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Properties

Label 16T99
16T99 1 2 1->2 3 1->3 9 1->9 4 2->4 10 2->10 3->4 5 3->5 11 3->11 6 4->6 12 4->12 5->6 5->6 7 5->7 8 6->8 7->8 7->8 7->10 8->9 9->10 9->11 10->12 11->12 14 11->14 13 12->13 13->14 13->14 15 13->15 16 14->16 15->1 15->16 15->16 16->2
Degree $16$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $\OD_{16}:C_2^2$

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Copy content magma:G := TransitiveGroup(16, 99);
 

Group invariants

Abstract group:  $\OD_{16}:C_2^2$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $99$
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:  $(1,3,5,7,10,12,13,15)(2,4,6,8,9,11,14,16)$, $(1,9)(2,10)(3,11)(4,12)(5,6)(7,8)(13,14)(15,16)$, $(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ 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, $C_4\times C_2^2$
$32$:  $(C_8:C_2):C_2$ x 2, $C_2 \times (C_2^2:C_4)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $(C_8:C_2):C_2$ x 2

Low degree siblings

16T72 x 4, 16T99 x 3, 32T60 x 4, 32T61, 32T100

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

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 8A1 8A-1 8B1 8B-1 8C1 8C-1 8D1 8D-1
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 2C 2C 2C 2C 4C 4C 4D 4D 4C 4C 4D 4D
Type
64.92.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.92.1b R 1 1 −1 −1 1 −1 −1 −1 1 1 −1 1 1 −1 −1 −1 1 −1 1 1 1 −1
64.92.1c R 1 1 −1 −1 1 −1 −1 −1 1 1 −1 1 1 −1 1 1 −1 1 −1 −1 −1 1
64.92.1d R 1 1 −1 −1 1 −1 1 1 −1 −1 −1 1 1 −1 −1 −1 1 1 1 −1 −1 1
64.92.1e R 1 1 −1 −1 1 −1 1 1 −1 −1 −1 1 1 −1 1 1 −1 −1 −1 1 1 −1
64.92.1f R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 1 −1 1 1 1
64.92.1g R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 −1 1 −1 −1 −1
64.92.1h R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
64.92.1i1 C 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 −1 1 −i i i i −i i −i −i
64.92.1i2 C 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 −1 1 i −i −i −i i −i i i
64.92.1j1 C 1 1 −1 −1 1 −1 1 −1 −1 1 1 −1 −1 1 −i i i −i −i −i i i
64.92.1j2 C 1 1 −1 −1 1 −1 1 −1 −1 1 1 −1 −1 1 i −i −i i i i −i −i
64.92.1k1 C 1 1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 −i i −i −i i i −i i
64.92.1k2 C 1 1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 i −i i i −i −i i −i
64.92.1l1 C 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 −i i −i i i −i i −i
64.92.1l2 C 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 i −i i −i −i i −i i
64.92.2a R 2 2 −2 −2 −2 2 0 0 0 0 −2 2 −2 2 0 0 0 0 0 0 0 0
64.92.2b R 2 2 −2 −2 −2 2 0 0 0 0 2 −2 2 −2 0 0 0 0 0 0 0 0
64.92.2c R 2 2 2 2 −2 −2 0 0 0 0 −2 −2 2 2 0 0 0 0 0 0 0 0
64.92.2d R 2 2 2 2 −2 −2 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0 0 0
64.92.4a R 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
64.92.4b 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