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Properties

Label 16T731
16T731 1 5 1->5 6 1->6 9 1->9 2 2->5 7 2->7 12 2->12 3 3->6 3->7 10 3->10 4 8 4->8 4->8 11 4->11 5->2 5->4 16 5->16 6->3 6->3 14 6->14 7->1 7->4 13 7->13 8->1 8->2 15 8->15 9->8 9->15 10->6 10->12 10->16 11->7 11->10 11->14 12->5 12->11 12->13 13->3 13->10 13->15 14->2 14->9 14->13 15->1 15->11 15->14 16->4 16->12
Degree $16$
Order $384$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^5:A_4$

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Copy content magma:G := TransitiveGroup(16, 731);
 

Group invariants

Abstract group:  $C_2^5:A_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $384=2^{7} \cdot 3$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $731$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,9,8)(2,12,5)(3,10,6)(4,11,7)(13,15,14)$, $(1,5,16,4,8,15)(2,7,13,3,6,14)(10,12,11)$, $(1,6,3,7)(2,5,4,8)(9,15,11,14)(10,16,12,13)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$ x 5
$24$:  $A_4\times C_2$ x 5
$48$:  $C_2^4:C_3$
$96$:  12T56

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 8: None

Low degree siblings

16T731 x 3, 24T730 x 2, 24T983 x 2, 24T997 x 2, 24T1024 x 4, 24T1025 x 2, 32T9324 x 2, 32T9338 x 4

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 $2^{8}$ $3$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$
2B $2^{6},1^{4}$ $4$ $2$ $6$ $( 5, 6)( 7, 8)( 9,12)(10,11)(13,16)(14,15)$
2C $2^{4},1^{8}$ $6$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2D $2^{8}$ $6$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
2E $2^{6},1^{4}$ $12$ $2$ $6$ $( 5, 6)( 7, 8)( 9,11)(10,12)(13,15)(14,16)$
2F $2^{8}$ $12$ $2$ $8$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5, 9)( 6,10)( 7,12)( 8,11)$
2G $2^{8}$ $12$ $2$ $8$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,11)( 6,12)( 7,10)( 8, 9)$
3A1 $3^{5},1$ $64$ $3$ $10$ $( 1, 2, 3)( 5,12,13)( 6, 9,16)( 7,10,14)( 8,11,15)$
3A-1 $3^{5},1$ $64$ $3$ $10$ $( 1, 3, 2)( 5,13,12)( 6,16, 9)( 7,14,10)( 8,15,11)$
4A $4^{4}$ $12$ $4$ $12$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,14,12,16)(10,13,11,15)$
4B $4^{4}$ $12$ $4$ $12$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,15,11,14)(10,16,12,13)$
4C $4^{4}$ $12$ $4$ $12$ $( 1,13, 3,16)( 2,14, 4,15)( 5,12, 8,10)( 6,11, 7, 9)$
4D $4^{4}$ $12$ $4$ $12$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,16,11,13)(10,15,12,14)$
4E $4^{4}$ $12$ $4$ $12$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,13,12,15)(10,14,11,16)$
4F $4^{4}$ $12$ $4$ $12$ $( 1,13, 3,16)( 2,14, 4,15)( 5,10, 8,12)( 6, 9, 7,11)$
6A1 $6^{2},3,1$ $64$ $6$ $12$ $( 1, 3, 2)( 5,16,12, 6,13, 9)( 7,15,10, 8,14,11)$
6A-1 $6^{2},3,1$ $64$ $6$ $12$ $( 1, 2, 3)( 5, 9,13, 6,12,16)( 7,11,14, 8,10,15)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A1 3A-1 4A 4B 4C 4D 4E 4F 6A1 6A-1
Size 1 3 4 6 6 12 12 12 64 64 12 12 12 12 12 12 64 64
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 2A 2A 2A 2A 2A 2A 3A1 3A-1
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 4A 4B 4C 4D 4E 4F 2B 2B
Type
384.18235.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
384.18235.1b R 1 1 −1 −1 −1 1 1 −1 1 1 1 −1 1 1 −1 −1 −1 −1
384.18235.1c1 C 1 1 1 1 1 1 1 1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1
384.18235.1c2 C 1 1 1 1 1 1 1 1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3
384.18235.1d1 C 1 1 −1 −1 −1 1 1 −1 ζ3−1 ζ3 1 −1 1 1 −1 −1 −ζ3 −ζ3−1
384.18235.1d2 C 1 1 −1 −1 −1 1 1 −1 ζ3 ζ3−1 1 −1 1 1 −1 −1 −ζ3−1 −ζ3
384.18235.3a R 3 3 3 −1 −1 −1 −1 −1 0 0 −1 −1 3 −1 3 −1 0 0
384.18235.3b R 3 3 3 −1 −1 −1 −1 −1 0 0 3 3 −1 −1 −1 −1 0 0
384.18235.3c R 3 3 3 −1 −1 −1 −1 3 0 0 −1 −1 −1 3 −1 −1 0 0
384.18235.3d R 3 3 3 −1 −1 −1 3 −1 0 0 −1 −1 −1 −1 −1 3 0 0
384.18235.3e R 3 3 3 3 3 3 −1 −1 0 0 −1 −1 −1 −1 −1 −1 0 0
384.18235.3f R 3 3 −3 −3 −3 3 −1 1 0 0 −1 1 −1 −1 1 1 0 0
384.18235.3g R 3 3 −3 1 1 −1 −1 −3 0 0 −1 1 −1 3 1 1 0 0
384.18235.3h R 3 3 −3 1 1 −1 −1 1 0 0 −1 1 3 −1 −3 1 0 0
384.18235.3i R 3 3 −3 1 1 −1 −1 1 0 0 3 −3 −1 −1 1 1 0 0
384.18235.3j R 3 3 −3 1 1 −1 3 1 0 0 −1 1 −1 −1 1 −3 0 0
384.18235.12a R 12 −4 0 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0
384.18235.12b R 12 −4 0 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed