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Properties

Label 16T82
16T82 1 2 1->2 6 1->6 10 1->10 16 1->16 5 2->5 9 2->9 15 2->15 3 11 3->11 3->16 4 12 4->12 4->15 5->1 14 5->14 6->2 13 6->13 7 7->9 7->10 7->12 7->15 8 8->9 8->10 8->11 8->16 9->1 9->14 10->2 10->13 11->4 11->7 11->14 12->3 12->8 12->13 13->5 13->9 14->6 14->10 15->3 15->8 15->16 16->4 16->7
Degree $16$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2:C_2^2$

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Copy content magma:G := TransitiveGroup(16, 82);
 

Group invariants

Abstract group:  $C_4^2:C_2^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $64=2^{6}$
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:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $82$
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:  $(1,2)(7,9)(8,10)(11,14)(12,13)(15,16)$, $(1,10,2,9)(3,11,4,12)(5,14,6,13)(7,15,8,16)$, $(1,6,2,5)(3,16,4,15)(7,12,8,11)(9,14,10,13)$, $(1,16)(2,15)(7,10)(8,9)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15
$16$:  $Q_8:C_2$ x 2, $C_2^4$
$32$:  $Q_8:C_2^2$ x 2, $C_2 \times (C_4\times C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8:C_2$, $Q_8:C_2^2$ x 2

Low degree siblings

16T82 x 3, 32T77 x 2, 32T78 x 2, 32T234 x 2

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

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A1 4A-1 4B1 4B-1 4C 4D 4E 4F 4G 4H 4I 4J
Size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2B 2C 2A 2C 2A 2B 2C 2C
Type
64.219.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.219.1b R 1 1 1 1 −1 −1 −1 −1 1 −1 1 −1 1 −1 1 1 1 −1 −1 1 −1 1
64.219.1c R 1 1 1 1 −1 −1 −1 −1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 1
64.219.1d R 1 1 1 1 −1 −1 −1 1 1 −1 −1 1 −1 1 −1 −1 1 −1 1 1 1 −1
64.219.1e R 1 1 1 1 −1 −1 −1 1 1 1 1 −1 1 −1 −1 1 −1 1 1 −1 −1 −1
64.219.1f R 1 1 1 1 −1 −1 1 −1 −1 −1 −1 1 −1 1 −1 1 −1 1 1 1 −1 1
64.219.1g R 1 1 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 −1 −1 1 −1 1 −1 1 1
64.219.1h R 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 1 −1
64.219.1i R 1 1 1 1 −1 −1 1 1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 −1
64.219.1j R 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 −1 1 −1
64.219.1k R 1 1 1 1 1 1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 1 1 −1 −1
64.219.1l R 1 1 1 1 1 1 −1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 1
64.219.1m R 1 1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 −1 1 −1 −1 −1 1 1 1
64.219.1n R 1 1 1 1 1 1 1 −1 1 −1 1 1 1 1 −1 1 −1 −1 −1 −1 1 −1
64.219.1o R 1 1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1 −1 −1 1 1 −1 1 −1 −1
64.219.1p R 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 −1 −1 1 −1 −1 1
64.219.2a1 C 2 −2 2 −2 −2 2 0 0 0 0 −2i −2i 2i 2i 0 0 0 0 0 0 0 0
64.219.2a2 C 2 −2 2 −2 −2 2 0 0 0 0 2i 2i −2i −2i 0 0 0 0 0 0 0 0
64.219.2b1 C 2 −2 2 −2 2 −2 0 0 0 0 −2i 2i 2i −2i 0 0 0 0 0 0 0 0
64.219.2b2 C 2 −2 2 −2 2 −2 0 0 0 0 2i −2i −2i 2i 0 0 0 0 0 0 0 0
64.219.4a R 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
64.219.4b R 4 4 −4 −4 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