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Properties

Label 16T273
16T273 1 7 1->7 1->7 10 1->10 2 8 2->8 2->8 9 2->9 3 3->2 4 3->4 3->9 4->1 4->10 5 5->4 12 5->12 14 5->14 6 6->3 11 6->11 13 6->13 7->5 7->5 15 7->15 8->6 8->6 16 8->16 9->15 9->15 10->16 10->16 11->2 11->9 11->12 12->1 12->10 13->3 13->11 14->4 14->12 15->13 15->13 16->14 16->14
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^5:C_4$

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Copy content magma:G := TransitiveGroup(16, 273);
 

Group invariants

Abstract group:  $C_2^5:C_4$
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 $t$:  $273$
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:  $(1,7,5,4)(2,8,6,3)(9,15,13,11)(10,16,14,12)$, $(1,10)(2,9)(3,4)(5,14)(6,13)(7,15)(8,16)(11,12)$, $(1,7,5,12)(2,8,6,11)(3,9,15,13)(4,10,16,14)$
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, $C_4\times C_2^2$
$32$:  $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
$64$:  $((C_8 : C_2):C_2):C_2$ x 2, 16T76

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Degree 8: $C_2^2:C_4$, $((C_8 : C_2):C_2):C_2$ x 2

Low degree siblings

16T227 x 4, 16T259 x 8, 16T261 x 4, 16T273 x 3, 16T283 x 4, 32T506, 32T507 x 2, 32T508 x 4, 32T595 x 4, 32T596 x 8, 32T597 x 2, 32T598 x 4, 32T599 x 4, 32T601, 32T602 x 2, 32T633, 32T657, 32T1130 x 2, 32T1796

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

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 2L 2M 4A 4B 4C 4D 4E1 4E-1 4F1 4F-1 8A1 8A-1 8B1 8B-1
Size 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2C 2C 2E 2E 2M 2M 2M 2M 4B 4B 4B 4B
Type
128.850.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.850.1b R 1 −1 −1 1 −1 1 −1 1 1 −1 1 −1 1 −1 −1 1 −1 −1 1 −1 1 1 1 1 −1 −1
128.850.1c R 1 −1 −1 1 −1 1 −1 1 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 1 1
128.850.1d R 1 −1 −1 1 −1 1 1 −1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 −1 1 1
128.850.1e R 1 −1 −1 1 −1 1 1 −1 1 −1 1 −1 −1 1 −1 1 1 1 −1 1 −1 −1 1 1 −1 −1
128.850.1f R 1 1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1
128.850.1g R 1 1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
128.850.1h R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1
128.850.1i1 C 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 1 −1 1 −1 −i i −1 1 i −i i −i −i i
128.850.1i2 C 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 1 −1 1 −1 i −i −1 1 −i i −i i i −i
128.850.1j1 C 1 −1 −1 1 −1 1 1 −1 1 −1 −1 1 −1 1 1 −1 −i i 1 −1 i −i −i i i −i
128.850.1j2 C 1 −1 −1 1 −1 1 1 −1 1 −1 −1 1 −1 1 1 −1 i −i 1 −1 −i i i −i −i i
128.850.1k1 C 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1 −i i 1 1 −i i i −i i −i
128.850.1k2 C 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1 i −i 1 1 i −i −i i −i i
128.850.1l1 C 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −i i −1 −1 −i i −i i −i i
128.850.1l2 C 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 i −i −1 −1 i −i i −i i −i
128.850.2a R 2 −2 −2 2 −2 2 0 0 −2 2 −2 2 0 0 −2 2 0 0 0 0 0 0 0 0 0 0
128.850.2b R 2 −2 −2 2 −2 2 0 0 −2 2 2 −2 0 0 2 −2 0 0 0 0 0 0 0 0 0 0
128.850.2c R 2 2 2 2 2 2 0 0 −2 −2 −2 −2 0 0 2 2 0 0 0 0 0 0 0 0 0 0
128.850.2d R 2 2 2 2 2 2 0 0 −2 −2 2 2 0 0 −2 −2 0 0 0 0 0 0 0 0 0 0
128.850.4a R 4 −4 −4 4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4b R 4 4 4 4 −4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4c R 4 −4 4 −4 0 0 −2 2 0 0 0 0 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4d R 4 −4 4 −4 0 0 2 −2 0 0 0 0 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4e R 4 4 −4 −4 0 0 −2 −2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4f R 4 4 −4 −4 0 0 2 2 0 0 0 0 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed