↑

Properties

Label 32T418
Degree $32$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^3:A_4$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(32, 418);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $418$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^3:A_4$
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:  (1,26,13,2,25,14)(3,30,11,4,29,12)(5,32,15,6,31,16)(7,28,9,8,27,10)(17,18)(19,22,23,20,21,24), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,23)(2,24)(3,21)(4,22)(5,19)(6,20)(7,17)(8,18)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$ x 5
$24$:  $A_4\times C_2$ x 5
$48$:  $C_2^4:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $A_4$ x 5

Degree 8: $A_4\times C_2$ x 5

Degree 16: 16T64

Low degree siblings

12T56 x 30, 24T176 x 5, 24T177 x 30, 24T178 x 10

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 $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B $2^{16}$ $3$ $2$ $16$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,19)( 6,20)( 7,17)( 8,18)( 9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)$
2C $2^{16}$ $3$ $2$ $16$ $( 1,24)( 2,23)( 3,22)( 4,21)( 5,20)( 6,19)( 7,18)( 8,17)( 9,28)(10,27)(11,26)(12,25)(13,32)(14,31)(15,30)(16,29)$
2D $2^{16}$ $3$ $2$ $16$ $( 1,27)( 2,28)( 3,25)( 4,26)( 5,31)( 6,32)( 7,29)( 8,30)( 9,23)(10,24)(11,21)(12,22)(13,19)(14,20)(15,17)(16,18)$
2E $2^{16}$ $3$ $2$ $16$ $( 1,28)( 2,27)( 3,26)( 4,25)( 5,32)( 6,31)( 7,30)( 8,29)( 9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)$
2F $2^{16}$ $3$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)(25,31)(26,32)(27,29)(28,30)$
2G $2^{16}$ $3$ $2$ $16$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,16)(10,15)(11,14)(12,13)(17,24)(18,23)(19,22)(20,21)(25,32)(26,31)(27,30)(28,29)$
2H $2^{16}$ $3$ $2$ $16$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)$
2I $2^{16}$ $3$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,22)( 6,21)( 7,24)( 8,23)( 9,30)(10,29)(11,32)(12,31)(13,26)(14,25)(15,28)(16,27)$
2J $2^{16}$ $3$ $2$ $16$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)$
2K $2^{16}$ $3$ $2$ $16$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)(17,30)(18,29)(19,32)(20,31)(21,26)(22,25)(23,28)(24,27)$
3A1 $3^{10},1^{2}$ $16$ $3$ $20$ $( 3, 5, 7)( 4, 6, 8)( 9,31,19)(10,32,20)(11,27,21)(12,28,22)(13,25,17)(14,26,18)(15,29,23)(16,30,24)$
3A-1 $3^{10},1^{2}$ $16$ $3$ $20$ $( 3, 7, 5)( 4, 8, 6)( 9,19,31)(10,20,32)(11,21,27)(12,22,28)(13,17,25)(14,18,26)(15,23,29)(16,24,30)$
6A1 $6^{5},2$ $16$ $6$ $26$ $( 1, 2)( 3, 8, 5, 4, 7, 6)( 9,20,31,10,19,32)(11,22,27,12,21,28)(13,18,25,14,17,26)(15,24,29,16,23,30)$
6A-1 $6^{5},2$ $16$ $6$ $26$ $( 1, 2)( 3, 6, 7, 4, 5, 8)( 9,32,19,10,31,20)(11,28,21,12,27,22)(13,26,17,14,25,18)(15,30,23,16,29,24)$

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:  96.229
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A1 3A-1 6A1 6A-1
Size 1 1 3 3 3 3 3 3 3 3 3 3 16 16 16 16
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1
3 P 1A 2A 2F 2G 2H 2I 2J 2K 2B 2C 2D 2E 1A 1A 2A 2A
Type
96.229.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.229.1b R 1 −1 1 −1 1 1 −1 −1 1 −1 1 −1 1 1 −1 −1
96.229.1c1 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
96.229.1c2 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
96.229.1d1 C 1 −1 1 −1 1 1 −1 −1 1 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1
96.229.1d2 C 1 −1 1 −1 1 1 −1 −1 1 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3
96.229.3a R 3 3 −1 −1 −1 −1 −1 −1 3 −1 −1 3 0 0 0 0
96.229.3b R 3 3 −1 −1 −1 −1 −1 3 −1 −1 3 −1 0 0 0 0
96.229.3c R 3 3 −1 −1 −1 3 −1 −1 −1 3 −1 −1 0 0 0 0
96.229.3d R 3 3 −1 −1 3 −1 3 −1 −1 −1 −1 −1 0 0 0 0
96.229.3e R 3 3 3 3 −1 −1 −1 −1 −1 −1 −1 −1 0 0 0 0
96.229.3f R 3 −3 −1 1 −1 −1 1 −3 −1 1 3 1 0 0 0 0
96.229.3g R 3 −3 −1 1 −1 −1 1 1 3 1 −1 −3 0 0 0 0
96.229.3h R 3 −3 −1 1 −1 3 1 1 −1 −3 −1 1 0 0 0 0
96.229.3i R 3 −3 −1 1 3 −1 −3 1 −1 1 −1 1 0 0 0 0
96.229.3j R 3 −3 3 −3 −1 −1 1 1 −1 1 −1 1 0 0 0 0

magma: CharacterTable(G);