Properties

Label 16T333
16T333 1 5 1->5 12 1->12 13 1->13 2 6 2->6 11 2->11 14 2->14 3 3->1 7 3->7 8 3->8 4 4->2 4->7 4->8 5->1 5->8 9 5->9 6->2 6->7 10 6->10 7->3 7->12 7->14 8->4 8->11 8->13 9->3 9->5 9->13 10->4 10->6 10->14 11->10 15 11->15 16 11->16 12->9 12->15 12->16 13->1 13->9 13->16 14->2 14->10 14->15 15->3 15->6 15->12 16->4 16->5 16->11
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^4:Q_8$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(16, 333);
 
Copy content sage:G = TransitiveGroup(16, 333)
 
Copy content oscar:G = transitive_group(16, 333)
 
Copy content gap:G := TransitiveGroup(16, 333);
 

Group invariants

Abstract group:  $C_2^4:Q_8$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $128=2^{7}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:  $3$
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $16$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $333$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
Parity:  $1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(16).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(16), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(16), G));
 
Generators:  $(1,13,9,5)(2,14,10,6)(3,7,12,15)(4,8,11,16)$, $(1,12,9,3)(2,11,10,4)(5,8,13,16)(6,7,14,15)$, $(1,5,9,13)(2,6,10,14)(3,8,4,7)(11,15,12,16)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Low degree siblings

16T333 x 3, 16T347 x 12, 32T758 x 12, 32T759 x 6, 32T760, 32T794 x 12, 32T795 x 3, 32T1551 x 3

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 $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,15)( 8,16)$
2D $2^{4},1^{8}$ $2$ $2$ $4$ $( 1,10)( 2, 9)( 3,11)( 4,12)$
2E $2^{4},1^{8}$ $2$ $2$ $4$ $( 1,10)( 2, 9)( 5,14)( 6,13)$
2F $2^{4},1^{8}$ $2$ $2$ $4$ $( 3,11)( 4,12)( 5,14)( 6,13)$
2G $2^{8}$ $2$ $2$ $8$ $( 1, 9)( 2,10)( 3, 4)( 5, 6)( 7,15)( 8,16)(11,12)(13,14)$
2H $2^{8}$ $2$ $2$ $8$ $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,10)(11,12)$
2I $2^{8}$ $2$ $2$ $8$ $( 1, 2)( 3,12)( 4,11)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$
2J $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3,12)( 4,11)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$
2K $2^{6},1^{4}$ $4$ $2$ $6$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 7,16)( 8,15)$
2L $2^{8}$ $4$ $2$ $8$ $( 1, 9)( 2,10)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)(11,12)$
2M $2^{2},1^{12}$ $4$ $2$ $2$ $( 5,14)( 6,13)$
4A $4^{4}$ $8$ $4$ $12$ $( 1,14, 2,13)( 3, 8, 4, 7)( 5, 9, 6,10)(11,15,12,16)$
4B $4^{4}$ $8$ $4$ $12$ $( 1,16, 9, 8)( 2,15,10, 7)( 3,14,12, 6)( 4,13,11, 5)$
4C $4^{4}$ $8$ $4$ $12$ $( 1,11, 9, 4)( 2,12,10, 3)( 5,16,13, 8)( 6,15,14, 7)$
4D $4^{4}$ $8$ $4$ $12$ $( 1,13, 9, 5)( 2,14,10, 6)( 3,16,12, 8)( 4,15,11, 7)$
4E $4^{4}$ $8$ $4$ $12$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,15,10,16)(11,14,12,13)$
4F $4^{4}$ $8$ $4$ $12$ $( 1,12, 2,11)( 3,10, 4, 9)( 5, 8, 6, 7)(13,16,14,15)$
4G1 $4^{4}$ $8$ $4$ $12$ $( 1,15, 9, 7)( 2,16,10, 8)( 3, 6, 4, 5)(11,13,12,14)$
4G-1 $4^{4}$ $8$ $4$ $12$ $( 1, 7, 9,15)( 2, 8,10,16)( 3, 5, 4, 6)(11,14,12,13)$
4H1 $4^{4}$ $8$ $4$ $12$ $( 1,12, 2,11)( 3,10, 4, 9)( 5,15,13, 7)( 6,16,14, 8)$
4H-1 $4^{4}$ $8$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7,13,15)( 6, 8,14,16)$
4I1 $4^{4}$ $8$ $4$ $12$ $( 1,13, 2,14)( 3,16,12, 8)( 4,15,11, 7)( 5,10, 6, 9)$
4I-1 $4^{4}$ $8$ $4$ $12$ $( 1, 6, 9,14)( 2, 5,10,13)( 3, 7, 4, 8)(11,16,12,15)$

Malle's constant $a(G)$:     $1/2$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G1 4G-1 4H1 4H-1 4I1 4I-1
Size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2B 2C 2C 2C 2B 2B 2G 2G 2H 2H 2I 2I
Type
128.761.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.761.1b R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 −1 −1 1 1 −1 1 1 1 −1
128.761.1c R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1 1 −1 −1 1 −1 −1 1 1
128.761.1d R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 1 1 −1 −1 1 1 1 −1 −1
128.761.1e R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 1 1 −1 1 1 −1 −1 −1 −1 1
128.761.1f R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
128.761.1g R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 1 −1 −1 −1 −1 −1 1 1 −1 1
128.761.1h R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 −1 1 −1
128.761.2a R 2 2 2 2 −2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0 0 −2 2 0 0 0 0 0
128.761.2b R 2 2 2 2 −2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0 0 2 −2 0 0 0 0 0
128.761.2c R 2 2 2 2 −2 2 −2 2 −2 −2 0 0 0 0 −2 0 0 2 0 0 0 0 0 0 0 0
128.761.2d R 2 2 2 2 −2 2 −2 2 −2 −2 0 0 0 0 2 0 0 −2 0 0 0 0 0 0 0 0
128.761.2e R 2 2 2 2 2 −2 −2 −2 2 −2 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 2
128.761.2f R 2 2 2 2 2 −2 −2 −2 2 −2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 −2
128.761.2g S 2 −2 −2 2 2 −2 2 2 −2 −2 −2 2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.761.2h S 2 −2 −2 2 2 −2 2 2 −2 −2 2 −2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
128.761.2i1 C 2 −2 −2 2 −2 −2 −2 2 2 2 0 0 0 0 0 −2i 0 0 0 0 0 0 0 0 2i 0
128.761.2i2 C 2 −2 −2 2 −2 −2 −2 2 2 2 0 0 0 0 0 2i 0 0 0 0 0 0 0 0 −2i 0
128.761.2j1 C 2 −2 −2 2 −2 2 2 −2 2 −2 0 0 0 0 0 0 0 0 −2i 0 0 2i 0 0 0 0
128.761.2j2 C 2 −2 −2 2 −2 2 2 −2 2 −2 0 0 0 0 0 0 0 0 2i 0 0 −2i 0 0 0 0
128.761.2k1 C 2 −2 −2 2 2 2 −2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0 −2i 2i 0 0
128.761.2k2 C 2 −2 −2 2 2 2 −2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0 2i −2i 0 0
128.761.4a R 4 −4 4 −4 0 0 0 0 0 0 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.761.4b R 4 −4 4 −4 0 0 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
128.761.4c R 4 4 −4 −4 0 0 0 0 0 0 −2 2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
128.761.4d R 4 4 −4 −4 0 0 0 0 0 0 2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

Data not computed