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Properties

Label 32T42
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_8:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 42);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_8:C_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $32$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,6,11,31)(2,5,12,32)(3,7,9,29)(4,8,10,30)(13,17,23,28)(14,18,24,27)(15,20,21,26)(16,19,22,25), (1,23,10,16,2,24,9,15)(3,21,11,13,4,22,12,14)(5,28,29,19,6,27,30,20)(7,25,31,18,8,26,32,17)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 6, $C_2^2$
$8$:  $C_4\times C_2$ x 3
$16$:  $C_8:C_2$ x 2, $C_4^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 6, $C_2^2$

Degree 8: $C_4\times C_2$ x 3, $C_8:C_2$ x 2

Degree 16: $C_4^2$, $C_8: C_2$ x 2

Low degree siblings

There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,32,30,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,31,30,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 5,11,32)( 2, 6,12,31)( 3, 8, 9,30)( 4, 7,10,29)(13,18,23,27)(14,17,24,28) (15,19,21,25)(16,20,22,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 7,12,30)( 2, 8,11,29)( 3, 5,10,31)( 4, 6, 9,32)(13,20,24,25)(14,19,23,26) (15,18,22,28)(16,17,21,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,30, 6,29)( 7,32, 8,31)(13,21,14,22)(15,24,16,23) (17,26,18,25)(19,28,20,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,29, 6,30)( 7,31, 8,32)(13,22,14,21)(15,23,16,24) (17,25,18,26)(19,27,20,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,32)( 6,31)( 7,29)( 8,30)(13,23)(14,24)(15,21) (16,22)(17,28)(18,27)(19,25)(20,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,31)( 6,32)( 7,30)( 8,29)(13,24)(14,23)(15,22) (16,21)(17,27)(18,28)(19,26)(20,25)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,13,10,22, 2,14, 9,21)( 3,15,11,23, 4,16,12,24)( 5,17,29,25, 6,18,30,26) ( 7,19,31,27, 8,20,32,28)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,15, 9,24, 2,16,10,23)( 3,14,12,22, 4,13,11,21)( 5,20,30,27, 6,19,29,28) ( 7,17,32,26, 8,18,31,25)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,17, 3,20, 2,18, 4,19)( 5,23, 8,21, 6,24, 7,22)( 9,26,12,27,10,25,11,28) (13,30,15,31,14,29,16,32)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,19, 4,18, 2,20, 3,17)( 5,22, 7,24, 6,21, 8,23)( 9,28,11,25,10,27,12,26) (13,32,16,29,14,31,15,30)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,21, 9,14, 2,22,10,13)( 3,24,12,16, 4,23,11,15)( 5,26,30,18, 6,25,29,17) ( 7,28,32,20, 8,27,31,19)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,23,10,16, 2,24, 9,15)( 3,21,11,13, 4,22,12,14)( 5,28,29,19, 6,27,30,20) ( 7,25,31,18, 8,26,32,17)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,25, 4,27, 2,26, 3,28)( 5,16, 7,14, 6,15, 8,13)( 9,17,11,19,10,18,12,20) (21,30,23,32,22,29,24,31)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,27, 3,25, 2,28, 4,26)( 5,14, 8,16, 6,13, 7,15)( 9,19,12,17,10,20,11,18) (21,32,24,30,22,31,23,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,29,12, 8)( 2,30,11, 7)( 3,32,10, 6)( 4,31, 9, 5)(13,26,24,19)(14,25,23,20) (15,27,22,17)(16,28,21,18)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,31,11, 6)( 2,32,12, 5)( 3,29, 9, 7)( 4,30,10, 8)(13,28,23,17)(14,27,24,18) (15,26,21,20)(16,25,22,19)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  32.4
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 8A1 8A-1 8B1 8B-1 8C1 8C-1 8D1 8D-1
Size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 2A 2A 2A 2B 2C 2C 2B 4A-1 4B-1 4A-1 4B-1 4A1 4A1 4B1 4B1
Type
32.4.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.4.1b R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1
32.4.1c R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 1 1 −1 −1
32.4.1d R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
32.4.1e1 C 1 1 −1 −1 −1 −1 1 1 −i i −i i i −i −1 −1 i −i 1 1
32.4.1e2 C 1 1 −1 −1 −1 −1 1 1 i −i i −i −i i −1 −1 −i i 1 1
32.4.1f1 C 1 1 −1 −1 −1 −1 1 1 −i i −i i −i i 1 1 −i i −1 −1
32.4.1f2 C 1 1 −1 −1 −1 −1 1 1 i −i i −i i −i 1 1 i −i −1 −1
32.4.1g1 C 1 1 −1 −1 1 1 −1 −1 −i i i −i −1 −1 −i i 1 1 i −i
32.4.1g2 C 1 1 −1 −1 1 1 −1 −1 i −i −i i −1 −1 i −i 1 1 −i i
32.4.1h1 C 1 1 −1 −1 1 1 −1 −1 −i i i −i 1 1 i −i −1 −1 −i i
32.4.1h2 C 1 1 −1 −1 1 1 −1 −1 i −i −i i 1 1 −i i −1 −1 i −i
32.4.1i1 C 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −i i i −i i −i i −i
32.4.1i2 C 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 i −i −i i −i i −i i
32.4.1j1 C 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −i i −i i i −i −i i
32.4.1j2 C 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 i −i i −i −i i i −i
32.4.2a1 C 2 −2 −2 2 −2i 2i −2i 2i 0 0 0 0 0 0 0 0 0 0 0 0
32.4.2a2 C 2 −2 −2 2 2i −2i 2i −2i 0 0 0 0 0 0 0 0 0 0 0 0
32.4.2b1 C 2 −2 2 −2 −2i 2i 2i −2i 0 0 0 0 0 0 0 0 0 0 0 0
32.4.2b2 C 2 −2 2 −2 2i −2i −2i 2i 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);