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Properties

Label 32T17
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2:Q_8$

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Copy content magma:G := TransitiveGroup(32, 17);
 

Group invariants

Abstract group:  $C_2^2:Q_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $32=2^{5}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$, $Q_8$ x 2
$16$:  $D_4\times C_2$, $Q_8:C_2$, $Q_8\times C_2$

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

Degree 16: $Q_8\times C_2$, $D_4\times C_2$, $Q_8 : C_2$, 16T31 x 2

Low degree siblings

16T31 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1,29)( 2,30)( 3,31)( 4,32)( 5,11)( 6,12)( 7, 9)( 8,10)(13,20)(14,19)(15,18)(16,17)(21,28)(22,27)(23,25)(24,26)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)$
2C $2^{16}$ $1$ $2$ $16$ $( 1,32)( 2,31)( 3,30)( 4,29)( 5, 9)( 6,10)( 7,11)( 8,12)(13,18)(14,17)(15,20)(16,19)(21,26)(22,25)(23,27)(24,28)$
2D $2^{16}$ $2$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,19)(14,20)(15,17)(16,18)(21,27)(22,28)(23,26)(24,25)(29,30)(31,32)$
2E $2^{16}$ $2$ $2$ $16$ $( 1,31)( 2,32)( 3,29)( 4,30)( 5,10)( 6, 9)( 7,12)( 8,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)$
4A $4^{8}$ $2$ $4$ $24$ $( 1, 9, 4,11)( 2,10, 3,12)( 5,29, 7,32)( 6,30, 8,31)(13,23,15,22)(14,24,16,21)(17,28,19,26)(18,27,20,25)$
4B $4^{8}$ $2$ $4$ $24$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,29,11,32)(10,30,12,31)(13,27,15,25)(14,28,16,26)(17,24,19,21)(18,23,20,22)$
4C1 $4^{8}$ $2$ $4$ $24$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,30, 7,31)( 6,29, 8,32)(13,26,15,28)(14,25,16,27)(17,22,19,23)(18,21,20,24)$
4C-1 $4^{8}$ $2$ $4$ $24$ $( 1,12, 4,10)( 2,11, 3, 9)( 5,31, 7,30)( 6,32, 8,29)(13,28,15,26)(14,27,16,25)(17,23,19,22)(18,24,20,21)$
4D $4^{8}$ $4$ $4$ $24$ $( 1,20,32,15)( 2,19,31,16)( 3,17,30,14)( 4,18,29,13)( 5,23, 9,27)( 6,24,10,28)( 7,22,11,25)( 8,21,12,26)$
4E $4^{8}$ $4$ $4$ $24$ $( 1,28, 4,26)( 2,27, 3,25)( 5,14, 7,16)( 6,13, 8,15)( 9,17,11,19)(10,18,12,20)(21,32,24,29)(22,31,23,30)$
4F $4^{8}$ $4$ $4$ $24$ $( 1,19, 4,17)( 2,20, 3,18)( 5,24, 7,21)( 6,23, 8,22)( 9,28,11,26)(10,27,12,25)(13,31,15,30)(14,32,16,29)$
4G $4^{8}$ $4$ $4$ $24$ $( 1,27,32,23)( 2,28,31,24)( 3,26,30,21)( 4,25,29,22)( 5,13, 9,18)( 6,14,10,17)( 7,15,11,20)( 8,16,12,19)$

Malle's constant $a(G)$:     $1/16$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 4A 4B 4C1 4C-1 4D 4E 4F 4G
Size 1 1 1 1 2 2 2 2 2 2 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2B 2B 2B 2B 2C 2B 2B 2C
Type
32.29.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.29.1b R 1 1 1 1 −1 −1 −1 −1 1 1 −1 1 1 −1
32.29.1c R 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1
32.29.1d R 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
32.29.1e R 1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1
32.29.1f R 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 1
32.29.1g R 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 1 −1
32.29.1h R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
32.29.2a R 2 −2 −2 2 0 0 0 0 −2 2 0 0 0 0
32.29.2b R 2 −2 −2 2 0 0 0 0 2 −2 0 0 0 0
32.29.2c S 2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0
32.29.2d S 2 −2 2 −2 2 −2 0 0 0 0 0 0 0 0
32.29.2e1 C 2 2 −2 −2 0 0 −2i 2i 0 0 0 0 0 0
32.29.2e2 C 2 2 −2 −2 0 0 2i −2i 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed