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Properties

Label 16T336
16T336 1 6 1->6 8 1->8 16 1->16 2 5 2->5 7 2->7 15 2->15 3 3->5 3->8 13 3->13 4 4->6 4->7 14 4->14 5->1 5->6 6->2 7->3 8->4 9 9->13 9->15 10 10->14 10->16 11 12 11->12 11->13 11->16 12->14 12->15 13->10 13->12 14->9 14->11 15->10 15->11 16->9 16->12
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2:D_4$

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Copy content magma:G := TransitiveGroup(16, 336);
 

Group invariants

Abstract group:  $C_4^2:D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $128=2^{7}$
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:  $4$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $336$
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:  $(1,8)(2,7)(3,5)(4,6)(9,13,10,14)(11,16,12,15)$, $(1,6,2,5)(3,8,4,7)(9,15,10,16)(11,13,12,14)$, $(1,16)(2,15)(3,13)(4,14)(5,6)(11,12)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$

Low degree siblings

16T340, 16T341 x 2, 16T352, 16T359, 16T366, 16T402, 32T767, 32T768, 32T769, 32T777, 32T778, 32T779, 32T780 x 2, 32T781, 32T807, 32T808, 32T820, 32T821, 32T894, 32T1569, 32T1694

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 $2^{8}$ $4$ $2$ $8$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5, 9)( 6,10)( 7,11)( 8,12)$
2D $2^{4},1^{8}$ $4$ $2$ $4$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
2E $2^{8}$ $8$ $2$ $8$ $( 1, 4)( 2, 3)( 5,11)( 6,12)( 7, 9)( 8,10)(13,16)(14,15)$
2F $2^{6},1^{4}$ $8$ $2$ $6$ $( 1,15)( 2,16)( 3,14)( 4,13)( 7, 8)( 9,10)$
2G $2^{8}$ $8$ $2$ $8$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,16)( 6,15)( 7,14)( 8,13)$
4A $4^{4}$ $4$ $4$ $12$ $( 1,16, 2,15)( 3,14, 4,13)( 5,10, 6, 9)( 7,12, 8,11)$
4B $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 2)( 5, 9, 6,10)( 7,12, 8,11)(15,16)$
4C $4^{2},2^{4}$ $8$ $4$ $10$ $( 1,13, 2,14)( 3,15, 4,16)( 5, 7)( 6, 8)( 9,12)(10,11)$
4D $4^{4}$ $8$ $4$ $12$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
4E $4^{2},2^{4}$ $8$ $4$ $10$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,12, 6,11)( 7, 9, 8,10)$
4F $4^{4}$ $8$ $4$ $12$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,16, 6,15)( 7,14, 8,13)$
4G $4^{4}$ $16$ $4$ $12$ $( 1,11,16, 7)( 2,12,15, 8)( 3, 9,14, 5)( 4,10,13, 6)$
4H $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,12, 2,11)( 3, 9, 4,10)( 5,14)( 6,13)( 7,15)( 8,16)$
8A $8^{2}$ $16$ $8$ $14$ $( 1,10,16, 6, 2, 9,15, 5)( 3,11,14, 7, 4,12,13, 8)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A
Size 1 1 2 4 4 8 8 8 4 8 8 8 8 8 16 16 16
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2B 2B 2A 2B 2A 2C 2D 4A
Type
128.932.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.932.1b R 1 1 1 1 1 −1 −1 −1 1 −1 −1 1 1 −1 1 −1 1
128.932.1c R 1 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 −1
128.932.1d R 1 1 1 1 1 −1 1 −1 1 −1 1 −1 −1 −1 −1 1 1
128.932.1e R 1 1 1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
128.932.1f R 1 1 1 1 1 1 −1 −1 1 −1 −1 1 1 1 −1 1 −1
128.932.1g R 1 1 1 1 1 1 −1 1 1 1 −1 −1 −1 1 −1 −1 1
128.932.1h R 1 1 1 1 1 1 1 −1 1 −1 1 −1 −1 1 1 −1 −1
128.932.2a R 2 2 2 −2 −2 0 0 −2 2 2 0 0 0 0 0 0 0
128.932.2b R 2 2 2 −2 −2 0 0 2 2 −2 0 0 0 0 0 0 0
128.932.2c R 2 2 2 −2 2 0 0 0 −2 0 0 −2 2 0 0 0 0
128.932.2d R 2 2 2 −2 2 0 0 0 −2 0 0 2 −2 0 0 0 0
128.932.2e R 2 2 2 2 −2 0 −2 0 −2 0 2 0 0 0 0 0 0
128.932.2f R 2 2 2 2 −2 0 2 0 −2 0 −2 0 0 0 0 0 0
128.932.4a R 4 4 −4 0 0 −2 0 0 0 0 0 0 0 2 0 0 0
128.932.4b R 4 4 −4 0 0 2 0 0 0 0 0 0 0 −2 0 0 0
128.932.8a R 8 −8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed