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Properties

Label 16T321
16T321 1 2 1->2 6 1->6 16 1->16 5 2->5 15 2->15 3 4 3->4 7 3->7 14 3->14 8 4->8 13 4->13 5->4 5->8 10 5->10 6->3 6->7 9 6->9 7->2 7->5 12 7->12 8->1 8->6 11 8->11 9->7 9->12 9->15 10->8 10->11 10->16 11->6 11->9 11->14 12->5 12->10 12->13 13->1 13->10 13->14 14->2 14->9 15->4 15->12 15->16 16->3 16->11
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2.D_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 321);
 

Group invariants

Abstract group:  $C_4^2.D_4$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $321$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2)(3,4)(5,8,6,7)(9,12,10,11)(13,14)(15,16)$, $(1,6,3,7,2,5,4,8)(9,15,12,13,10,16,11,14)$, $(1,16,3,14,2,15,4,13)(5,10,8,11,6,9,7,12)$
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, $Q_8:C_2$ x 4, $C_4\times C_2^2$
$32$:  $C_4^2:C_2$, $C_4 \times D_4$ x 2, 16T27, 16T30, 16T34, 16T37
$64$:  32T200

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8:C_2$

Low degree siblings

16T321, 32T735, 32T736 x 2, 32T737, 32T1913, 32T2000

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
2D $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)(13,14)(15,16)$
2E $2^{8}$ $8$ $2$ $8$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,16)(10,15)(11,13)(12,14)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$
4C $4^{4}$ $2$ $4$ $12$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$
4D $4^{4}$ $2$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
4E1 $4^{2},2^{4}$ $2$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,14)(15,16)$
4E-1 $4^{2},2^{4}$ $2$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$
4F1 $4^{2},1^{8}$ $2$ $4$ $6$ $( 5, 7, 6, 8)( 9,11,10,12)$
4F-1 $4^{2},1^{8}$ $2$ $4$ $6$ $( 5, 8, 6, 7)( 9,12,10,11)$
4G1 $4^{2},2^{2},1^{4}$ $4$ $4$ $8$ $( 5, 7, 6, 8)( 9,12,10,11)(13,14)(15,16)$
4G-1 $4^{2},2^{2},1^{4}$ $4$ $4$ $8$ $( 5, 8, 6, 7)( 9,11,10,12)(13,14)(15,16)$
4H $4^{4}$ $8$ $4$ $12$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,15,10,16)(11,14,12,13)$
4I1 $4^{2},2^{4}$ $8$ $4$ $10$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,14, 6,13)( 7,16, 8,15)$
4I-1 $4^{2},2^{4}$ $8$ $4$ $10$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,13)( 6,14)( 7,15)( 8,16)$
8A1 $8^{2}$ $4$ $8$ $14$ $( 1,13, 4,15, 2,14, 3,16)( 5,10, 8,11, 6, 9, 7,12)$
8A-1 $8^{2}$ $4$ $8$ $14$ $( 1,15, 3,13, 2,16, 4,14)( 5,11, 7,10, 6,12, 8, 9)$
8B1 $8^{2}$ $4$ $8$ $14$ $( 1,13, 4,15, 2,14, 3,16)( 5, 9, 8,12, 6,10, 7,11)$
8B-1 $8^{2}$ $4$ $8$ $14$ $( 1,15, 3,13, 2,16, 4,14)( 5,12, 7, 9, 6,11, 8,10)$
8C1 $8^{2}$ $4$ $8$ $14$ $( 1,14, 4,16, 2,13, 3,15)( 5,11, 7,10, 6,12, 8, 9)$
8C-1 $8^{2}$ $4$ $8$ $14$ $( 1,16, 3,14, 2,15, 4,13)( 5,10, 8,11, 6, 9, 7,12)$
8D1 $8^{2}$ $4$ $8$ $14$ $( 1,16, 3,14, 2,15, 4,13)( 5, 9, 8,12, 6,10, 7,11)$
8D-1 $8^{2}$ $4$ $8$ $14$ $( 1,14, 4,16, 2,13, 3,15)( 5,12, 7, 9, 6,11, 8,10)$
8E $8^{2}$ $8$ $8$ $14$ $( 1,12, 3,10, 2,11, 4, 9)( 5,16, 7,13, 6,15, 8,14)$
8F $8^{2}$ $8$ $8$ $14$ $( 1,12, 4, 9, 2,11, 3,10)( 5,15, 8,13, 6,16, 7,14)$
8G1 $8^{2}$ $8$ $8$ $14$ $( 1, 8, 4, 5, 2, 7, 3, 6)( 9,14,11,16,10,13,12,15)$
8G-1 $8^{2}$ $8$ $8$ $14$ $( 1, 8, 3, 6, 2, 7, 4, 5)( 9,13,12,16,10,14,11,15)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

32 x 32 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed