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Properties

Label 16T974
16T974 1 8 1->8 9 1->9 15 1->15 2 7 2->7 10 2->10 16 2->16 3 5 3->5 11 3->11 14 3->14 4 6 4->6 12 4->12 13 4->13 5->9 5->13 5->14 6->10 6->13 6->14 7->12 7->15 7->15 8->11 8->16 8->16 9->3 9->7 10->4 10->8 11->1 11->5 12->2 12->6 13->1 13->5 13->9 14->2 14->6 14->10 15->3 15->8 15->12 16->4 16->7 16->11
Degree $16$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^3.D_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 974);
 

Group invariants

Abstract group:  $C_4^3.D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $512=2^{9}$
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 $t$:  $974$
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:  $(1,8,16,11)(2,7,15,12)(3,5,13,9)(4,6,14,10)$, $(1,15,3,14,2,16,4,13)(5,9,7,12,6,10,8,11)$, $(1,9)(2,10)(3,11)(4,12)(5,14,6,13)(7,15,8,16)$
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, $C_2^3$, $Q_8$ x 2
$16$:  $D_4\times C_2$ x 3, $Q_8:C_2$ x 3, $Q_8\times C_2$
$32$:  $C_2^2 \wr C_2$, 16T30 x 3, 16T31 x 3
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 32T304
$128$:  16T333, 16T342, 16T344
$256$:  32T6140

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

16T974 x 3, 32T10875 x 2, 32T10876 x 2, 32T10877 x 2, 32T20066 x 2, 32T20067 x 2, 32T25936, 32T26004 x 2

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
2C $2^{4},1^{8}$ $4$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2D $2^{8}$ $8$ $2$ $8$ $( 1,16)( 2,15)( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
2E $2^{8}$ $8$ $2$ $8$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5, 9)( 6,10)( 7,12)( 8,11)$
2F $2^{6},1^{4}$ $16$ $2$ $6$ $( 1,15)( 2,16)( 3,13)( 4,14)( 7, 8)( 9,10)$
2G $2^{8}$ $16$ $2$ $8$ $( 1,15)( 2,16)( 3,13)( 4,14)( 5, 8)( 6, 7)( 9,11)(10,12)$
4A $4^{4}$ $2$ $4$ $12$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$
4C $4^{4}$ $4$ $4$ $12$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
4D $4^{2},2^{4}$ $4$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$
4E $4^{2},1^{8}$ $4$ $4$ $6$ $( 5, 7, 6, 8)( 9,11,10,12)$
4F $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,15,14,16)$
4G $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 5, 8, 6, 7)( 9,11,10,12)(13,14)(15,16)$
4H $4^{2},1^{8}$ $8$ $4$ $6$ $( 9,11,10,12)(13,16,14,15)$
4I $4^{4}$ $8$ $4$ $12$ $( 1,15, 2,16)( 3,14, 4,13)( 5, 9, 6,10)( 7,12, 8,11)$
4J $4^{4}$ $8$ $4$ $12$ $( 1,14, 2,13)( 3,16, 4,15)( 5, 9, 6,10)( 7,12, 8,11)$
4K1 $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,10)(11,12)$
4K-1 $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 2)( 3, 4)( 9,11,10,12)(13,15,14,16)$
4L $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,13, 2,14)( 3,16, 4,15)( 5, 8)( 6, 7)( 9,11)(10,12)$
4M $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1,16, 2,15)( 3,14, 4,13)( 7, 8)(11,12)$
4N $4^{2},2^{4}$ $32$ $4$ $10$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,16)( 6,15)( 7,14)( 8,13)$
4O $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$
4P1 $4^{4}$ $32$ $4$ $12$ $( 1,12,15, 7)( 2,11,16, 8)( 3,10,14, 6)( 4, 9,13, 5)$
4P-1 $4^{4}$ $32$ $4$ $12$ $( 1, 7,15,12)( 2, 8,16,11)( 3, 6,14,10)( 4, 5,13, 9)$
8A1 $8^{2}$ $16$ $8$ $14$ $( 1, 6, 4, 8, 2, 5, 3, 7)( 9,15,12,13,10,16,11,14)$
8A-1 $8^{2}$ $16$ $8$ $14$ $( 1, 7, 3, 5, 2, 8, 4, 6)( 9,13,11,15,10,14,12,16)$
8B1 $8^{2}$ $16$ $8$ $14$ $( 1, 5, 4, 7, 2, 6, 3, 8)( 9,14,12,15,10,13,11,16)$
8B-1 $8^{2}$ $16$ $8$ $14$ $( 1, 8, 3, 6, 2, 7, 4, 5)( 9,15,11,14,10,16,12,13)$
8C $8^{2}$ $32$ $8$ $14$ $( 1,15, 4,13, 2,16, 3,14)( 5,11, 8,10, 6,12, 7, 9)$
8D1 $8,2^{3},1^{2}$ $32$ $8$ $10$ $( 1, 3)( 2, 4)( 5,10, 7,12, 6, 9, 8,11)(15,16)$
8D-1 $8,2^{3},1^{2}$ $32$ $8$ $10$ $( 1, 3)( 2, 4)( 5,11, 8, 9, 6,12, 7,10)(15,16)$
8E1 $8^{2}$ $32$ $8$ $14$ $( 1, 9,14, 6, 2,10,13, 5)( 3,12,16, 8, 4,11,15, 7)$
8E-1 $8^{2}$ $32$ $8$ $14$ $( 1, 5,13,10, 2, 6,14, 9)( 3, 7,15,11, 4, 8,16,12)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

35 x 35 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed