↑

Properties

Label 16T292
16T292 1 8 1->8 9 1->9 10 1->10 2 7 2->7 2->9 2->10 3 5 3->5 11 3->11 12 3->12 4 6 4->6 4->11 4->12 5->4 15 5->15 16 5->16 6->3 6->15 6->16 7->1 13 7->13 14 7->14 8->2 8->13 8->14 9->6 9->6 9->14 10->5 10->5 10->13 11->8 11->8 11->15 12->7 12->7 12->16 13->3 13->4 13->9 14->3 14->4 14->10 15->1 15->2 15->12 16->1 16->2 16->11
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2^3\times C_4):C_4$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 292);
 

Group invariants

Abstract group:  $(C_2^3\times C_4):C_4$
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 $t$:  $292$
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:  $(1,8,2,7)(3,5,4,6)(9,14,10,13)(11,15,12,16)$, $(1,9,6,15)(2,10,5,16)(3,11,8,13)(4,12,7,14)$, $(1,10,5,15,2,9,6,16)(3,12,7,13,4,11,8,14)$
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, $C_2 \times (C_2^2:C_4)$
$64$:  16T76

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$

Low degree siblings

16T243 x 2, 16T280 x 2, 16T292, 32T553, 32T554, 32T555 x 2, 32T651, 32T652, 32T674 x 2, 32T675, 32T676 x 2, 32T677, 32T1137 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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2C $2^{4},1^{8}$ $4$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
2D $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
2E $2^{2},1^{12}$ $4$ $2$ $2$ $(13,14)(15,16)$
2F $2^{8}$ $4$ $2$ $8$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,14)(10,13)(11,15)(12,16)$
2G $2^{8}$ $4$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,16)(10,15)(11,14)(12,13)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
4C $4^{4}$ $4$ $4$ $12$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
4D $4^{4}$ $4$ $4$ $12$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,14,10,13)(11,15,12,16)$
4E $4^{4}$ $4$ $4$ $12$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,15,10,16)(11,13,12,14)$
4F1 $4^{4}$ $4$ $4$ $12$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
4F-1 $4^{4}$ $4$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
4G $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,16,10,15)(11,14,12,13)$
4H $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,14)(10,13)(11,15)(12,16)$
4I1 $4^{4}$ $8$ $4$ $12$ $( 1,16, 5, 9)( 2,15, 6,10)( 3,14, 7,11)( 4,13, 8,12)$
4I-1 $4^{4}$ $8$ $4$ $12$ $( 1, 9, 5,16)( 2,10, 6,15)( 3,11, 7,14)( 4,12, 8,13)$
4J1 $4^{4}$ $8$ $4$ $12$ $( 1,14, 6,12)( 2,13, 5,11)( 3,15, 8, 9)( 4,16, 7,10)$
4J-1 $4^{4}$ $8$ $4$ $12$ $( 1,11, 6,13)( 2,12, 5,14)( 3,10, 8,16)( 4, 9, 7,15)$
8A1 $8^{2}$ $8$ $8$ $14$ $( 1,10, 5,16, 2, 9, 6,15)( 3,12, 7,14, 4,11, 8,13)$
8A-1 $8^{2}$ $8$ $8$ $14$ $( 1,16, 6, 9, 2,15, 5,10)( 3,14, 8,11, 4,13, 7,12)$
8B1 $8^{2}$ $8$ $8$ $14$ $( 1,12, 6,13, 2,11, 5,14)( 3, 9, 8,16, 4,10, 7,15)$
8B-1 $8^{2}$ $8$ $8$ $14$ $( 1,14, 5,12, 2,13, 6,11)( 3,15, 7, 9, 4,16, 8,10)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A1 4A-1 4B 4C 4D 4E 4F1 4F-1 4G 4H 4I1 4I-1 4J1 4J-1 8A1 8A-1 8B1 8B-1
Size 1 1 2 4 4 4 4 4 1 1 2 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 2A 2A 2A 2A 2A 2A 2A 2A 2B 2B 2G 2G 2G 2G 4E 4E 4E 4E
Type
128.852.1a 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
128.852.1b 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
128.852.1c 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
128.852.1d 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
128.852.1e 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
128.852.1f 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
128.852.1g 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
128.852.1h 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
128.852.1i1 C 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1 −1 −1 −i 1 −i 1 i i −i i −i i
128.852.1i2 C 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1 −1 −1 i 1 i 1 −i −i i −i i −i
128.852.1j1 C 1 1 1 1 −1 −1 −1 1 −1 −1 −1 1 1 −1 1 −1 −i −1 i 1 −i i i −i −i i
128.852.1j2 C 1 1 1 1 −1 −1 −1 1 −1 −1 −1 1 1 −1 1 −1 i −1 −i 1 i −i −i i i −i
128.852.1k1 C 1 1 1 1 −1 1 1 −1 1 1 1 −1 1 1 1 −1 −i −1 −i −1 i i i −i i −i
128.852.1k2 C 1 1 1 1 −1 1 1 −1 1 1 1 −1 1 1 1 −1 i −1 i −1 −i −i −i i −i i
128.852.1l1 C 1 1 1 1 −1 1 1 1 −1 −1 −1 1 −1 −1 −1 −1 −i 1 i −1 −i i −i i i −i
128.852.1l2 C 1 1 1 1 −1 1 1 1 −1 −1 −1 1 −1 −1 −1 −1 i 1 −i −1 i −i i −i −i i
128.852.2a R 2 2 2 −2 −2 0 0 −2 −2 −2 −2 2 0 2 0 2 0 0 0 0 0 0 0 0 0 0
128.852.2b R 2 2 2 −2 −2 0 0 2 2 2 2 −2 0 −2 0 2 0 0 0 0 0 0 0 0 0 0
128.852.2c R 2 2 2 −2 2 0 0 −2 2 2 2 2 0 −2 0 −2 0 0 0 0 0 0 0 0 0 0
128.852.2d R 2 2 2 −2 2 0 0 2 −2 −2 −2 −2 0 2 0 −2 0 0 0 0 0 0 0 0 0 0
128.852.4a R 4 4 −4 0 0 0 0 0 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.852.4b R 4 4 −4 0 0 0 0 0 4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.852.4c1 C 4 −4 0 0 0 −2 2 0 −4i 4i 0 0 −2i 0 2i 0 0 0 0 0 0 0 0 0 0 0
128.852.4c2 C 4 −4 0 0 0 −2 2 0 4i −4i 0 0 2i 0 −2i 0 0 0 0 0 0 0 0 0 0 0
128.852.4d1 C 4 −4 0 0 0 2 −2 0 −4i 4i 0 0 2i 0 −2i 0 0 0 0 0 0 0 0 0 0 0
128.852.4d2 C 4 −4 0 0 0 2 −2 0 4i −4i 0 0 −2i 0 2i 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed