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Properties

Label 32T2627
Degree $32$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4^2:C_2^2$

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Copy content magma:G := TransitiveGroup(32, 2627);
 

Group invariants

Abstract group:  $D_4^2:C_2^2$
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 $t$:  $2627$
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:  $(1,10,31,17)(2,9,32,18)(3,11,29,20)(4,12,30,19)(5,15,26,21)(6,16,25,22)(7,13,28,23)(8,14,27,24)$, $(1,5,3,8)(2,6,4,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,32,28,30)(26,31,27,29)$, $(1,31,3,29)(2,32,4,30)(5,26,8,27)(6,25,7,28)(9,23,12,22)(10,24,11,21)(13,19,16,18)(14,20,15,17)$, $(1,16)(2,15)(3,13)(4,14)(5,9)(6,10)(7,11)(8,12)(17,25)(18,26)(19,27)(20,28)(21,32)(22,31)(23,29)(24,30)$
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
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T105
$128$:  16T245

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, 16T529 x 2

Low degree siblings

16T529 x 4, 16T653 x 8, 32T2627, 32T2628 x 8, 32T2629 x 4, 32T2630 x 2, 32T2631 x 2, 32T3191 x 4, 32T3192 x 4, 32T3193 x 4, 32T3194 x 4, 32T3195 x 4, 32T3196 x 4, 32T4770 x 8

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

Malle's constant $a(G)$:     $1/8$

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Character table

34 x 34 character table

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Regular extensions

Data not computed