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Properties

Label 32T101
Degree $32$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_8:C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(32, 101);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $101$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_8:C_2^2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $16$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,15)(2,16)(3,13)(4,14)(5,9)(6,10)(7,12)(8,11)(17,29)(18,30)(19,31)(20,32)(21,26)(22,25)(23,28)(24,27), (1,22)(2,21)(3,23)(4,24)(5,18)(6,17)(7,20)(8,19)(9,26)(10,25)(11,28)(12,27)(13,29)(14,30)(15,31)(16,32), (1,16,3,14)(2,15,4,13)(5,10,7,11)(6,9,8,12)(17,32,19,30)(18,31,20,29)(21,28,24,25)(22,27,23,26), (1,5,3,7)(2,6,4,8)(9,15,12,13)(10,16,11,14)(17,24,19,21)(18,23,20,22)(25,32,28,30)(26,31,27,29)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $D_{4}$ x 4, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 6, $C_2^4$
$32$:  $C_2^2 \times D_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, 16T100, 16T118 x 2

Low degree siblings

16T100, 16T118 x 4, 32T101, 32T102, 32T127 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $64=2^{6}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
Label:  64.256
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 4A1 4A-1 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D
Size 1 1 2 2 2 4 4 4 4 1 1 2 2 2 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 2A 4B 4B 4B 4B
Type
64.256.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.256.1b R 1 1 −1 −1 1 −1 −1 −1 1 −1 −1 1 1 −1 1 1 −1 1 −1 −1 1 1
64.256.1c R 1 1 −1 −1 1 −1 1 −1 −1 −1 −1 1 1 −1 1 1 1 −1 1 1 −1 −1
64.256.1d R 1 1 −1 −1 1 1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 −1 −1
64.256.1e R 1 1 −1 −1 1 1 1 1 −1 −1 −1 1 1 −1 −1 −1 1 −1 −1 −1 1 1
64.256.1f R 1 1 −1 1 −1 −1 −1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1
64.256.1g R 1 1 −1 1 −1 −1 1 1 −1 1 1 1 −1 −1 1 −1 −1 1 −1 1 1 −1
64.256.1h R 1 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1 1 −1 −1 1 1 −1
64.256.1i R 1 1 −1 1 −1 1 1 −1 −1 1 1 1 −1 −1 −1 1 −1 1 1 −1 −1 1
64.256.1j R 1 1 1 −1 −1 −1 −1 1 −1 −1 −1 1 −1 1 −1 1 1 1 −1 1 −1 1
64.256.1k R 1 1 1 −1 −1 −1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 −1
64.256.1l R 1 1 1 −1 −1 1 −1 −1 −1 −1 −1 1 −1 1 1 −1 1 1 1 −1 1 −1
64.256.1m R 1 1 1 −1 −1 1 1 −1 1 −1 −1 1 −1 1 1 −1 −1 −1 −1 1 −1 1
64.256.1n R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1
64.256.1o R 1 1 1 1 1 −1 1 −1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
64.256.1p R 1 1 1 1 1 1 −1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
64.256.2a R 2 2 −2 −2 2 0 0 0 0 2 2 −2 −2 2 0 0 0 0 0 0 0 0
64.256.2b R 2 2 −2 2 −2 0 0 0 0 −2 −2 −2 2 2 0 0 0 0 0 0 0 0
64.256.2c R 2 2 2 −2 −2 0 0 0 0 2 2 −2 2 −2 0 0 0 0 0 0 0 0
64.256.2d R 2 2 2 2 2 0 0 0 0 −2 −2 −2 −2 −2 0 0 0 0 0 0 0 0
64.256.4a1 C 4 −4 0 0 0 0 0 0 0 −4i 4i 0 0 0 0 0 0 0 0 0 0 0
64.256.4a2 C 4 −4 0 0 0 0 0 0 0 4i −4i 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);