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Properties

Label 16T580
16T580 1 4 1->4 12 1->12 2 3 2->3 11 2->11 3->1 10 3->10 4->2 9 4->9 5 5->1 5->10 6 6->2 6->9 7 7->4 7->12 8 8->3 8->11 9->8 14 9->14 10->7 13 10->13 11->5 16 11->16 12->6 15 12->15 13->5 13->15 14->6 14->16 15->7 15->14 16->8 16->13
Degree $16$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_8^2:C_4$

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Copy content magma:G := TransitiveGroup(16, 580);
 

Group invariants

Abstract group:  $C_8^2:C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $256=2^{8}$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $580$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,12,15,7,4,9,14,6,2,11,16,8,3,10,13,5)$, $(1,4,2,3)(5,10,7,12,6,9,8,11)(13,15,14,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 6, $C_2^2$
$8$:  $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$
$16$:  $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3
$32$:  $C_4\wr C_2$ x 2, 32T41
$64$:  16T74, 16T90, 16T121
$128$:  32T1094

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_4\wr C_2$

Low degree siblings

16T580 x 3, 32T2851 x 2, 32T2852 x 2, 32T2853 x 4

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
2C $2^{4},1^{8}$ $4$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
4C1 $4^{2},1^{8}$ $2$ $4$ $6$ $( 5, 8, 6, 7)( 9,12,10,11)$
4C-1 $4^{2},1^{8}$ $2$ $4$ $6$ $( 5, 7, 6, 8)( 9,11,10,12)$
4D1 $4^{2},2^{4}$ $2$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$
4D-1 $4^{2},2^{4}$ $2$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,14)(15,16)$
4E $4^{4}$ $4$ $4$ $12$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$
4F $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,12,10,11)$
4G1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14)(10,13)(11,16)(12,15)$
4G-1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14)(10,13)(11,16)(12,15)$
8A1 $8,1^{8}$ $4$ $8$ $7$ $( 5,11, 8, 9, 6,12, 7,10)$
8A-1 $8,1^{8}$ $4$ $8$ $7$ $( 5, 9, 7,11, 6,10, 8,12)$
8B1 $8^{2}$ $4$ $8$ $14$ $( 1,16, 3,14, 2,15, 4,13)( 5, 9, 7,11, 6,10, 8,12)$
8B-1 $8^{2}$ $4$ $8$ $14$ $( 1,16, 3,14, 2,15, 4,13)( 5,10, 7,12, 6, 9, 8,11)$
8C1 $8,2^{4}$ $4$ $8$ $11$ $( 1, 2)( 3, 4)( 5,12, 8,10, 6,11, 7, 9)(13,14)(15,16)$
8C-1 $8,2^{4}$ $4$ $8$ $11$ $( 1, 2)( 3, 4)( 5,10, 7,12, 6, 9, 8,11)(13,14)(15,16)$
8D1 $8,4^{2}$ $4$ $8$ $13$ $( 1, 3, 2, 4)( 5,10, 7,12, 6, 9, 8,11)(13,16,14,15)$
8D-1 $8,4^{2}$ $4$ $8$ $13$ $( 1,16, 3,14, 2,15, 4,13)( 5, 7, 6, 8)( 9,11,10,12)$
8E1 $8^{2}$ $4$ $8$ $14$ $( 1,16, 3,14, 2,15, 4,13)( 5,12, 8,10, 6,11, 7, 9)$
8E-1 $8^{2}$ $4$ $8$ $14$ $( 1,14, 4,16, 2,13, 3,15)( 5, 9, 7,11, 6,10, 8,12)$
8F1 $8,4^{2}$ $4$ $8$ $13$ $( 1, 3, 2, 4)( 5,11, 8, 9, 6,12, 7,10)(13,16,14,15)$
8F-1 $8,4^{2}$ $4$ $8$ $13$ $( 1, 4, 2, 3)( 5, 9, 7,11, 6,10, 8,12)(13,15,14,16)$
8G1 $8^{2}$ $4$ $8$ $14$ $( 1,15, 4,14, 2,16, 3,13)( 5,12, 7, 9, 6,11, 8,10)$
8G-1 $8^{2}$ $4$ $8$ $14$ $( 1,13, 3,16, 2,14, 4,15)( 5,10, 8,11, 6, 9, 7,12)$
8H $8^{2}$ $8$ $8$ $14$ $( 1,15, 4,14, 2,16, 3,13)( 5,10, 8,11, 6, 9, 7,12)$
8I1 $8,2^{2},1^{4}$ $8$ $8$ $9$ $( 1, 2)( 3, 4)( 5,12, 7, 9, 6,11, 8,10)$
8I-1 $8,2^{2},1^{4}$ $8$ $8$ $9$ $( 1, 2)( 3, 4)( 5,10, 8,11, 6, 9, 7,12)$
8J1 $8,4^{2}$ $8$ $8$ $13$ $( 1, 4, 2, 3)( 5, 9, 8,12, 6,10, 7,11)(13,16,14,15)$
8J-1 $8,4^{2}$ $8$ $8$ $13$ $( 1, 4, 2, 3)( 5,12, 7, 9, 6,11, 8,10)(13,16,14,15)$
8K1 $8^{2}$ $16$ $8$ $14$ $( 1, 7, 4, 6, 2, 8, 3, 5)( 9,14,12,15,10,13,11,16)$
8K-1 $8^{2}$ $16$ $8$ $14$ $( 1, 8, 4, 5, 2, 7, 3, 6)( 9,14,12,15,10,13,11,16)$
16A1 $16$ $16$ $16$ $15$ $( 1,12,15, 7, 4, 9,14, 6, 2,11,16, 8, 3,10,13, 5)$
16A-1 $16$ $16$ $16$ $15$ $( 1, 9,14, 7, 3,12,15, 5, 2,10,13, 8, 4,11,16, 6)$
16B1 $16$ $16$ $16$ $15$ $( 1,10,13, 8, 3,11,16, 6, 2, 9,14, 7, 4,12,15, 5)$
16B-1 $16$ $16$ $16$ $15$ $( 1,11,16, 8, 4,10,13, 5, 2,12,15, 7, 3, 9,14, 6)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

40 x 40 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed