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Properties

Label 16T900
16T900 1 8 1->8 1->8 16 1->16 2 7 2->7 2->7 15 2->15 3 5 3->5 3->5 14 3->14 4 6 4->6 4->6 13 4->13 9 5->9 11 5->11 12 5->12 10 6->10 6->11 6->12 7->9 7->10 7->12 8->9 8->10 8->11 9->8 9->13 9->14 10->7 10->13 10->14 11->5 11->15 11->16 12->6 12->15 12->16 13->2 13->3 13->4 14->1 14->3 14->4 15->1 15->2 15->4 16->1 16->2 16->3
Degree $16$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^6.D_4$

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Copy content magma:G := TransitiveGroup(16, 900);
 

Group invariants

Abstract group:  $C_2^6.D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $512=2^{9}$
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:  $5$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $900$
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:  $(1,8,10,14,4,6,12,16)(2,7,9,13,3,5,11,15)$, $(1,16,3,14)(2,15,4,13)(5,9,8,11)(6,10,7,12)$, $(1,8,9,14,3,5,12,15)(2,7,10,13,4,6,11,16)$
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, $C_4\times C_2^2$
$32$:  $Z_8 : Z_8^\times$, $C_4\wr C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$, 16T32
$64$:  $((C_8 : C_2):C_2):C_2$ x 2, 16T76, 16T111, 32T264
$128$:  16T227, 16T234, 16T254
$256$:  32T4016

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

16T834 x 4, 16T862 x 4, 16T897 x 8, 16T900 x 3, 16T921 x 4, 32T10002 x 4, 32T10003 x 4, 32T10004 x 4, 32T10005 x 2, 32T10006 x 4, 32T10007 x 4, 32T10008 x 2, 32T10009 x 2, 32T10010 x 2, 32T10011 x 2, 32T10012 x 2, 32T10013 x 4, 32T10014 x 2, 32T10015 x 2, 32T10016 x 4, 32T10017 x 8, 32T10018 x 2, 32T10019 x 2, 32T10020 x 2, 32T10021 x 2, 32T10022 x 4, 32T10023 x 4, 32T10024 x 4, 32T10262 x 2, 32T10263 x 2, 32T10264 x 2, 32T10265 x 2, 32T10266 x 2, 32T10267 x 2, 32T10448 x 4, 32T10449 x 4, 32T10450 x 4, 32T10451 x 4, 32T10452 x 4, 32T10453 x 4, 32T10462 x 4, 32T10463 x 4, 32T10464 x 4, 32T10584 x 4, 32T10585 x 4, 32T10586 x 4, 32T22167 x 2, 32T22180 x 2

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)(13,14)(15,16)$
2C $2^{8}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$
2D $2^{8}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
2E $2^{4},1^{8}$ $4$ $2$ $4$ $( 5, 7)( 6, 8)(13,15)(14,16)$
2F $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)$
2G $2^{2},1^{12}$ $4$ $2$ $2$ $(13,14)(15,16)$
2H $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$
2I $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)(13,16)(14,15)$
2J $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 8)( 6, 7)(13,15)(14,16)$
2K $2^{4},1^{8}$ $4$ $2$ $4$ $( 5, 7)( 6, 8)(13,16)(14,15)$
2L $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,16)(14,15)$
2M $2^{8}$ $4$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)$
2N $2^{4},1^{8}$ $4$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
2O $2^{8}$ $8$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)$
2P $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 3)( 2, 4)( 9,12)(10,11)(13,14)(15,16)$
2Q $2^{8}$ $16$ $2$ $8$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,10)( 6, 9)( 7,11)( 8,12)$
4A1 $4^{4}$ $8$ $4$ $12$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,13, 7,15)( 6,14, 8,16)$
4A-1 $4^{4}$ $8$ $4$ $12$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,15, 7,13)( 6,16, 8,14)$
4B1 $4^{4}$ $8$ $4$ $12$ $( 1,12, 4,10)( 2,11, 3, 9)( 5,14, 7,16)( 6,13, 8,15)$
4B-1 $4^{4}$ $8$ $4$ $12$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,16, 7,14)( 6,15, 8,13)$
4C $4^{4}$ $16$ $4$ $12$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,15, 7,13)( 6,16, 8,14)$
4D $4^{4}$ $16$ $4$ $12$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,16, 8,13)( 6,15, 7,14)$
4E $4^{4}$ $16$ $4$ $12$ $( 1,14, 3,16)( 2,13, 4,15)( 5,11, 7,10)( 6,12, 8, 9)$
4F $4^{4}$ $16$ $4$ $12$ $( 1,14, 4,15)( 2,13, 3,16)( 5,12, 8,10)( 6,11, 7, 9)$
4G $4^{4}$ $16$ $4$ $12$ $( 1,14, 2,13)( 3,16, 4,15)( 5, 9, 6,10)( 7,12, 8,11)$
4H1 $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1, 2)( 5,14, 7,15)( 6,13, 8,16)(11,12)$
4H-1 $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1, 2)( 5,15, 7,14)( 6,16, 8,13)(11,12)$
4I1 $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1, 2)( 5,15, 8,13)( 6,16, 7,14)(11,12)$
4I-1 $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1, 2)( 5,14, 8,16)( 6,13, 7,15)(11,12)$
4J1 $4^{4}$ $16$ $4$ $12$ $( 1, 3, 2, 4)( 5,14, 7,15)( 6,13, 8,16)( 9,11,10,12)$
4J-1 $4^{4}$ $16$ $4$ $12$ $( 1, 3, 2, 4)( 5,15, 7,14)( 6,16, 8,13)( 9,11,10,12)$
4K1 $4^{4}$ $16$ $4$ $12$ $( 1, 3, 2, 4)( 5,15, 8,13)( 6,16, 7,14)( 9,11,10,12)$
4K-1 $4^{4}$ $16$ $4$ $12$ $( 1, 3, 2, 4)( 5,14, 8,16)( 6,13, 7,15)( 9,11,10,12)$
4L $4^{4}$ $32$ $4$ $12$ $( 1,14, 4,15)( 2,13, 3,16)( 5,11, 7,10)( 6,12, 8, 9)$
4M $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,10, 6, 9)( 7,11, 8,12)$
8A1 $8^{2}$ $32$ $8$ $14$ $( 1,16,12, 5, 4,14,10, 7)( 2,15,11, 6, 3,13, 9, 8)$
8A-1 $8^{2}$ $32$ $8$ $14$ $( 1, 7,10,14, 4, 5,12,16)( 2, 8, 9,13, 3, 6,11,15)$
8B1 $8^{2}$ $32$ $8$ $14$ $( 1,14,10, 5, 4,16,12, 7)( 2,13, 9, 6, 3,15,11, 8)$
8B-1 $8^{2}$ $32$ $8$ $14$ $( 1, 7,12,16, 4, 5,10,14)( 2, 8,11,15, 3, 6, 9,13)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

41 x 41 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed