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Properties

Label 16T181
Degree $16$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4\times S_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 181);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $181$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\times S_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,11,2,12)(3,9,4,10)(5,13,6,14)(7,16,8,15), (1,10,5,12,4,8,2,9,6,11,3,7)(13,16,14,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$24$:  $S_4$, $S_3 \times C_4$
$48$:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $S_4$

Degree 8: $S_4\times C_2$

Low degree siblings

12T53 x 2, 16T181, 24T129, 24T130, 24T167, 24T168 x 2, 24T169 x 2, 32T387

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $96=2^{5} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  96.186
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 4A1 4A-1 4B1 4B-1 4C 4D 4E1 4E-1 4F1 4F-1 6A 12A1 12A-1
Size 1 1 3 3 6 6 8 1 1 3 3 6 6 6 6 6 6 8 8 8
2 P 1A 1A 1A 1A 1A 1A 3A 2A 2A 2A 2A 2B 2C 2B 2A 2A 2C 3A 6A 6A
3 P 1A 2A 2B 2C 2D 2E 1A 4A-1 4A1 4B-1 4B1 4D 4E-1 4C 4F-1 4F1 4E1 2A 4A1 4A-1
Type
96.186.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.186.1b R 1 1 1 1 −1 −1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 −1 −1
96.186.1c R 1 1 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1
96.186.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1 1 −1 −1
96.186.1e1 C 1 −1 1 −1 −1 1 1 −i i i −i −1 1 −i i −i i −1 i −i
96.186.1e2 C 1 −1 1 −1 −1 1 1 i −i −i i −1 1 i −i i −i −1 −i i
96.186.1f1 C 1 −1 1 −1 1 −1 1 −i i i −i 1 −1 i −i i −i −1 i −i
96.186.1f2 C 1 −1 1 −1 1 −1 1 i −i −i i 1 −1 −i i −i i −1 −i i
96.186.2a R 2 2 2 2 0 0 −1 2 2 2 2 0 0 0 0 0 0 −1 −1 −1
96.186.2b R 2 2 2 2 0 0 −1 −2 −2 −2 −2 0 0 0 0 0 0 −1 1 1
96.186.2c1 C 2 −2 2 −2 0 0 −1 −2i 2i 2i −2i 0 0 0 0 0 0 1 −i i
96.186.2c2 C 2 −2 2 −2 0 0 −1 2i −2i −2i 2i 0 0 0 0 0 0 1 i −i
96.186.3a R 3 3 −1 −1 1 1 0 3 3 −1 −1 −1 −1 −1 −1 1 1 0 0 0
96.186.3b R 3 3 −1 −1 −1 −1 0 3 3 −1 −1 1 1 1 1 −1 −1 0 0 0
96.186.3c R 3 3 −1 −1 −1 −1 0 −3 −3 1 1 1 1 −1 −1 1 1 0 0 0
96.186.3d R 3 3 −1 −1 1 1 0 −3 −3 1 1 −1 −1 1 1 −1 −1 0 0 0
96.186.3e1 C 3 −3 −1 1 −1 1 0 −3i 3i −i i 1 −1 i −i −i i 0 0 0
96.186.3e2 C 3 −3 −1 1 −1 1 0 3i −3i i −i 1 −1 −i i i −i 0 0 0
96.186.3f1 C 3 −3 −1 1 1 −1 0 −3i 3i −i i −1 1 −i i i −i 0 0 0
96.186.3f2 C 3 −3 −1 1 1 −1 0 3i −3i i −i −1 1 i −i −i i 0 0 0

magma: CharacterTable(G);