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Properties

Label 16T21
16T21 1 5 1->5 11 1->11 13 1->13 2 6 2->6 12 2->12 14 2->14 3 7 3->7 9 3->9 16 3->16 4 8 4->8 10 4->10 15 4->15 5->7 5->9 6->8 6->10 7->3 7->4 7->11 8->3 8->4 8->12 9->5 9->6 9->13 10->5 10->6 10->14 11->15 11->16 12->15 12->16 13->1 13->2 14->1 14->2 15->12 15->14 16->11 16->13
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2 \times (C_2^2:C_4)$

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Copy content magma:G := TransitiveGroup(16, 21);
 

Group invariants

Abstract group:  $C_2 \times (C_2^2:C_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $32=2^{5}$
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:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,11,16,13)(2,12,15,14)(3,9,5,7)(4,10,6,8)$, $(1,13,2,14)(3,7,4,8)(5,9,6,10)(11,15,12,16)$, $(1,5)(2,6)(3,16)(4,15)(7,11)(8,12)(9,13)(10,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4

Degree 8: $C_4\times C_2$, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4

Low degree siblings

16T21 x 3, 32T7

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 $2^{8}$ $1$ $2$ $8$ $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,11)( 8,12)( 9,13)(10,14)$
2C $2^{8}$ $1$ $2$ $8$ $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$
2D $2^{8}$ $1$ $2$ $8$ $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,14)( 8,13)( 9,12)(10,11)$
2E $2^{8}$ $1$ $2$ $8$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,12)( 8,11)( 9,14)(10,13)$
2F $2^{8}$ $1$ $2$ $8$ $( 1,16)( 2,15)( 3, 5)( 4, 6)( 7, 9)( 8,10)(11,13)(12,14)$
2G $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2H $2^{4},1^{8}$ $2$ $2$ $4$ $( 7,10)( 8, 9)(11,14)(12,13)$
2I $2^{8}$ $2$ $2$ $8$ $( 1,16)( 2,15)( 3, 5)( 4, 6)( 7, 8)( 9,10)(11,12)(13,14)$
2J $2^{8}$ $2$ $2$ $8$ $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,14)( 8,13)( 9,12)(10,11)$
2K $2^{8}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,12)( 8,11)( 9,14)(10,13)$
4A1 $4^{4}$ $2$ $4$ $12$ $( 1, 7,16, 9)( 2, 8,15,10)( 3,13, 5,11)( 4,14, 6,12)$
4A-1 $4^{4}$ $2$ $4$ $12$ $( 1, 9,16, 7)( 2,10,15, 8)( 3,11, 5,13)( 4,12, 6,14)$
4B1 $4^{4}$ $2$ $4$ $12$ $( 1, 7, 2, 8)( 3,13, 4,14)( 5,11, 6,12)( 9,15,10,16)$
4B-1 $4^{4}$ $2$ $4$ $12$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,13, 6,14)( 7,15, 8,16)$
4C1 $4^{4}$ $2$ $4$ $12$ $( 1,11,16,13)( 2,12,15,14)( 3, 9, 5, 7)( 4,10, 6, 8)$
4C-1 $4^{4}$ $2$ $4$ $12$ $( 1,13,16,11)( 2,14,15,12)( 3, 7, 5, 9)( 4, 8, 6,10)$
4D1 $4^{4}$ $2$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,15,14,16)$
4D-1 $4^{4}$ $2$ $4$ $12$ $( 1,13, 2,14)( 3, 7, 4, 8)( 5, 9, 6,10)(11,15,12,16)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1
Size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2F 2F 2G 2G 2F 2F 2G 2G
Type
32.22.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.22.1b R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 −1 1 1 1 1 −1 −1 −1
32.22.1c R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1
32.22.1d R 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 −1 1 −1 1
32.22.1e R 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1 −1 1 −1
32.22.1f R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 −1 1 1 −1 1
32.22.1g R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 1 1 −1 −1 1 −1
32.22.1h R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
32.22.1i1 C 1 −1 1 −1 −1 1 1 −1 −1 1 −1 1 −i i −i i −i −i i i
32.22.1i2 C 1 −1 1 −1 −1 1 1 −1 −1 1 −1 1 i −i i −i i i −i −i
32.22.1j1 C 1 −1 1 −1 −1 1 1 −1 1 −1 1 −1 −i −i −i i i i i −i
32.22.1j2 C 1 −1 1 −1 −1 1 1 −1 1 −1 1 −1 i i i −i −i −i −i i
32.22.1k1 C 1 1 −1 −1 −1 1 −1 1 −1 1 1 −1 −i i i −i −i i i −i
32.22.1k2 C 1 1 −1 −1 −1 1 −1 1 −1 1 1 −1 i −i −i i i −i −i i
32.22.1l1 C 1 1 −1 −1 −1 1 −1 1 1 −1 −1 1 −i −i i −i i −i i i
32.22.1l2 C 1 1 −1 −1 −1 1 −1 1 1 −1 −1 1 i i −i i −i i −i −i
32.22.2a R 2 −2 −2 −2 2 −2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
32.22.2b R 2 −2 2 2 −2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
32.22.2c R 2 2 −2 2 −2 −2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
32.22.2d R 2 2 2 −2 2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed