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Properties

Label 16T297
16T297 1 7 1->7 14 1->14 16 1->16 2 8 2->8 13 2->13 15 2->15 3 3->2 10 3->10 3->16 4 4->1 9 4->9 4->15 5 5->4 5->10 12 5->12 6 6->3 6->9 11 6->11 7->5 7->11 7->13 8->6 8->12 8->14 9->8 9->8 10->7 10->7 11->2 11->5 11->16 12->1 12->6 12->15 13->3 13->4 13->14 14->3 14->4 15->2 15->6 15->11 16->1 16->5 16->12
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4.D_4$

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Copy content magma:G := TransitiveGroup(16, 297);
 

Group invariants

Abstract group:  $C_2^4.D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $128=2^{7}$
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:  $4$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $297$
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:  $(1,16,5,12)(2,15,6,11)(3,10,7,13)(4,9,8,14)$, $(1,7,5,4)(2,8,6,3)(11,16,12,15)(13,14)$, $(1,14,3,16)(2,13,4,15)(5,10,7,11)(6,9,8,12)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$:  $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
$64$:  16T76

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

16T234 x 2, 16T250 x 2, 16T297 x 3, 16T310 x 2, 32T524 x 2, 32T525, 32T526 x 2, 32T569 x 2, 32T570, 32T689 x 4, 32T690 x 4, 32T718 x 2, 32T719

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $2^{8}$ $2$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$
2C $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2D $2^{8}$ $2$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
2E $2^{8}$ $4$ $2$ $8$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,16)(10,15)(11,13)(12,14)$
2F $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)(11,12)(15,16)$
2G $2^{8}$ $4$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$
2H $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)(11,12)(15,16)$
2I $2^{8}$ $8$ $2$ $8$ $( 1,16)( 2,15)( 3, 9)( 4,10)( 5,12)( 6,11)( 7,14)( 8,13)$
4A $4^{4}$ $4$ $4$ $12$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,15,10,16)(11,13,12,14)$
4B $4^{4}$ $4$ $4$ $12$ $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,12,10,11)(13,15,14,16)$
4C $4^{4}$ $8$ $4$ $12$ $( 1,10, 5,13)( 2, 9, 6,14)( 3,15, 7,11)( 4,16, 8,12)$
4D $4^{4}$ $8$ $4$ $12$ $( 1,13, 6, 9)( 2,14, 5,10)( 3,11, 8,16)( 4,12, 7,15)$
4E $4^{4}$ $8$ $4$ $12$ $( 1,12, 2,11)( 3,14, 4,13)( 5,16, 6,15)( 7, 9, 8,10)$
4F1 $4^{4}$ $8$ $4$ $12$ $( 1,12, 4,10)( 2,11, 3, 9)( 5,15, 8,14)( 6,16, 7,13)$
4F-1 $4^{4}$ $8$ $4$ $12$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,14, 8,15)( 6,13, 7,16)$
4G1 $4^{3},2,1^{2}$ $8$ $4$ $10$ $( 1, 4, 5, 7)( 2, 3, 6, 8)(11,15,12,16)(13,14)$
4G-1 $4^{3},2,1^{2}$ $8$ $4$ $10$ $( 1, 5, 2, 6)( 7, 8)( 9,15,13,11)(10,16,14,12)$
4H1 $4^{4}$ $8$ $4$ $12$ $( 1,15, 4,14)( 2,16, 3,13)( 5,12, 8,10)( 6,11, 7, 9)$
4H-1 $4^{4}$ $8$ $4$ $12$ $( 1,14, 4,15)( 2,13, 3,16)( 5,10, 8,12)( 6, 9, 7,11)$
4I1 $4^{3},2,1^{2}$ $8$ $4$ $10$ $( 1, 7, 6, 3)( 2, 8, 5, 4)( 9,14,10,13)(15,16)$
4I-1 $4^{3},2,1^{2}$ $8$ $4$ $10$ $( 1, 2)( 3, 8, 4, 7)( 9,11,14,16)(10,12,13,15)$

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 4F1 4F-1 4G1 4G-1 4H1 4H-1 4I1 4I-1
Size 1 1 2 2 2 4 4 4 4 8 4 4 8 8 8 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2D 2D 2A 2G 2G 2H 2H 2G 2G 2H 2H
Type
128.859.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.859.1b R 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1
128.859.1c R 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 1 1 1 1 −1 1 −1 −1 −1 1 −1
128.859.1d R 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 1 −1 1 1 −1 1
128.859.1e R 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 1 −1 −1 1 −1 1 1 −1 1 −1
128.859.1f R 1 1 1 1 1 1 1 1 1 −1 1 1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
128.859.1g R 1 1 1 1 1 1 1 1 1 −1 1 1 −1 1 −1 −1 −1 −1 −1 −1 1 1 1
128.859.1h R 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 1 −1 −1 −1
128.859.1i1 C 1 1 −1 −1 1 1 1 −1 −1 −1 1 −1 −1 −i 1 i −i −i i 1 i i −i
128.859.1i2 C 1 1 −1 −1 1 1 1 −1 −1 −1 1 −1 −1 i 1 −i i i −i 1 −i −i i
128.859.1j1 C 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 1 −i −1 −i i i −i −1 i i −i
128.859.1j2 C 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 1 i −1 i −i −i i −1 −i −i i
128.859.1k1 C 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 −i −1 −i −i i i 1 −i i i
128.859.1k2 C 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 i −1 i i −i −i 1 i −i −i
128.859.1l1 C 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −i 1 i i −i −i −1 −i i i
128.859.1l2 C 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 i 1 −i −i i i −1 i −i −i
128.859.2a R 2 2 −2 −2 2 −2 2 2 −2 0 −2 2 0 0 0 0 0 0 0 0 0 0 0
128.859.2b R 2 2 −2 −2 2 2 −2 −2 2 0 −2 2 0 0 0 0 0 0 0 0 0 0 0
128.859.2c R 2 2 2 2 2 −2 2 −2 2 0 −2 −2 0 0 0 0 0 0 0 0 0 0 0
128.859.2d R 2 2 2 2 2 2 −2 2 −2 0 −2 −2 0 0 0 0 0 0 0 0 0 0 0
128.859.4a R 4 4 −4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.859.4b R 4 4 4 −4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.859.8a R 8 −8 0 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