↑

Properties

Label 16T712
16T712 1 5 1->5 2 6 2->6 3 10 3->10 11 3->11 4 9 4->9 12 4->12 8 5->8 7 6->7 15 7->15 16 7->16 8->15 8->16 9->3 9->7 10->4 10->8 11->4 11->9 12->3 12->10 13 13->12 14 14->11 15->6 15->13 16->5 16->14
Degree $16$
Order $336$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_8:C_6$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 712);
 

Group invariants

Abstract group:  $F_8:C_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $336=2^{4} \cdot 3 \cdot 7$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $712$
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:  $(3,11,9)(4,12,10)(5,8,16)(6,7,15)$, $(1,5)(2,6)(3,10,8,15,13,12)(4,9,7,16,14,11)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$21$:  $C_7:C_3$
$42$:  $(C_7:C_3) \times C_2$
$168$:  $C_2^3:(C_7: C_3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 8: $C_2^3:(C_7: C_3)$

Low degree siblings

14T18, 28T44, 42T67

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 $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B $2^{8}$ $7$ $2$ $8$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,14)( 8,13)( 9,11)(10,12)$
2C $2^{8}$ $7$ $2$ $8$ $( 1,16)( 2,15)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,14)(12,13)$
3A1 $3^{4},1^{4}$ $28$ $3$ $8$ $( 1, 7,10)( 2, 8, 9)( 3,13,11)( 4,14,12)$
3A-1 $3^{4},1^{4}$ $28$ $3$ $8$ $( 1,10, 7)( 2, 9, 8)( 3,11,13)( 4,12,14)$
6A1 $6^{2},2^{2}$ $28$ $6$ $12$ $( 1,12, 7, 4,10,14)( 2,11, 8, 3, 9,13)( 5,16)( 6,15)$
6A-1 $6^{2},2^{2}$ $28$ $6$ $12$ $( 1,14,10, 4, 7,12)( 2,13, 9, 3, 8,11)( 5,16)( 6,15)$
6B1 $6^{2},2^{2}$ $28$ $6$ $12$ $( 1, 5,15, 2, 6,16)( 3, 4)( 7,13,12, 8,14,11)( 9,10)$
6B-1 $6^{2},2^{2}$ $28$ $6$ $12$ $( 1,16, 6, 2,15, 5)( 3, 4)( 7,11,14, 8,12,13)( 9,10)$
6C1 $6^{2},2^{2}$ $28$ $6$ $12$ $( 1, 3,14,16, 6,11)( 2, 4,13,15, 5,12)( 7, 9)( 8,10)$
6C-1 $6^{2},2^{2}$ $28$ $6$ $12$ $( 1,11, 6,16,14, 3)( 2,12, 5,15,13, 4)( 7, 9)( 8,10)$
7A1 $7^{2},1^{2}$ $24$ $7$ $12$ $( 3,13, 5,16,11, 8, 9)( 4,14, 6,15,12, 7,10)$
7A-1 $7^{2},1^{2}$ $24$ $7$ $12$ $( 3, 9, 8,11,16, 5,13)( 4,10, 7,12,15, 6,14)$
14A1 $14,2$ $24$ $14$ $14$ $( 1, 2)( 3,12,13, 7, 5,10,16, 4,11,14, 8, 6, 9,15)$
14A-1 $14,2$ $24$ $14$ $14$ $( 1, 2)( 3,15, 9, 6, 8,14,11, 4,16,10, 5, 7,13,12)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A1 7A-1 14A1 14A-1
Size 1 1 7 7 28 28 28 28 28 28 28 28 24 24 24 24
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1 7A1 7A-1 7A1 7A-1
3 P 1A 2A 2B 2C 1A 1A 2B 2B 2A 2A 2C 2C 7A-1 7A1 14A-1 14A1
7 P 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 1A 1A 2A 2A
Type
336.210.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
336.210.1b R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
336.210.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1
336.210.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1
336.210.1d1 C 1 −1 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 1 1 −1 −1
336.210.1d2 C 1 −1 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 1 1 −1 −1
336.210.3a1 C 3 3 3 3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
336.210.3a2 C 3 3 3 3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
336.210.3b1 C 3 −3 3 −3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
336.210.3b2 C 3 −3 3 −3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
336.210.7a R 7 7 −1 −1 1 1 −1 −1 1 1 −1 −1 0 0 0 0
336.210.7b R 7 −7 −1 1 1 1 −1 −1 −1 −1 1 1 0 0 0 0
336.210.7c1 C 7 7 −1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 0 0 0 0
336.210.7c2 C 7 7 −1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 0 0 0 0
336.210.7d1 C 7 −7 −1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 0 0 0 0
336.210.7d2 C 7 −7 −1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed