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Properties

Label 16T1036
Degree $16$
Order $672$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\SL(2,7):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 1036);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1036$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\SL(2,7):C_2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4,13,5,7,16,11,9,2,3,14,6,8,15,12,10), (1,8,16,5,11,4,10)(2,7,15,6,12,3,9)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$336$:  $\PGL(2,7)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 8: $\PGL(2,7)$

Low degree siblings

16T1036, 32T34613

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $56$ $2$ $( 1, 2)( 5, 7)( 6, 8)( 9,13)(10,14)(11,16)(12,15)$
$ 6, 6, 2, 1, 1 $ $56$ $6$ $( 1, 2)( 5,10,15, 7,14,12)( 6, 9,16, 8,13,11)$
$ 6, 6, 2, 1, 1 $ $56$ $6$ $( 3, 4)( 5, 9,15, 8,14,11)( 6,10,16, 7,13,12)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $56$ $3$ $( 5,14,15)( 6,13,16)( 7,10,12)( 8, 9,11)$
$ 6, 6, 2, 2 $ $56$ $6$ $( 1, 2)( 3, 4)( 5,13,15, 6,14,16)( 7, 9,12, 8,10,11)$
$ 7, 7, 1, 1 $ $48$ $7$ $( 3, 6,13,11,16, 9, 8)( 4, 5,14,12,15,10, 7)$
$ 14, 2 $ $48$ $14$ $( 1, 2)( 3, 5,13,12,16,10, 8, 4, 6,14,11,15, 9, 7)$
$ 4, 4, 4, 4 $ $42$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,14,12,13)$
$ 16 $ $42$ $16$ $( 1, 4, 6,11, 8,14,16, 9, 2, 3, 5,12, 7,13,15,10)$
$ 16 $ $42$ $16$ $( 1, 3, 6,12, 8,13,16,10, 2, 4, 5,11, 7,14,15, 9)$
$ 8, 8 $ $42$ $8$ $( 1, 3, 5,13, 2, 4, 6,14)( 7,12,10,16, 8,11, 9,15)$
$ 8, 8 $ $42$ $8$ $( 1, 4, 5,14, 2, 3, 6,13)( 7,11,10,15, 8,12, 9,16)$
$ 16 $ $42$ $16$ $( 1, 4, 6,15,12,13, 9, 8, 2, 3, 5,16,11,14,10, 7)$
$ 16 $ $42$ $16$ $( 1, 3, 6,16,12,14, 9, 7, 2, 4, 5,15,11,13,10, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $672=2^{5} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  672.1044
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 6A 6B1 6B-1 7A 8A1 8A3 14A 16A1 16A-1 16A3 16A-3
Size 1 1 56 56 42 56 56 56 48 42 42 48 42 42 42 42
2 P 1A 1A 1A 3A 2A 3A 3A 3A 7A 4A 4A 7A 8A3 8A3 8A1 8A1
3 P 1A 2A 2B 1A 4A 2A 2B 2B 7A 8A3 8A1 14A 16A1 16A-1 16A-3 16A3
7 P 1A 2A 2B 3A 4A 6A 6B1 6B-1 1A 8A1 8A3 2A 16A-3 16A3 16A-1 16A1
Type
672.1044.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
672.1044.1b R 1 1 −1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
672.1044.6a R 6 6 0 0 −2 0 0 0 −1 2 2 −1 0 0 0 0
672.1044.6b1 R 6 6 0 0 2 0 0 0 −1 0 0 −1 −ζ8−1−ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8−1+ζ8
672.1044.6b2 R 6 6 0 0 2 0 0 0 −1 0 0 −1 ζ8−1+ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−1−ζ8
672.1044.6c1 C 6 −6 0 0 0 0 0 0 −1 −ζ16−2−ζ162 ζ16−2+ζ162 1 −ζ16−ζ167 ζ16+ζ167 −ζ163−ζ165 ζ163+ζ165
672.1044.6c2 C 6 −6 0 0 0 0 0 0 −1 −ζ16−2−ζ162 ζ16−2+ζ162 1 ζ16+ζ167 −ζ16−ζ167 ζ163+ζ165 −ζ163−ζ165
672.1044.6c3 C 6 −6 0 0 0 0 0 0 −1 ζ16−2+ζ162 −ζ16−2−ζ162 1 −ζ163−ζ165 ζ163+ζ165 ζ16+ζ167 −ζ16−ζ167
672.1044.6c4 C 6 −6 0 0 0 0 0 0 −1 ζ16−2+ζ162 −ζ16−2−ζ162 1 ζ163+ζ165 −ζ163−ζ165 −ζ16−ζ167 ζ16+ζ167
672.1044.7a R 7 7 1 1 −1 1 1 1 0 −1 −1 0 −1 −1 −1 −1
672.1044.7b R 7 7 −1 1 −1 1 −1 −1 0 −1 −1 0 1 1 1 1
672.1044.8a R 8 −8 0 2 0 −2 0 0 1 0 0 −1 0 0 0 0
672.1044.8b R 8 8 2 −1 0 −1 −1 −1 1 0 0 1 0 0 0 0
672.1044.8c R 8 8 −2 −1 0 −1 1 1 1 0 0 1 0 0 0 0
672.1044.8d1 C 8 −8 0 −1 0 1 −1−2ζ3 1+2ζ3 1 0 0 −1 0 0 0 0
672.1044.8d2 C 8 −8 0 −1 0 1 1+2ζ3 −1−2ζ3 1 0 0 −1 0 0 0 0

magma: CharacterTable(G);