↑

Properties

Label 16T227
16T227 1 5 1->5 16 1->16 2 6 2->6 15 2->15 3 4 3->4 3->6 4->5 11 5->11 14 5->14 12 6->12 13 6->13 7 8 7->8 9 7->9 7->16 10 8->10 8->15 9->8 9->13 10->7 10->14 11->12 11->13 12->14 13->4 14->3 15->1 15->16 16->2
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^5:C_4$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 227);
 

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 $t$:  $227$
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:  $(1,16,2,15)(3,6,12,14)(4,5,11,13)(7,9,8,10)$, $(1,5)(2,6)(3,4)(7,8)(9,13)(10,14)(11,12)(15,16)$, $(5,14)(6,13)(7,16)(8,15)$
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: $C_2^2:C_4$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Low degree siblings

16T227 x 3, 16T259 x 8, 16T261 x 4, 16T273 x 4, 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 $2^{8}$ $2$ $2$ $8$ $( 1, 9)( 2,10)( 3, 4)( 5,13)( 6,14)( 7, 8)(11,12)(15,16)$
2E $2^{4},1^{8}$ $2$ $2$ $4$ $( 1,10)( 2, 9)( 5,14)( 6,13)$
2F $2^{8}$ $4$ $2$ $8$ $( 1, 5)( 2, 6)( 3,12)( 4,11)( 7,15)( 8,16)( 9,13)(10,14)$
2G $2^{8}$ $4$ $2$ $8$ $( 1, 6)( 2, 5)( 3,11)( 4,12)( 7,16)( 8,15)( 9,14)(10,13)$
2H $2^{8}$ $4$ $2$ $8$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5, 9)( 6,10)( 7,12)( 8,11)$
2I $2^{8}$ $4$ $2$ $8$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,10)( 6, 9)( 7,11)( 8,12)$
2J $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,10)(11,16)(12,15)(13,14)$
2K $2^{4},1^{8}$ $4$ $2$ $4$ $( 3, 8)( 4, 7)(11,15)(12,16)$
2L $2^{4},1^{8}$ $4$ $2$ $4$ $( 5,14)( 6,13)( 7,16)( 8,15)$
2M $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,10)(11,12)$
4A $4^{4}$ $4$ $4$ $12$ $( 1, 6,10,13)( 2, 5, 9,14)( 3, 8,11,15)( 4, 7,12,16)$
4B $4^{4}$ $4$ $4$ $12$ $( 1, 5,10,14)( 2, 6, 9,13)( 3, 7,11,16)( 4, 8,12,15)$
4C $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1,13,10, 6)( 2,14, 9, 5)( 3,11)( 4,12)$
4D $4^{2},2^{4}$ $8$ $4$ $10$ $( 1,14,10, 5)( 2,13, 9, 6)( 3,12)( 4,11)( 7, 8)(15,16)$
4E1 $4^{4}$ $8$ $4$ $12$ $( 1, 3, 2, 4)( 5,16,13, 8)( 6,15,14, 7)( 9,12,10,11)$
4E-1 $4^{4}$ $8$ $4$ $12$ $( 1, 4, 2, 3)( 5, 8,13,16)( 6, 7,14,15)( 9,11,10,12)$
4F1 $4^{4}$ $8$ $4$ $12$ $( 1, 4, 2, 3)( 5,15,13, 7)( 6,16,14, 8)( 9,11,10,12)$
4F-1 $4^{4}$ $8$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7,13,15)( 6, 8,14,16)( 9,12,10,11)$
8A1 $8^{2}$ $8$ $8$ $14$ $( 1,16, 5, 3,10, 7,14,11)( 2,15, 6, 4, 9, 8,13,12)$
8A-1 $8^{2}$ $8$ $8$ $14$ $( 1, 8, 5, 4,10,15,14,12)( 2, 7, 6, 3, 9,16,13,11)$
8B1 $8^{2}$ $8$ $8$ $14$ $( 1,15, 5, 4,10, 8,14,12)( 2,16, 6, 3, 9, 7,13,11)$
8B-1 $8^{2}$ $8$ $8$ $14$ $( 1, 7, 5, 3,10,16,14,11)( 2, 8, 6, 4, 9,15,13,12)$

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