2 −2 2 −1 0 0 0 0 0 0 0 1 1 −1 −1 −1 1 1 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8
768.1085346.2d2 S 2 2 −2 −2 −2 2 −2 2 −1 0 0 0 0 0 0 0 1 1 −1 −1 −1 1 1 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8
768.1085346.3a R 3 3 3 3 3 3 3 3 0 −1 −1 1 1 1 1 −1 0 0 0 0 0 0 0 −1 −1 −1 −1
768.1085346.3b R 3 3 3 3 3 3 3 3 0 −1 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 1 1 1 1
768.1085346.3c1 C 3 3 3 −3 −3 3 3 −3 0 1 1 −i i −i i −1 0 0 0 0 0 0 0 i i −i −i
768.1085346.3c2 C 3 3 3 −3 −3 3 3 −3 0 1 1 i −i i −i −1 0 0 0 0 0 0 0 −i −i i i
768.1085346.4a R 4 4 −4 4 4 4 −4 −4 1 0 0 0 0 0 0 0 −1 1 1 −1 −1 1 −1 0 0 0 0
768.1085346.4b S 4 4 −4 −4 −4 4 −4 4 1 0 0 0 0 0 0 0 −1 −1 1 1 1 −1 −1 0 0 0 0
768.1085346.6a1 C 6 6 −6 0 0 −2 2 0 0 −2i 2i −1+i −1−i 1−i 1+i 0 0 0 0 0 0 0 0 0 0 0 0
768.1085346.6a2 C 6 6 −6 0 0 −2 2 0 0 2i −2i −1−i −1+i 1+i 1−i 0 0 0 0 0 0 0 0 0 0 0 0
768.1085346.6b1 C 6 6 −6 0 0 −2 2 0 0 −2i 2i 1−i 1+i −1+i −1−i 0 0 0 0 0 0 0 0 0 0 0 0
768.1085346.6b2 C 6 6 −6 0 0 −2 2 0 0 2i −2i 1+i 1−i −1−i −1+i 0 0 0 0 0 0 0 0 0 0 0 0
768.1085346.8a R 8 −8 0 −4 4 0 0 0 2 0 0 0 0 0 0 0 0 2 −2 0 0 −2 0 0 0 0 0
768.1085346.8b R 8 −8 0 4 −4 0 0 0 2 0 0 0 0 0 0 0 0 −2 −2 0 0 2 0 0 0 0 0
768.1085346.8c1 C 8 −8 0 −4 4 0 0 0 −1 0 0 0 0 0 0 0 −1−2ζ3 −1 1 −1−2ζ3 1+2ζ3 1 1+2ζ3 0 0 0 0
768.1085346.8c2 C 8 −8 0 −4 4 0 0 0 −1 0 0 0 0 0 0 0 1+2ζ3 −1 1 1+2ζ3 −1−2ζ3 1 −1−2ζ3 0 0 0 0
768.1085346.8d1 C 8 −8 0 4 −4 0 0 0 −1 0 0 0 0 0 0 0 −1−2ζ3 1 1 1+2ζ3 −1−2ζ3 −1 1+2ζ3 0 0 0 0
768.1085346.8d2 C 8 −8 0 4 −4 0 0 0 −1 0 0 0 0 0 0 0 1+2ζ3 1 1 −1−2ζ3 1+2ζ3 −1 −1−2ζ3 0 0 0 0
768.1085346.12a R 12 12 12 0 0 −4 −4 0 0 0 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