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Properties

Label 32T1479
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2\times Q_{16}):C_2^2$

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Copy content magma:G := TransitiveGroup(32, 1479);
 

Group invariants

Abstract group:  $(C_2\times Q_{16}):C_2^2$
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 $t$:  $1479$
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:  $(1,7,3,5)(2,8,4,6)(9,13,11,15)(10,14,12,16)(17,29,19,31)(18,30,20,32)(21,27,23,25)(22,28,24,26)$, $(1,30,14,25,3,32,16,27)(2,29,13,26,4,31,15,28)(5,23,12,18,7,21,10,20)(6,24,11,17,8,22,9,19)$, $(1,6,15,10,3,8,13,12)(2,5,16,9,4,7,14,11)(17,30,23,28,19,32,21,26)(18,29,24,27,20,31,22,25)$, $(1,26,14,31,3,28,16,29)(2,25,13,32,4,27,15,30)(5,17,12,22,7,19,10,24)(6,18,11,21,8,20,9,23)$
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 12, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 18, $C_2^4$
$32$:  $C_2^2 \wr C_2$ x 4, $C_2^2 \times D_4$ x 3
$64$:  16T105, 16T116 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 6

Degree 8: $D_4$, $D_4\times C_2$ x 2, $C_2^2 \wr C_2$ x 4

Degree 16: 16T39, 16T116 x 2

Low degree siblings

32T1385 x 8, 32T1479 x 3

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