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Properties

Label 32T2
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $\OD_{16}:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(32, 2);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7

Degree 8: $C_4\times C_2$ x 6, $C_2^3$

Degree 16: $C_4\times C_2^2$, $(C_8:C_2):C_2$ x 3

Low degree siblings

16T16 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  32.38
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 4A1 4A-1 4B 4C 4D 8A1 8A-1 8A3 8A-3 8B1 8B-1 8C1 8C-1 8D1 8D-1
Size 1 1 2 2 2 1 1 2 2 2 1 1 1 1 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 4A1 4A-1 4A-1 4A1 4A1 4A-1 4A1 4A-1 4A-1 4A1
Type
32.38.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.38.1b R 1 1 −1 −1 1 1 1 −1 1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1
32.38.1c R 1 1 −1 −1 1 1 1 −1 1 −1 1 1 1 1 −1 −1 1 1 −1 −1
32.38.1d R 1 1 −1 1 −1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 −1 −1
32.38.1e R 1 1 −1 1 −1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1
32.38.1f R 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 −1 −1 −1 1 1 1 1
32.38.1g R 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
32.38.1h R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
32.38.1i1 C 1 1 −1 −1 −1 −1 −1 1 1 1 −i i i −i i −i i −i −i i
32.38.1i2 C 1 1 −1 −1 −1 −1 −1 1 1 1 i −i −i i −i i −i i i −i
32.38.1j1 C 1 1 −1 1 1 −1 −1 −1 −1 1 −i i i −i i −i −i i i −i
32.38.1j2 C 1 1 −1 1 1 −1 −1 −1 −1 1 i −i −i i −i i i −i −i i
32.38.1k1 C 1 1 1 −1 1 −1 −1 1 −1 −1 −i i i −i −i i −i i −i i
32.38.1k2 C 1 1 1 −1 1 −1 −1 1 −1 −1 i −i −i i i −i i −i i −i
32.38.1l1 C 1 1 1 1 −1 −1 −1 −1 1 −1 −i i i −i −i i i −i i −i
32.38.1l2 C 1 1 1 1 −1 −1 −1 −1 1 −1 i −i −i i i −i −i i −i i
32.38.2a1 C 2 −2 0 0 0 −2ζ82 2ζ82 0 0 0 2ζ83 −2ζ8 2ζ8 −2ζ83 0 0 0 0 0 0
32.38.2a2 C 2 −2 0 0 0 2ζ82 −2ζ82 0 0 0 −2ζ8 2ζ83 −2ζ83 2ζ8 0 0 0 0 0 0
32.38.2a3 C 2 −2 0 0 0 −2ζ82 2ζ82 0 0 0 −2ζ83 2ζ8 −2ζ8 2ζ83 0 0 0 0 0 0
32.38.2a4 C 2 −2 0 0 0 2ζ82 −2ζ82 0 0 0 2ζ8 −2ζ83 2ζ83 −2ζ8 0 0 0 0 0 0

magma: CharacterTable(G);