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Properties

Label 32T1334
32T1334 1 2 1->2 3 1->3 15 1->15 32 1->32 4 2->4 16 2->16 31 2->31 3->4 13 3->13 30 3->30 14 4->14 29 4->29 5 6 5->6 8 5->8 10 5->10 26 5->26 7 6->7 9 6->9 25 6->25 7->8 11 7->11 27 7->27 12 8->12 28 8->28 9->10 17 9->17 18 10->18 11->12 20 11->20 19 12->19 13->14 24 13->24 23 14->23 15->16 21 15->21 22 16->22 17->14 17->19 17->20 17->32 18->13 18->19 18->20 18->31 19->15 19->29 20->16 20->30 21->9 21->23 21->24 21->28 22->10 22->23 22->24 22->27 23->12 23->25 24->11 24->26 25->1 25->28 26->2 26->27 27->3 28->4 29->6 29->32 30->5 30->31 31->7 32->8
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4.D_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(32, 1334);
 

Group invariants

Abstract group:  $C_2^4.D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $128=2^{7}$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1334$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,32,8,28,4,29,6,25)(2,31,7,27,3,30,5,26)(9,17,14,23,12,19,15,21)(10,18,13,24,11,20,16,22)$, $(1,15)(2,16)(3,13)(4,14)(5,10)(6,9)(7,11)(8,12)(17,32)(18,31)(19,29)(20,30)(21,28)(22,27)(23,25)(24,26)$, $(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)$, $(1,2)(3,4)(5,6)(7,8)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$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, $C_2^2 \times D_4$, 16T44 x 2
$64$:  16T87, 16T100, 32T282

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $Q_8:C_2^2$ x 2

Degree 16: 16T47, 16T100, 16T119

Low degree siblings

32T1334 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)(29,31)(30,32)$
2C $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2D $2^{16}$ $2$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)$
2E $2^{8},1^{16}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)(25,27)(26,28)(29,31)(30,32)$
2F $2^{12},1^{8}$ $4$ $2$ $12$ $( 9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2G $2^{12},1^{8}$ $4$ $2$ $12$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)$
2H $2^{16}$ $8$ $2$ $16$ $( 1,22)( 2,21)( 3,23)( 4,24)( 5,17)( 6,18)( 7,19)( 8,20)( 9,25)(10,26)(11,27)(12,28)(13,30)(14,29)(15,32)(16,31)$
2I $2^{16}$ $8$ $2$ $16$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5, 9)( 6,10)( 7,12)( 8,11)(17,31)(18,32)(19,30)(20,29)(21,27)(22,28)(23,26)(24,25)$
4A $4^{8}$ $2$ $4$ $24$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,13,12,16)(10,14,11,15)(17,24,19,22)(18,23,20,21)(25,31,28,30)(26,32,27,29)$
4B $4^{8}$ $2$ $4$ $24$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,14,12,15)(10,13,11,16)(17,23,19,21)(18,24,20,22)(25,32,28,29)(26,31,27,30)$
4C1 $4^{8}$ $2$ $4$ $24$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,15,12,14)(10,16,11,13)(17,24,19,22)(18,23,20,21)(25,32,28,29)(26,31,27,30)$
4C-1 $4^{8}$ $2$ $4$ $24$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,14,12,15)(10,13,11,16)(17,22,19,24)(18,21,20,23)(25,29,28,32)(26,30,27,31)$
4D1 $4^{8}$ $2$ $4$ $24$ $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,13,12,16)(10,14,11,15)(17,24,19,22)(18,23,20,21)(25,29,28,32)(26,30,27,31)$
4D-1 $4^{8}$ $2$ $4$ $24$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,16,12,13)(10,15,11,14)(17,22,19,24)(18,21,20,23)(25,32,28,29)(26,31,27,30)$
4E1 $4^{8}$ $2$ $4$ $24$ $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,14,12,15)(10,13,11,16)(17,22,19,24)(18,21,20,23)(25,31,28,30)(26,32,27,29)$
4E-1 $4^{8}$ $2$ $4$ $24$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,15,12,14)(10,16,11,13)(17,24,19,22)(18,23,20,21)(25,30,28,31)(26,29,27,32)$
4F $4^{8}$ $8$ $4$ $24$ $( 1,17, 3,20)( 2,18, 4,19)( 5,24, 8,21)( 6,23, 7,22)( 9,29,11,31)(10,30,12,32)(13,27,15,25)(14,28,16,26)$
4G $4^{8}$ $8$ $4$ $24$ $( 1, 9, 3,11)( 2,10, 4,12)( 5,13, 8,15)( 6,14, 7,16)(17,26,20,28)(18,25,19,27)(21,31,24,29)(22,32,23,30)$
4H $4^{8}$ $8$ $4$ $24$ $( 1,24, 2,23)( 3,21, 4,22)( 5,19, 6,20)( 7,17, 8,18)( 9,25,10,26)(11,27,12,28)(13,30,14,29)(15,32,16,31)$
4I $4^{8}$ $8$ $4$ $24$ $( 1,17, 4,19)( 2,18, 3,20)( 5,24, 7,22)( 6,23, 8,21)( 9,32,12,29)(10,31,11,30)(13,26,16,27)(14,25,15,28)$
4J $4^{8}$ $8$ $4$ $24$ $( 1,13, 4,16)( 2,14, 3,15)( 5,12, 7, 9)( 6,11, 8,10)(17,29,19,32)(18,30,20,31)(21,25,23,28)(22,26,24,27)$
4K $4^{8}$ $8$ $4$ $24$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,13, 6,14)( 7,16, 8,15)(17,25,18,26)(19,28,20,27)(21,32,22,31)(23,29,24,30)$
8A1 $8^{4}$ $4$ $8$ $28$ $( 1,28, 7,30, 4,25, 5,31)( 2,27, 8,29, 3,26, 6,32)( 9,24,13,19,12,22,16,17)(10,23,14,20,11,21,15,18)$
8A3 $8^{4}$ $4$ $8$ $28$ $( 1,25, 7,31, 4,28, 5,30)( 2,26, 8,32, 3,27, 6,29)( 9,22,13,17,12,24,16,19)(10,21,14,18,11,23,15,20)$
8B1 $8^{4}$ $4$ $8$ $28$ $( 1,31, 8,27, 4,30, 6,26)( 2,32, 7,28, 3,29, 5,25)( 9,18,14,24,12,20,15,22)(10,17,13,23,11,19,16,21)$
8B-1 $8^{4}$ $4$ $8$ $28$ $( 1,30, 8,26, 4,31, 6,27)( 2,29, 7,25, 3,32, 5,28)( 9,20,14,22,12,18,15,24)(10,19,13,21,11,17,16,23)$
8C1 $8^{4}$ $4$ $8$ $28$ $( 1,32, 5,26, 4,29, 7,27)( 2,31, 6,25, 3,30, 8,28)( 9,19,16,24,12,17,13,22)(10,20,15,23,11,18,14,21)$
8C3 $8^{4}$ $4$ $8$ $28$ $( 1,29, 5,27, 4,32, 7,26)( 2,30, 6,28, 3,31, 8,25)( 9,17,16,22,12,19,13,24)(10,18,15,21,11,20,14,23)$
8D1 $8^{4}$ $4$ $8$ $28$ $( 1,26, 6,30, 4,27, 8,31)( 2,25, 5,29, 3,28, 7,32)( 9,23,15,17,12,21,14,19)(10,24,16,18,11,22,13,20)$
8D-1 $8^{4}$ $4$ $8$ $28$ $( 1,27, 6,31, 4,26, 8,30)( 2,28, 5,32, 3,25, 7,29)( 9,21,15,19,12,23,14,17)(10,22,16,20,11,24,13,18)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

32 x 32 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed