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Properties

Label 32T183
Degree $32$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^3:C_8$

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Copy content magma:G := TransitiveGroup(32, 183);
 

Group invariants

Abstract group:  $C_2^3:C_8$
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:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $183$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $16$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,6,9,13,19,22,27,29)(2,5,10,14,20,21,28,30)(3,8,11,15,17,24,25,31)(4,7,12,16,18,23,26,32)$, $(1,32,10,8,19,16,28,24)(2,31,9,7,20,15,27,23)(3,30,12,6,17,14,26,22)(4,29,11,5,18,13,25,21)$, $(1,14,10,22,19,30,28,6)(2,13,9,21,20,29,27,5)(3,16,12,24,17,32,26,8)(4,15,11,23,18,31,25,7)$
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 4, $C_8$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_8:C_2$ x 2, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$, $C_8\times C_2$ x 6
$32$:  $C_2 \times (C_8:C_2)$, $C_2 \times (C_2^2:C_4)$, $C_2^2 : C_8$ x 4, 32T37

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4

Degree 8: $C_8$ x 2, $C_4\times C_2$, $C_8:C_2$ x 2, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4

Degree 16: $C_8\times C_2$, $C_2 \times (C_8:C_2)$, $C_2 \times (C_2^2:C_4)$, $C_2^2 : C_8$ x 4

Low degree siblings

32T183 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 $2^{16}$ $1$ $2$ $16$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,22)( 6,21)( 7,24)( 8,23)( 9,28)(10,27)(11,26)(12,25)(13,30)(14,29)(15,32)(16,31)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
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 $2^{16}$ $1$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,24)( 6,23)( 7,22)( 8,21)( 9,26)(10,25)(11,28)(12,27)(13,32)(14,31)(15,30)(16,29)$
2E $2^{16}$ $1$ $2$ $16$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,23)( 6,24)( 7,21)( 8,22)( 9,25)(10,26)(11,27)(12,28)(13,31)(14,32)(15,29)(16,30)$
2F $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2G $2^{16}$ $1$ $2$ $16$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,21)( 6,22)( 7,23)( 8,24)( 9,27)(10,28)(11,25)(12,26)(13,29)(14,30)(15,31)(16,32)$
2H $2^{16}$ $2$ $2$ $16$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5, 6)( 7, 8)( 9,27)(10,28)(11,25)(12,26)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$
2I $2^{8},1^{16}$ $2$ $2$ $8$ $( 5,22)( 6,21)( 7,24)( 8,23)(13,30)(14,29)(15,32)(16,31)$
2J $2^{16}$ $2$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5, 7)( 6, 8)( 9,26)(10,25)(11,28)(12,27)(13,15)(14,16)(21,23)(22,24)(29,31)(30,32)$
2K $2^{16}$ $2$ $2$ $16$ $( 1, 4)( 2, 3)( 5,23)( 6,24)( 7,21)( 8,22)( 9,12)(10,11)(13,31)(14,32)(15,29)(16,30)(17,20)(18,19)(25,28)(26,27)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1,26,19,12)( 2,25,20,11)( 3,28,17,10)( 4,27,18, 9)( 5,31,21,15)( 6,32,22,16)( 7,29,23,13)( 8,30,24,14)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1,12,19,26)( 2,11,20,25)( 3,10,17,28)( 4, 9,18,27)( 5,15,21,31)( 6,16,22,32)( 7,13,23,29)( 8,14,24,30)$
4B1 $4^{8}$ $1$ $4$ $24$ $( 1,11,19,25)( 2,12,20,26)( 3, 9,17,27)( 4,10,18,28)( 5,16,21,32)( 6,15,22,31)( 7,14,23,30)( 8,13,24,29)$
4B-1 $4^{8}$ $1$ $4$ $24$ $( 1,25,19,11)( 2,26,20,12)( 3,27,17, 9)( 4,28,18,10)( 5,32,21,16)( 6,31,22,15)( 7,30,23,14)( 8,29,24,13)$
4C1 $4^{8}$ $1$ $4$ $24$ $( 1,27,19, 9)( 2,28,20,10)( 3,25,17,11)( 4,26,18,12)( 5,30,21,14)( 6,29,22,13)( 7,32,23,16)( 8,31,24,15)$
4C-1 $4^{8}$ $1$ $4$ $24$ $( 1, 9,19,27)( 2,10,20,28)( 3,11,17,25)( 4,12,18,26)( 5,14,21,30)( 6,13,22,29)( 7,16,23,32)( 8,15,24,31)$
4D1 $4^{8}$ $1$ $4$ $24$ $( 1,10,19,28)( 2, 9,20,27)( 3,12,17,26)( 4,11,18,25)( 5,13,21,29)( 6,14,22,30)( 7,15,23,31)( 8,16,24,32)$
4D-1 $4^{8}$ $1$ $4$ $24$ $( 1,28,19,10)( 2,27,20, 9)( 3,26,17,12)( 4,25,18,11)( 5,29,21,13)( 6,30,22,14)( 7,31,23,15)( 8,32,24,16)$
4E1 $4^{8}$ $2$ $4$ $24$ $( 1, 9,19,27)( 2,10,20,28)( 3,11,17,25)( 4,12,18,26)( 5,29,21,13)( 6,30,22,14)( 7,31,23,15)( 8,32,24,16)$
4E-1 $4^{8}$ $2$ $4$ $24$ $( 1,27,19, 9)( 2,28,20,10)( 3,25,17,11)( 4,26,18,12)( 5,13,21,29)( 6,14,22,30)( 7,15,23,31)( 8,16,24,32)$
4F1 $4^{8}$ $2$ $4$ $24$ $( 1,12,19,26)( 2,11,20,25)( 3,10,17,28)( 4, 9,18,27)( 5,32,21,16)( 6,31,22,15)( 7,30,23,14)( 8,29,24,13)$
4F-1 $4^{8}$ $2$ $4$ $24$ $( 1,26,19,12)( 2,25,20,11)( 3,28,17,10)( 4,27,18, 9)( 5,16,21,32)( 6,15,22,31)( 7,14,23,30)( 8,13,24,29)$
8A1 $8^{4}$ $2$ $8$ $28$ $( 1,30,27,21,19,14, 9, 5)( 2,29,28,22,20,13,10, 6)( 3,32,25,23,17,16,11, 7)( 4,31,26,24,18,15,12, 8)$
8A-1 $8^{4}$ $2$ $8$ $28$ $( 1, 5, 9,14,19,21,27,30)( 2, 6,10,13,20,22,28,29)( 3, 7,11,16,17,23,25,32)( 4, 8,12,15,18,24,26,31)$
8A3 $8^{4}$ $2$ $8$ $28$ $( 1,21, 9,30,19, 5,27,14)( 2,22,10,29,20, 6,28,13)( 3,23,11,32,17, 7,25,16)( 4,24,12,31,18, 8,26,15)$
8A-3 $8^{4}$ $2$ $8$ $28$ $( 1,14,27, 5,19,30, 9,21)( 2,13,28, 6,20,29,10,22)( 3,16,25, 7,17,32,11,23)( 4,15,26, 8,18,31,12,24)$
8B1 $8^{4}$ $2$ $8$ $28$ $( 1,14,10,22,19,30,28, 6)( 2,13, 9,21,20,29,27, 5)( 3,16,12,24,17,32,26, 8)( 4,15,11,23,18,31,25, 7)$
8B-1 $8^{4}$ $2$ $8$ $28$ $( 1,21,28,13,19, 5,10,29)( 2,22,27,14,20, 6, 9,30)( 3,23,26,15,17, 7,12,31)( 4,24,25,16,18, 8,11,32)$
8B3 $8^{4}$ $2$ $8$ $28$ $( 1, 5,28,29,19,21,10,13)( 2, 6,27,30,20,22, 9,14)( 3, 7,26,31,17,23,12,15)( 4, 8,25,32,18,24,11,16)$
8B-3 $8^{4}$ $2$ $8$ $28$ $( 1,30,10, 6,19,14,28,22)( 2,29, 9, 5,20,13,27,21)( 3,32,12, 8,17,16,26,24)( 4,31,11, 7,18,15,25,23)$
8C1 $8^{4}$ $2$ $8$ $28$ $( 1,31,27,24,19,15, 9, 8)( 2,32,28,23,20,16,10, 7)( 3,29,25,22,17,13,11, 6)( 4,30,26,21,18,14,12, 5)$
8C-1 $8^{4}$ $2$ $8$ $28$ $( 1, 8, 9,15,19,24,27,31)( 2, 7,10,16,20,23,28,32)( 3, 6,11,13,17,22,25,29)( 4, 5,12,14,18,21,26,30)$
8C3 $8^{4}$ $2$ $8$ $28$ $( 1,24, 9,31,19, 8,27,15)( 2,23,10,32,20, 7,28,16)( 3,22,11,29,17, 6,25,13)( 4,21,12,30,18, 5,26,14)$
8C-3 $8^{4}$ $2$ $8$ $28$ $( 1,15,27, 8,19,31, 9,24)( 2,16,28, 7,20,32,10,23)( 3,13,25, 6,17,29,11,22)( 4,14,26, 5,18,30,12,21)$
8D1 $8^{4}$ $2$ $8$ $28$ $( 1,15,10,23,19,31,28, 7)( 2,16, 9,24,20,32,27, 8)( 3,13,12,21,17,29,26, 5)( 4,14,11,22,18,30,25, 6)$
8D-1 $8^{4}$ $2$ $8$ $28$ $( 1,24,28,16,19, 8,10,32)( 2,23,27,15,20, 7, 9,31)( 3,22,26,14,17, 6,12,30)( 4,21,25,13,18, 5,11,29)$
8D3 $8^{4}$ $2$ $8$ $28$ $( 1, 8,28,32,19,24,10,16)( 2, 7,27,31,20,23, 9,15)( 3, 6,26,30,17,22,12,14)( 4, 5,25,29,18,21,11,13)$
8D-3 $8^{4}$ $2$ $8$ $28$ $( 1,31,10, 7,19,15,28,23)( 2,32, 9, 8,20,16,27,24)( 3,29,12, 5,17,13,26,21)( 4,30,11, 6,18,14,25,22)$

Malle's constant $a(G)$:     $1/8$

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Character table

40 x 40 character table

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Regular extensions

Data not computed