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Properties

Label 32T8
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $\OD_{32}$

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Show commands: Magma

magma: G := TransitiveGroup(32, 8);
 

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

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

Degree 16: $C_8\times C_2$, $C_{16} : C_2$

Low degree siblings

16T22

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

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:  32.17
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 4A1 4A-1 4B 8A1 8A-1 8A3 8A-3 8B1 8B-1 16A1 16A-1 16A3 16A-3 16B1 16B-1 16B3 16B-3
Size 1 1 2 1 1 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 2A 2A 2A 4A-1 4A1 4A-1 4A1 4A1 4A-1 8A-3 8A-1 8A-1 8A1 8A3 8A3 8A1 8A-3
Type
32.17.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.17.1b R 1 1 −1 1 1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1
32.17.1c R 1 1 −1 1 1 −1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
32.17.1d R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
32.17.1e1 C 1 1 −1 1 1 −1 −1 −1 −1 −1 1 1 −i i i −i −i i i −i
32.17.1e2 C 1 1 −1 1 1 −1 −1 −1 −1 −1 1 1 i −i −i i i −i −i i
32.17.1f1 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −i i i −i i −i −i i
32.17.1f2 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 i −i −i i −i i i −i
32.17.1g1 C 1 1 −1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ83 −ζ8 ζ8 −ζ83 −ζ8 ζ83 −ζ83 ζ8
32.17.1g2 C 1 1 −1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8 ζ83 −ζ8 ζ8 −ζ83
32.17.1g3 C 1 1 −1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83 ζ8 −ζ83 ζ83 −ζ8
32.17.1g4 C 1 1 −1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8 −ζ83 ζ8 −ζ8 ζ83
32.17.1h1 C 1 1 1 −1 −1 −1 −ζ82 ζ82 ζ82 −ζ82 −ζ82 ζ82 ζ83 −ζ8 ζ8 −ζ83 ζ8 −ζ83 ζ83 −ζ8
32.17.1h2 C 1 1 1 −1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 ζ82 −ζ82 −ζ8 ζ83 −ζ83 ζ8 −ζ83 ζ8 −ζ8 ζ83
32.17.1h3 C 1 1 1 −1 −1 −1 −ζ82 ζ82 ζ82 −ζ82 −ζ82 ζ82 −ζ83 ζ8 −ζ8 ζ83 −ζ8 ζ83 −ζ83 ζ8
32.17.1h4 C 1 1 1 −1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 ζ82 −ζ82 ζ8 −ζ83 ζ83 −ζ8 ζ83 −ζ8 ζ8 −ζ83
32.17.2a1 C 2 −2 0 −2ζ82 2ζ82 0 2ζ83 −2ζ8 2ζ8 −2ζ83 0 0 0 0 0 0 0 0 0 0
32.17.2a2 C 2 −2 0 2ζ82 −2ζ82 0 −2ζ8 2ζ83 −2ζ83 2ζ8 0 0 0 0 0 0 0 0 0 0
32.17.2a3 C 2 −2 0 −2ζ82 2ζ82 0 −2ζ83 2ζ8 −2ζ8 2ζ83 0 0 0 0 0 0 0 0 0 0
32.17.2a4 C 2 −2 0 2ζ82 −2ζ82 0 2ζ8 −2ζ83 2ζ83 −2ζ8 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);