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Properties

Label 16T1800
16T1800 1 15 1->15 16 1->16 2 2->15 2->16 3 12 3->12 4 11 4->11 5 7 5->7 10 5->10 6 8 6->8 9 6->9 7->10 14 7->14 8->9 13 8->13 9->6 9->7 10->5 10->8 11->1 12->2 13->1 13->14 14->2 15->5 15->11 16->6 16->12
Degree $16$
Order $21504$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^7:F_8:C_3$

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Copy content magma:G := TransitiveGroup(16, 1800);
 

Group invariants

Abstract group:  $C_2^7:F_8:C_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $21504=2^{10} \cdot 3 \cdot 7$
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 $t$:  $1800$
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:  $(1,15,5,10,8,13)(2,16,6,9,7,14)(3,12)(4,11)$, $(1,16,12,2,15,11)(5,7,10)(6,8,9)(13,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$21$:  $C_7:C_3$
$168$:  $C_2^3:(C_7: C_3)$ x 3
$1344$:  24T2951
$10752$:  56T?

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

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

Low degree siblings

There are no siblings 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 $2^{4},1^{8}$ $14$ $2$ $4$ $( 3, 4)( 5, 6)(11,12)(13,14)$
2C $2^{2},1^{12}$ $28$ $2$ $2$ $(11,12)(15,16)$
2D $2^{6},1^{4}$ $28$ $2$ $6$ $( 1, 2)( 3, 4)( 7, 8)(11,12)(13,14)(15,16)$
2E $2^{4},1^{8}$ $56$ $2$ $4$ $( 7, 8)(11,12)(13,14)(15,16)$
2F $2^{8}$ $56$ $2$ $8$ $( 1, 7)( 2, 8)( 3,13)( 4,14)( 5,12)( 6,11)( 9,16)(10,15)$
2G $2^{8}$ $56$ $2$ $8$ $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,14)(10,13)$
3A1 $3^{4},1^{4}$ $448$ $3$ $8$ $( 3,13,15)( 4,14,16)( 5, 7,12)( 6, 8,11)$
3A-1 $3^{4},1^{4}$ $448$ $3$ $8$ $( 3,15,13)( 4,16,14)( 5,12, 7)( 6,11, 8)$
4A $4^{4}$ $56$ $4$ $12$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,14, 6,13)( 7,16, 8,15)$
4B $4^{4}$ $56$ $4$ $12$ $( 1,12, 2,11)( 3, 9, 4,10)( 5, 8, 6, 7)(13,15,14,16)$
4C $4^{2},2^{4}$ $336$ $4$ $10$ $( 1, 4, 2, 3)( 5,16)( 6,15)( 7,14, 8,13)( 9,12)(10,11)$
4D $4^{2},2^{4}$ $336$ $4$ $10$ $( 1,14)( 2,13)( 3, 7, 4, 8)( 5,10)( 6, 9)(11,16,12,15)$
6A1 $6^{2},2^{2}$ $448$ $6$ $12$ $( 1, 2)( 3,16,13, 4,15,14)( 5,11, 7, 6,12, 8)( 9,10)$
6A-1 $6^{2},2^{2}$ $448$ $6$ $12$ $( 1, 2)( 3,14,15, 4,13,16)( 5, 8,12, 6, 7,11)( 9,10)$
6B1 $3^{4},2^{2}$ $448$ $6$ $10$ $( 1, 9, 4)( 2,10, 3)( 5, 7,14)( 6, 8,13)(11,12)(15,16)$
6B-1 $3^{4},2^{2}$ $448$ $6$ $10$ $( 1, 4, 9)( 2, 3,10)( 5,14, 7)( 6,13, 8)(11,12)(15,16)$
6C1 $6^{2},1^{4}$ $448$ $6$ $10$ $( 1, 4,11, 2, 3,12)( 7,15,14, 8,16,13)$
6C-1 $6^{2},1^{4}$ $448$ $6$ $10$ $( 1,12, 3, 2,11, 4)( 7,13,16, 8,14,15)$
6D1 $6,3^{2},2,1^{2}$ $896$ $6$ $10$ $( 1, 9, 7)( 2,10, 8)( 3,12,14, 4,11,13)( 5, 6)$
6D-1 $6,3^{2},2,1^{2}$ $896$ $6$ $10$ $( 1, 7, 9)( 2, 8,10)( 3,13,11, 4,14,12)( 5, 6)$
6E1 $6,3^{2},2,1^{2}$ $896$ $6$ $10$ $( 3, 5,10)( 4, 6, 9)( 7, 8)(11,16,13,12,15,14)$
6E-1 $6,3^{2},2,1^{2}$ $896$ $6$ $10$ $( 3,10, 5)( 4, 9, 6)( 7, 8)(11,14,15,12,13,16)$
6F1 $6^{2},2^{2}$ $896$ $6$ $12$ $( 1, 4,15, 7,14,10)( 2, 3,16, 8,13, 9)( 5,12)( 6,11)$
6F-1 $6^{2},2^{2}$ $896$ $6$ $12$ $( 1,10,14, 7,15, 4)( 2, 9,13, 8,16, 3)( 5,12)( 6,11)$
6G1 $6^{2},2^{2}$ $896$ $6$ $12$ $( 1,11, 3, 5, 8,16)( 2,12, 4, 6, 7,15)( 9,14)(10,13)$
6G-1 $6^{2},2^{2}$ $896$ $6$ $12$ $( 1,16, 8, 5, 3,11)( 2,15, 7, 6, 4,12)( 9,14)(10,13)$
7A1 $7^{2},1^{2}$ $1536$ $7$ $12$ $( 1,10, 4,15, 5,11,13)( 2, 9, 3,16, 6,12,14)$
7A-1 $7^{2},1^{2}$ $1536$ $7$ $12$ $( 1,13,11, 5,15, 4,10)( 2,14,12, 6,16, 3, 9)$
12A1 $12,4$ $896$ $12$ $14$ $( 1,10, 2, 9)( 3, 6,16,12,13, 8, 4, 5,15,11,14, 7)$
12A-1 $12,4$ $896$ $12$ $14$ $( 1, 9, 2,10)( 3, 7,14,11,15, 5, 4, 8,13,12,16, 6)$
12B1 $12,4$ $896$ $12$ $14$ $( 1,11, 2,12)( 3, 7,16, 9, 5,13, 4, 8,15,10, 6,14)$
12B-1 $12,4$ $896$ $12$ $14$ $( 1,12, 2,11)( 3,14, 6,10,15, 8, 4,13, 5, 9,16, 7)$
14A1 $14,2$ $1536$ $14$ $14$ $( 1, 6,10,12, 4,14,15, 2, 5, 9,11, 3,13,16)( 7, 8)$
14A-1 $14,2$ $1536$ $14$ $14$ $( 1,16,13, 3,11, 9, 5, 2,15,14, 4,12,10, 6)( 7, 8)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

36 x 36 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed