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Properties

Label 32T85
Degree $32$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^3:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 85);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $85$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^3:D_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,26,31,23)(2,25,32,24)(3,27,29,22)(4,28,30,21)(5,15,9,17)(6,16,10,18)(7,14,12,20)(8,13,11,19), (1,13,29,18)(2,14,30,17)(3,16,31,19)(4,15,32,20)(5,21,12,25)(6,22,11,26)(7,24,9,28)(8,23,10,27), (1,17)(2,18)(3,20)(4,19)(5,27)(6,28)(7,26)(8,25)(9,22)(10,21)(11,24)(12,23)(13,30)(14,29)(15,31)(16,32), (1,6,3,8)(2,5,4,7)(9,30,12,32)(10,29,11,31)(13,27,16,26)(14,28,15,25)(17,24,20,21)(18,23,19,22)
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, $C_2^4$
$32$:  $Q_8:C_2^2$ x 2, $C_2^2 \times D_4$

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: $C_2^3$, $D_4\times C_2$ x 6, $Q_8:C_2^2$ x 4

Degree 16: $Q_8 : C_2^2$ x 2, $C_2^2 \times D_4$, 16T87 x 4

Low degree siblings

16T87 x 8, 16T119 x 4, 32T85 x 3, 32T86 x 4, 32T128 x 2

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

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:  $2$
Label:  64.215
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H
Size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2B 2C 2A 2C 2A 2A 2B
Type
64.215.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.215.1b R 1 1 1 1 −1 −1 1 1 −1 −1 −1 1 −1 −1 −1 1 1 1 1 1 −1 −1
64.215.1c R 1 1 1 1 −1 −1 1 1 −1 −1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1
64.215.1d R 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 1 −1 −1 1 −1 1 1 1
64.215.1e R 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 1 1 1 1 −1 −1 1 −1 −1 −1
64.215.1f R 1 1 1 1 −1 1 −1 −1 1 −1 −1 1 −1 1 1 −1 −1 −1 1 1 1 −1
64.215.1g R 1 1 1 1 −1 1 −1 −1 1 −1 −1 1 1 −1 1 1 −1 1 −1 −1 −1 1
64.215.1h R 1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 1 −1 1 1 −1 −1 1 −1 1
64.215.1i R 1 1 1 1 −1 1 −1 −1 1 −1 1 −1 1 −1 −1 −1 1 1 1 −1 1 −1
64.215.1j R 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 −1 1 1 −1 1 1 1 −1 −1 1
64.215.1k R 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 1 −1 1 1 1 −1 −1 1 1 −1
64.215.1l R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 1 −1 1 −1 1 −1 −1 1 −1
64.215.1m R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1 1
64.215.1n R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 −1 1 −1 1 1
64.215.1o R 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 1 −1 1 −1 −1
64.215.1p R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 1 −1 1 −1 −1 −1 −1 −1
64.215.2a R 2 2 −2 −2 −2 −2 2 −2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
64.215.2b R 2 2 −2 −2 −2 2 −2 2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
64.215.2c R 2 2 −2 −2 2 −2 −2 2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
64.215.2d R 2 2 −2 −2 2 2 2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
64.215.4a R 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
64.215.4b R 4 −4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);