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Properties

Label 32T1167
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2\times D_4).D_4$

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Copy content magma:G := TransitiveGroup(32, 1167);
 

Group invariants

Abstract group:  $(C_2\times D_4).D_4$
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 $t$:  $1167$
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:  $(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$, $(1,26,4,27)(2,25,3,28)(5,22,8,23)(6,21,7,24)(9,20,12,17)(10,19,11,18)(13,30,16,31)(14,29,15,32)$, $(1,28,16,31,3,26,14,29)(2,27,15,32,4,25,13,30)(5,19,10,22,7,17,12,24)(6,20,9,21,8,18,11,23)$, $(1,18,3,20)(2,17,4,19)(5,29,7,31)(6,30,8,32)(9,27,11,25)(10,28,12,26)(13,22,15,24)(14,21,16,23)$
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, $Q_8:C_2$ x 4, $C_2^4$
$32$:  $C_2 \times (C_4\times C_2):C_2$ x 2, $C_2^2 \times D_4$, 16T37 x 4
$64$:  16T80, 16T116, 32T205

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$, $Q_8:C_2$ x 2

Degree 16: 16T37, 16T80, 16T116

Low degree siblings

32T1167 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 $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
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 $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2D $2^{8},1^{16}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2E $2^{16}$ $2$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2F $2^{16}$ $4$ $2$ $16$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,28)( 6,27)( 7,26)( 8,25)( 9,32)(10,31)(11,30)(12,29)(13,17)(14,18)(15,19)(16,20)$
2G $2^{16}$ $4$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,29)( 6,30)( 7,31)( 8,32)( 9,27)(10,28)(11,25)(12,26)(13,22)(14,21)(15,24)(16,23)$
2H $2^{14},1^{4}$ $8$ $2$ $14$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5, 8)( 6, 7)( 9,10)(11,12)(17,24)(18,23)(19,22)(20,21)(25,27)(26,28)$
2I $2^{14},1^{4}$ $8$ $2$ $14$ $( 1,15)( 2,16)( 3,13)( 4,14)( 5, 6)( 7, 8)( 9,12)(10,11)(17,24)(18,23)(19,22)(20,21)(25,27)(26,28)$
4A $4^{8}$ $2$ $4$ $24$ $( 1,14, 3,16)( 2,13, 4,15)( 5,12, 7,10)( 6,11, 8, 9)(17,24,19,22)(18,23,20,21)(25,30,27,32)(26,29,28,31)$
4B $4^{8}$ $2$ $4$ $24$ $( 1,13, 3,15)( 2,14, 4,16)( 5,11, 7, 9)( 6,12, 8,10)(17,23,19,21)(18,24,20,22)(25,29,27,31)(26,30,28,32)$
4C $4^{8}$ $2$ $4$ $24$ $( 1,15, 3,13)( 2,16, 4,14)( 5, 9, 7,11)( 6,10, 8,12)(17,23,19,21)(18,24,20,22)(25,29,27,31)(26,30,28,32)$
4D $4^{8}$ $2$ $4$ $24$ $( 1,16, 3,14)( 2,15, 4,13)( 5,10, 7,12)( 6, 9, 8,11)(17,24,19,22)(18,23,20,21)(25,30,27,32)(26,29,28,31)$
4E $4^{8}$ $4$ $4$ $24$ $( 1,20, 3,18)( 2,19, 4,17)( 5,31, 7,29)( 6,32, 8,30)( 9,25,11,27)(10,26,12,28)(13,24,15,22)(14,23,16,21)$
4F $4^{8}$ $4$ $4$ $24$ $( 1,21, 3,23)( 2,22, 4,24)( 5,26, 7,28)( 6,25, 8,27)( 9,30,11,32)(10,29,12,31)(13,19,15,17)(14,20,16,18)$
4G1 $4^{8}$ $4$ $4$ $24$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,16,10,15)(11,14,12,13)(17,30,18,29)(19,32,20,31)(21,28,22,27)(23,26,24,25)$
4G-1 $4^{8}$ $4$ $4$ $24$ $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,15,10,16)(11,13,12,14)(17,29,18,30)(19,31,20,32)(21,27,22,28)(23,25,24,26)$
4H1 $4^{8}$ $4$ $4$ $24$ $( 1,26, 2,25)( 3,28, 4,27)( 5,22, 6,21)( 7,24, 8,23)( 9,20,10,19)(11,18,12,17)(13,30,14,29)(15,32,16,31)$
4H-1 $4^{8}$ $4$ $4$ $24$ $( 1,29, 2,30)( 3,31, 4,32)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)(13,27,14,28)(15,25,16,26)$
4I1 $4^{8}$ $4$ $4$ $24$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)(17,30,18,29)(19,32,20,31)(21,28,22,27)(23,26,24,25)$
4I-1 $4^{8}$ $4$ $4$ $24$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)(17,29,18,30)(19,31,20,32)(21,27,22,28)(23,25,24,26)$
4J1 $4^{8}$ $4$ $4$ $24$ $( 1,28, 4,25)( 2,27, 3,26)( 5,24, 8,21)( 6,23, 7,22)( 9,18,12,19)(10,17,11,20)(13,32,16,29)(14,31,15,30)$
4J-1 $4^{8}$ $4$ $4$ $24$ $( 1,31, 4,30)( 2,32, 3,29)( 5,19, 8,18)( 6,20, 7,17)( 9,23,12,22)(10,24,11,21)(13,25,16,28)(14,26,15,27)$
4K $4^{8}$ $8$ $4$ $24$ $( 1,18, 2,17)( 3,20, 4,19)( 5,26, 6,25)( 7,28, 8,27)( 9,32,10,31)(11,30,12,29)(13,24,14,23)(15,22,16,21)$
4L $4^{8}$ $8$ $4$ $24$ $( 1,20, 4,17)( 2,19, 3,18)( 5,28, 8,25)( 6,27, 7,26)( 9,30,12,31)(10,29,11,32)(13,22,16,23)(14,21,15,24)$
8A1 $8^{4}$ $4$ $8$ $28$ $( 1, 5,15, 9, 3, 7,13,11)( 2, 6,16,10, 4, 8,14,12)(17,29,23,27,19,31,21,25)(18,30,24,28,20,32,22,26)$
8A3 $8^{4}$ $4$ $8$ $28$ $( 1, 7,15,11, 3, 5,13, 9)( 2, 8,16,12, 4, 6,14,10)(17,31,23,25,19,29,21,27)(18,32,24,26,20,30,22,28)$
8B1 $8^{4}$ $4$ $8$ $28$ $( 1, 7,15,11, 3, 5,13, 9)( 2, 8,16,12, 4, 6,14,10)(17,29,23,27,19,31,21,25)(18,30,24,28,20,32,22,26)$
8B-1 $8^{4}$ $4$ $8$ $28$ $( 1, 5,15, 9, 3, 7,13,11)( 2, 6,16,10, 4, 8,14,12)(17,31,23,25,19,29,21,27)(18,32,24,26,20,30,22,28)$
8C $8^{4}$ $8$ $8$ $28$ $( 1,27,16,32, 3,25,14,30)( 2,28,15,31, 4,26,13,29)( 5,20,10,21, 7,18,12,23)( 6,19, 9,22, 8,17,11,24)$
8D $8^{4}$ $8$ $8$ $28$ $( 1,25,14,32, 3,27,16,30)( 2,26,13,31, 4,28,15,29)( 5,18,12,21, 7,20,10,23)( 6,17,11,22, 8,19, 9,24)$

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