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Properties

Label 32T397
Degree $32$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4:S_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 397);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $397$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4:S_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,28,5,16)(2,27,6,15)(3,26,8,13)(4,25,7,14)(9,22,29,17)(10,21,30,18)(11,24,32,20)(12,23,31,19), (1,7,12,29)(2,8,11,30)(3,6,10,32)(4,5,9,31)(13,25,21,17)(14,26,22,18)(15,28,24,19)(16,27,23,20)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2, $S_4$

Degree 8: $D_4$, $S_4$, $S_4\times C_2$ x 2

Degree 16: 16T61, 16T191 x 2

Low degree siblings

12T54 x 2, 16T191 x 2, 24T128 x 2, 24T170, 24T171 x 2, 24T172 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 $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)$
2B $2^{16}$ $3$ $2$ $16$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,30)( 6,29)( 7,32)( 8,31)(13,23)(14,24)(15,22)(16,21)(17,27)(18,28)(19,26)(20,25)$
2C $2^{16}$ $3$ $2$ $16$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,31)( 6,32)( 7,29)( 8,30)(13,21)(14,22)(15,24)(16,23)(17,25)(18,26)(19,28)(20,27)$
2D $2^{16}$ $12$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,31)(10,32)(11,30)(12,29)(13,25)(14,26)(15,28)(16,27)(17,18)(19,20)(21,22)(23,24)$
2E $2^{16}$ $12$ $2$ $16$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,28)( 6,27)( 7,25)( 8,26)( 9,14)(10,13)(11,15)(12,16)(17,29)(18,30)(19,31)(20,32)$
3A $3^{8},1^{8}$ $8$ $3$ $16$ $( 5,12,31)( 6,11,32)( 7, 9,29)( 8,10,30)(13,26,18)(14,25,17)(15,27,20)(16,28,19)$
4A $4^{8}$ $2$ $4$ $24$ $( 1,24, 3,22)( 2,23, 4,21)( 5,20, 8,17)( 6,19, 7,18)( 9,13,11,16)(10,14,12,15)(25,31,27,30)(26,32,28,29)$
4B $4^{8}$ $6$ $4$ $24$ $( 1,15, 3,14)( 2,16, 4,13)( 5,27, 8,25)( 6,28, 7,26)( 9,21,11,23)(10,22,12,24)(17,31,20,30)(18,32,19,29)$
4C $4^{8}$ $12$ $4$ $24$ $( 1,29, 5, 9)( 2,30, 6,10)( 3,32, 8,11)( 4,31, 7,12)(13,22,26,17)(14,21,25,18)(15,23,27,19)(16,24,28,20)$
4D $4^{8}$ $12$ $4$ $24$ $( 1,28,12,19)( 2,27,11,20)( 3,26,10,18)( 4,25, 9,17)( 5,23,31,16)( 6,24,32,15)( 7,22,29,14)( 8,21,30,13)$
6A $6^{4},2^{4}$ $8$ $6$ $24$ $( 1, 3)( 2, 4)( 5,30,12, 8,31,10)( 6,29,11, 7,32, 9)(13,19,26,16,18,28)(14,20,25,15,17,27)(21,23)(22,24)$
12A1 $12^{2},4^{2}$ $8$ $12$ $28$ $( 1,24, 3,22)( 2,23, 4,21)( 5,15,30,17,12,27, 8,14,31,20,10,25)( 6,16,29,18,11,28, 7,13,32,19, 9,26)$
12A5 $12^{2},4^{2}$ $8$ $12$ $28$ $( 1,24, 3,22)( 2,23, 4,21)( 5,27,10,17,31,15, 8,25,12,20,30,14)( 6,28, 9,18,32,16, 7,26,11,19,29,13)$

Malle's constant $a(G)$:     $1/16$

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:  96.187
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 4A 4B 4C 4D 6A 12A1 12A5
Size 1 1 3 3 12 12 8 2 6 12 12 8 8 8
2 P 1A 1A 1A 1A 1A 1A 3A 2A 2A 2C 2C 3A 6A 6A
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 4C 4D 2A 4A 4A
Type
96.187.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.187.1b R 1 1 1 1 −1 −1 1 1 1 −1 −1 1 1 1
96.187.1c R 1 1 1 1 −1 1 1 −1 −1 −1 1 1 −1 −1
96.187.1d R 1 1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1
96.187.2a R 2 2 2 2 0 0 −1 2 2 0 0 −1 −1 −1
96.187.2b R 2 −2 2 −2 0 0 2 0 0 0 0 −2 0 0
96.187.2c R 2 2 2 2 0 0 −1 −2 −2 0 0 −1 1 1
96.187.2d1 R 2 −2 2 −2 0 0 −1 0 0 0 0 1 −ζ12−1−ζ12 ζ12−1+ζ12
96.187.2d2 R 2 −2 2 −2 0 0 −1 0 0 0 0 1 ζ12−1+ζ12 −ζ12−1−ζ12
96.187.3a R 3 3 −1 −1 1 1 0 3 −1 −1 −1 0 0 0
96.187.3b R 3 3 −1 −1 −1 −1 0 3 −1 1 1 0 0 0
96.187.3c R 3 3 −1 −1 −1 1 0 −3 1 1 −1 0 0 0
96.187.3d R 3 3 −1 −1 1 −1 0 −3 1 −1 1 0 0 0
96.187.6a R 6 −6 −2 2 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);