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Properties

Label 32T2144
Degree $32$
Order $192$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^3.S_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 2144);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2144$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^3.S_4$
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:  (1,15,18,27)(2,16,17,28)(3,10,20,30)(4,9,19,29)(5,12,22,31)(6,11,21,32)(7,13,23,25)(8,14,24,26), (1,9,30,20)(2,10,29,19)(3,13,31,24)(4,14,32,23)(5,12,25,17)(6,11,26,18)(7,16,27,22)(8,15,28,21)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$
$24$:  $S_4$
$48$:  12T27

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $S_4$

Degree 8: $S_4$

Degree 16: 16T430

Low degree siblings

16T430, 24T374, 24T378, 24T384, 24T387

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 $2^{16}$ $3$ $2$ $16$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,16)(12,15)(17,21)(18,22)(19,24)(20,23)(25,30)(26,29)(27,31)(28,32)$
2B $2^{12},1^{8}$ $4$ $2$ $12$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15)(10,16)(11,13)(12,14)(25,28)(26,27)(29,31)(30,32)$
2C $2^{16}$ $12$ $2$ $16$ $( 1,24)( 2,23)( 3,21)( 4,22)( 5,19)( 6,20)( 7,17)( 8,18)( 9,27)(10,28)(11,25)(12,26)(13,32)(14,31)(15,29)(16,30)$
3A $3^{10},1^{2}$ $32$ $3$ $20$ $( 1,28,10)( 2,27, 9)( 3,29,14)( 4,30,13)( 5,25,16)( 6,26,15)( 7,31,12)( 8,32,11)(17,21,23)(18,22,24)$
4A $4^{8}$ $12$ $4$ $24$ $( 1,24, 5,19)( 2,23, 6,20)( 3,21, 7,17)( 4,22, 8,18)( 9,26,14,29)(10,25,13,30)(11,28,16,32)(12,27,15,31)$
4B1 $4^{8}$ $24$ $4$ $24$ $( 1,27,19,14)( 2,28,20,13)( 3,30,17,11)( 4,29,18,12)( 5,31,24, 9)( 6,32,23,10)( 7,25,21,16)( 8,26,22,15)$
4B-1 $4^{8}$ $24$ $4$ $24$ $( 1,26,22, 9)( 2,25,21,10)( 3,32,23,16)( 4,31,24,15)( 5,29,18,14)( 6,30,17,13)( 7,28,20,11)( 8,27,19,12)$
4C1 $4^{6},2^{4}$ $24$ $4$ $22$ $( 1, 7, 5, 3)( 2, 8, 6, 4)( 9,32,15,30)(10,31,16,29)(11,26,13,27)(12,25,14,28)(17,22)(18,21)(19,20)(23,24)$
4C-1 $4^{6},2^{4}$ $24$ $4$ $22$ $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,30,15,32)(10,29,16,31)(11,27,13,26)(12,28,14,25)(17,22)(18,21)(19,20)(23,24)$
6A $6^{4},3^{2},1^{2}$ $32$ $6$ $24$ $( 1,25,10, 5,28,16)( 2,26, 9, 6,27,15)( 3,31,14, 7,29,12)( 4,32,13, 8,30,11)(17,21,23)(18,22,24)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $192=2^{6} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  192.185
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A 4B1 4B-1 4C1 4C-1 6A
Size 1 3 4 12 32 12 24 24 24 24 32
2 P 1A 1A 1A 1A 3A 2A 2C 2C 2B 2B 3A
3 P 1A 2A 2B 2C 1A 4A 4B-1 4B1 4C-1 4C1 2B
Type
192.185.1a R 1 1 1 1 1 1 1 1 1 1 1
192.185.1b R 1 1 1 1 1 1 −1 −1 −1 −1 1
192.185.1c1 C 1 1 −1 −1 1 1 −i i −i i −1
192.185.1c2 C 1 1 −1 −1 1 1 i −i i −i −1
192.185.2a R 2 2 2 2 −1 2 0 0 0 0 −1
192.185.2b S 2 2 −2 −2 −1 2 0 0 0 0 1
192.185.3a R 3 3 3 −1 0 −1 −1 −1 1 1 0
192.185.3b R 3 3 3 −1 0 −1 1 1 −1 −1 0
192.185.3c1 C 3 3 −3 1 0 −1 −i i i −i 0
192.185.3c2 C 3 3 −3 1 0 −1 i −i −i i 0
192.185.12a R 12 −4 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);