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Properties

Label 16T362
16T362 1 6 1->6 10 1->10 2 5 2->5 9 2->9 3 16 3->16 4 15 4->15 13 5->13 14 6->14 7 8 7->8 12 7->12 11 8->11 9->7 9->10 10->8 11->1 11->16 12->2 12->15 13->3 14->4 15->6 16->5
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2.(C_2\times C_4)$

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Copy content magma:G := TransitiveGroup(16, 362);
 

Group invariants

Abstract group:  $C_4^2.(C_2\times C_4)$
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:  $5$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $362$
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:  $(1,10,8,11)(2,9,7,12)(3,16,5,13)(4,15,6,14)$, $(1,6)(2,5)(7,8)(9,10)(11,16)(12,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_2^3: C_4$

Low degree siblings

16T362, 32T826 x 2, 32T827, 32T1679, 32T1947

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

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 4A 4B 4C 4D1 4D-1 8A 8B 8C1 8C-1
Size 1 1 2 8 16 4 8 8 16 16 8 8 16 16
2 P 1A 1A 1A 1A 1A 2A 2B 2A 2C 2C 4A 4A 4C 4C
Type
128.138.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.138.1b R 1 1 1 1 −1 1 1 1 −1 −1 −1 −1 1 1
128.138.1c R 1 1 1 1 −1 1 1 1 1 1 −1 −1 −1 −1
128.138.1d R 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1
128.138.1e1 C 1 1 1 −1 −1 1 1 −1 −i i 1 1 i −i
128.138.1e2 C 1 1 1 −1 −1 1 1 −1 i −i 1 1 −i i
128.138.1f1 C 1 1 1 −1 1 1 1 −1 −i i −1 −1 −i i
128.138.1f2 C 1 1 1 −1 1 1 1 −1 i −i −1 −1 i −i
128.138.2a R 2 2 2 −2 0 2 −2 2 0 0 0 0 0 0
128.138.2b R 2 2 2 2 0 2 −2 −2 0 0 0 0 0 0
128.138.4a R 4 4 4 0 0 −4 0 0 0 0 0 0 0 0
128.138.4b R 4 4 −4 0 0 0 0 0 0 0 −2 2 0 0
128.138.4c R 4 4 −4 0 0 0 0 0 0 0 2 −2 0 0
128.138.8a R 8 −8 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed