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Properties

Label 32T298
32T298 1 5 1->5 7 1->7 10 1->10 2 6 2->6 8 2->8 9 2->9 3 3->5 3->7 11 3->11 4 4->6 4->8 12 4->12 5->6 29 9->29 32 9->32 30 10->30 31 10->31 11->29 11->32 12->30 12->31 13 26 13->26 27 13->27 13->32 14 25 14->25 28 14->28 14->31 15 15->26 15->27 15->30 16 16->25 16->28 16->29 17 21 17->21 24 17->24 17->26 18 22 18->22 23 18->23 18->25 19 19->21 19->24 19->28 20 20->22 20->23 20->27 23->24
Degree $32$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_{16}:C_2$

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Copy content magma:G := TransitiveGroup(32, 298);
 

Group invariants

Abstract group:  $D_{16}:C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $64=2^{6}$
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:  $4$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $298$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,10)(2,9)(3,11)(4,12)(5,6)(13,32)(14,31)(15,30)(16,29)(17,26)(18,25)(19,28)(20,27)(23,24)$, $(1,7)(2,8)(3,5)(4,6)(9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,26)(16,25)(17,21)(18,22)(19,24)(20,23)$, $(1,5)(2,6)(3,7)(4,8)(9,29)(10,30)(11,32)(12,31)(13,26)(14,25)(15,27)(16,28)(17,24)(18,23)(19,21)(20,22)$
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, $C_2^3$
$16$:  $D_{8}$ x 2, $D_4\times C_2$
$32$:  16T29

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_{8}$ x 2, $D_4\times C_2$

Degree 16: 16T29

Low degree siblings

16T134 x 2, 32T145, 32T146 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 $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B $2^{16}$ $2$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2C $2^{16}$ $8$ $2$ $16$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,29)(10,30)(11,32)(12,31)(13,26)(14,25)(15,27)(16,28)(17,24)(18,23)(19,21)(20,22)$
2D $2^{14},1^{4}$ $8$ $2$ $14$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 7, 8)(13,31)(14,32)(15,29)(16,30)(17,25)(18,26)(19,27)(20,28)(21,22)$
2E $2^{16}$ $8$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,26)(16,25)(17,21)(18,22)(19,24)(20,23)$
4A $4^{8}$ $2$ $4$ $24$ $( 1,17, 2,18)( 3,19, 4,20)( 5,23, 6,24)( 7,22, 8,21)( 9,25,10,26)(11,28,12,27)(13,30,14,29)(15,31,16,32)$
4B $4^{8}$ $2$ $4$ $24$ $( 1,19, 2,20)( 3,17, 4,18)( 5,22, 6,21)( 7,23, 8,24)( 9,28,10,27)(11,25,12,26)(13,31,14,32)(15,30,16,29)$
4C $4^{8}$ $8$ $4$ $24$ $( 1,12, 2,11)( 3, 9, 4,10)( 5, 8, 6, 7)(13,29,14,30)(15,31,16,32)(17,27,18,28)(19,25,20,26)(21,24,22,23)$
8A1 $8^{4}$ $2$ $8$ $28$ $( 1, 9,19,28, 2,10,20,27)( 3,11,17,25, 4,12,18,26)( 5,15,22,30, 6,16,21,29)( 7,13,23,31, 8,14,24,32)$
8A3 $8^{4}$ $2$ $8$ $28$ $( 1,28,20, 9, 2,27,19,10)( 3,25,18,11, 4,26,17,12)( 5,30,21,15, 6,29,22,16)( 7,31,24,13, 8,32,23,14)$
8B $8^{4}$ $4$ $8$ $28$ $( 1,11,19,25, 2,12,20,26)( 3, 9,17,28, 4,10,18,27)( 5,13,22,31, 6,14,21,32)( 7,15,23,30, 8,16,24,29)$
16A1 $16^{2}$ $4$ $16$ $30$ $( 1, 5, 9,15,19,22,28,30, 2, 6,10,16,20,21,27,29)( 3, 8,11,14,17,24,25,32, 4, 7,12,13,18,23,26,31)$
16A3 $16^{2}$ $4$ $16$ $30$ $( 1,22,10,29,19, 6,27,15, 2,21, 9,30,20, 5,28,16)( 3,24,12,31,17, 7,26,14, 4,23,11,32,18, 8,25,13)$
16B1 $16^{2}$ $4$ $16$ $30$ $( 1, 8,10,13,19,24,27,31, 2, 7, 9,14,20,23,28,32)( 3, 5,12,16,17,22,26,29, 4, 6,11,15,18,21,25,30)$
16B3 $16^{2}$ $4$ $16$ $30$ $( 1,24, 9,32,19, 7,28,13, 2,23,10,31,20, 8,27,14)( 3,22,11,30,17, 6,25,16, 4,21,12,29,18, 5,26,15)$

Malle's constant $a(G)$:     $1/14$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 4A 4B 4C 8A1 8A3 8B 16A1 16A3 16B1 16B3
Size 1 1 2 8 8 8 2 2 8 2 2 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2A 2A 2A 4B 4B 4B 8A1 8A3 8A3 8A1
Type
64.190.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.190.1b R 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1
64.190.1c R 1 1 −1 −1 1 1 −1 1 −1 1 1 −1 1 −1 −1 1
64.190.1d R 1 1 −1 1 −1 −1 −1 1 1 1 1 −1 1 −1 −1 1
64.190.1e R 1 1 −1 1 −1 1 −1 1 −1 1 1 −1 −1 1 1 −1
64.190.1f R 1 1 1 −1 −1 −1 1 1 −1 1 1 1 1 1 1 1
64.190.1g R 1 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
64.190.1h R 1 1 1 1 1 −1 1 1 −1 1 1 1 −1 −1 −1 −1
64.190.2a R 2 2 −2 0 0 0 −2 2 0 −2 −2 2 0 0 0 0
64.190.2b R 2 2 2 0 0 0 2 2 0 −2 −2 −2 0 0 0 0
64.190.2c1 R 2 2 −2 0 0 0 2 −2 0 0 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8
64.190.2c2 R 2 2 −2 0 0 0 2 −2 0 0 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8
64.190.2d1 R 2 2 2 0 0 0 −2 −2 0 0 0 0 −ζ8−1−ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8−1+ζ8
64.190.2d2 R 2 2 2 0 0 0 −2 −2 0 0 0 0 ζ8−1+ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−1−ζ8
64.190.4a1 R 4 −4 0 0 0 0 0 0 0 −2ζ8−1−2ζ8 2ζ8−1+2ζ8 0 0 0 0 0
64.190.4a2 R 4 −4 0 0 0 0 0 0 0 2ζ8−1+2ζ8 −2ζ8−1−2ζ8 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed