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Properties

Label 32T4050
32T4050 1 2 1->2 1->2 28 1->28 27 2->27 3 4 3->4 26 3->26 25 4->25 5 8 5->8 23 5->23 6 7 6->7 24 6->24 7->5 21 7->21 8->6 22 8->22 9 12 9->12 17 9->17 10 11 10->11 18 10->18 11->9 19 11->19 12->10 20 12->20 13 14 13->14 32 13->32 31 14->31 15 16 15->16 15->16 30 15->30 29 16->29 17->8 17->22 17->27 18->7 18->21 18->28 19->6 19->24 19->26 20->5 20->23 20->25 21->10 21->32 22->9 22->31 23->12 23->29 24->11 24->30 25->15 25->17 25->29 26->16 26->18 26->30 27->13 27->20 27->31 28->14 28->19 28->32 29->3 29->22 30->4 30->21 31->1 31->23 32->2 32->24
Degree $32$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^5.D_4$

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Copy content magma:G := TransitiveGroup(32, 4050);
 

Group invariants

Abstract group:  $C_2^5.D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $256=2^{8}$
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 $t$:  $4050$
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:  $(1,28,14,31)(2,27,13,32)(3,26,16,29)(4,25,15,30)(5,23,12,20)(6,24,11,19)(7,21,10,18)(8,22,9,17)$, $(1,2)(3,4)(13,14)(15,16)(17,22)(18,21)(19,24)(20,23)(25,29)(26,30)(27,31)(28,32)$, $(1,2)(5,8,6,7)(9,12,10,11)(15,16)(17,27,20,25)(18,28,19,26)(21,32,24,30)(22,31,23,29)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 8, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$, $C_2^3 : C_4 $ x 4, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76 x 2, 32T239
$128$:  16T230, 16T234, 16T235

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Degree 8: $C_2^2 \wr C_2$, $C_2^3 : C_4 $ x 2

Degree 16: 16T143, 16T230, 16T297

Low degree siblings

32T4050 x 7, 32T4105 x 4, 32T4110 x 4, 32T4370 x 2, 32T4371 x 2, 32T4376 x 4, 32T5322 x 2, 32T5325 x 2, 32T5332 x 4, 32T5342 x 4, 32T5344 x 4

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 $2^{16}$ $1$ $2$ $16$ $( 1,15)( 2,16)( 3,13)( 4,14)( 5, 9)( 6,10)( 7,11)( 8,12)(17,23)(18,24)(19,21)(20,22)(25,31)(26,32)(27,29)(28,30)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2C $2^{16}$ $1$ $2$ $16$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,10)( 6, 9)( 7,12)( 8,11)(17,24)(18,23)(19,22)(20,21)(25,32)(26,31)(27,30)(28,29)$
2D $2^{8},1^{16}$ $2$ $2$ $8$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,10)( 6, 9)( 7,12)( 8,11)$
2E $2^{16}$ $2$ $2$ $16$ $( 1,15)( 2,16)( 3,13)( 4,14)( 5, 9)( 6,10)( 7,11)( 8,12)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2F $2^{16}$ $2$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2G $2^{16}$ $2$ $2$ $16$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,11)( 6,12)( 7, 9)( 8,10)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)$
2H $2^{12},1^{8}$ $4$ $2$ $12$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,11)( 6,12)( 7, 9)( 8,10)(17,18)(19,20)(21,22)(23,24)$
2I $2^{16}$ $4$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,23)(18,24)(19,21)(20,22)(25,32)(26,31)(27,30)(28,29)$
2J $2^{16}$ $4$ $2$ $16$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,11)( 6,12)( 7, 9)( 8,10)(17,23)(18,24)(19,21)(20,22)(25,32)(26,31)(27,30)(28,29)$
2K $2^{16}$ $4$ $2$ $16$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,11)( 6,12)( 7, 9)( 8,10)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2L $2^{12},1^{8}$ $4$ $2$ $12$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)$
2M $2^{16}$ $4$ $2$ $16$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,14)( 6,13)( 7,16)( 8,15)(17,27)(18,28)(19,25)(20,26)(21,31)(22,32)(23,29)(24,30)$
2N $2^{16}$ $4$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,13)(10,14)(11,15)(12,16)(17,30)(18,29)(19,32)(20,31)(21,26)(22,25)(23,28)(24,27)$
2O $2^{16}$ $8$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,13)(10,14)(11,15)(12,16)(17,27)(18,28)(19,25)(20,26)(21,31)(22,32)(23,29)(24,30)$
2P $2^{16}$ $16$ $2$ $16$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,28)( 6,27)( 7,26)( 8,25)( 9,30)(10,29)(11,32)(12,31)(13,19)(14,20)(15,17)(16,18)$
4A $4^{8}$ $4$ $4$ $24$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,15, 6,16)( 7,13, 8,14)(17,28,18,27)(19,26,20,25)(21,32,22,31)(23,30,24,29)$
4B $4^{8}$ $4$ $4$ $24$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,16,10,15)(11,14,12,13)(17,29,18,30)(19,31,20,32)(21,25,22,26)(23,27,24,28)$
4C $4^{8}$ $8$ $4$ $24$ $( 1,21, 4,24)( 2,22, 3,23)( 5,25, 8,28)( 6,26, 7,27)( 9,31,12,30)(10,32,11,29)(13,17,16,20)(14,18,15,19)$
4D $4^{8}$ $8$ $4$ $24$ $( 1,20,13,24)( 2,19,14,23)( 3,18,15,22)( 4,17,16,21)( 5,32,11,28)( 6,31,12,27)( 7,30, 9,26)( 8,29,10,25)$
4E $4^{8}$ $8$ $4$ $24$ $( 1,23, 4,22)( 2,24, 3,21)( 5,27, 8,26)( 6,28, 7,25)( 9,29,12,32)(10,30,11,31)(13,19,16,18)(14,20,15,17)$
4F $4^{8}$ $8$ $4$ $24$ $( 1,18,13,22)( 2,17,14,21)( 3,20,15,24)( 4,19,16,23)( 5,30,11,26)( 6,29,12,25)( 7,32, 9,28)( 8,31,10,27)$
4G $4^{8}$ $8$ $4$ $24$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,16,10,15)(11,14,12,13)(17,28,18,27)(19,26,20,25)(21,32,22,31)(23,30,24,29)$
4H1 $4^{6},2^{2},1^{4}$ $8$ $4$ $20$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,15, 8,13)( 6,16, 7,14)(17,20,18,19)(21,23,22,24)(27,28)(29,30)$
4H-1 $4^{6},2^{2},1^{4}$ $8$ $4$ $20$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,13, 8,15)( 6,14, 7,16)(17,19,18,20)(21,24,22,23)(27,28)(29,30)$
4I1 $4^{6},2^{2},1^{4}$ $8$ $4$ $20$ $( 1, 7, 3, 6)( 2, 8, 4, 5)( 9,16,12,14)(10,15,11,13)(17,20,18,19)(21,23,22,24)(27,28)(29,30)$
4I-1 $4^{6},2^{2},1^{4}$ $8$ $4$ $20$ $( 1, 6, 3, 7)( 2, 5, 4, 8)( 9,14,12,16)(10,13,11,15)(17,19,18,20)(21,24,22,23)(27,28)(29,30)$
4J1 $4^{6},2^{4}$ $8$ $4$ $22$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,15, 8,13)( 6,16, 7,14)(17,21,18,22)(19,24,20,23)(25,32)(26,31)(27,29)(28,30)$
4J-1 $4^{6},2^{4}$ $8$ $4$ $22$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,13, 8,15)( 6,14, 7,16)(17,22,18,21)(19,23,20,24)(25,32)(26,31)(27,29)(28,30)$
4K1 $4^{6},2^{4}$ $8$ $4$ $22$ $( 1, 7, 3, 6)( 2, 8, 4, 5)( 9,16,12,14)(10,15,11,13)(17,21,18,22)(19,24,20,23)(25,32)(26,31)(27,29)(28,30)$
4K-1 $4^{6},2^{4}$ $8$ $4$ $22$ $( 1, 6, 3, 7)( 2, 5, 4, 8)( 9,14,12,16)(10,13,11,15)(17,22,18,21)(19,23,20,24)(25,32)(26,31)(27,29)(28,30)$
4L $4^{8}$ $16$ $4$ $24$ $( 1,18,16,23)( 2,17,15,24)( 3,20,14,21)( 4,19,13,22)( 5,29,10,28)( 6,30, 9,27)( 7,31,12,26)( 8,32,11,25)$
4M1 $4^{8}$ $16$ $4$ $24$ $( 1,21,12,31)( 2,22,11,32)( 3,24,10,30)( 4,23, 9,29)( 5,27,14,17)( 6,28,13,18)( 7,26,16,20)( 8,25,15,19)$
4M-1 $4^{8}$ $16$ $4$ $24$ $( 1,24, 9,31)( 2,23,10,32)( 3,21,11,30)( 4,22,12,29)( 5,25,15,18)( 6,26,16,17)( 7,28,13,19)( 8,27,14,20)$
4N1 $4^{8}$ $16$ $4$ $24$ $( 1,20, 7,31)( 2,19, 8,32)( 3,17, 5,30)( 4,18, 6,29)( 9,28,13,23)(10,27,14,24)(11,25,15,22)(12,26,16,21)$
4N-1 $4^{8}$ $16$ $4$ $24$ $( 1,17, 6,31)( 2,18, 5,32)( 3,20, 8,30)( 4,19, 7,29)( 9,26,16,24)(10,25,15,23)(11,27,14,21)(12,28,13,22)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

37 x 37 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed