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Properties

Label 32T4769
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, 4769);
 

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 $t$:  $4769$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,19,7,16,2,20,8,15)(3,18,5,14,4,17,6,13)(9,25,31,22,10,26,32,21)(11,27,30,23,12,28,29,24)$, $(1,12,2,11)(3,10,4,9)(5,31,6,32)(7,30,8,29)(13,25,14,26)(15,27,16,28)(17,22,18,21)(19,24,20,23)$, $(1,18,29,21,2,17,30,22)(3,19,31,23,4,20,32,24)(5,15,10,27,6,16,9,28)(7,13,11,26,8,14,12,25)$, $(1,5,11,32)(2,6,12,31)(3,8,10,29)(4,7,9,30)(13,28,21,19)(14,27,22,20)(15,26,23,17)(16,25,24,18)$
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 2

Degree 8: $D_4\times C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Degree 16: 16T271

Low degree siblings

16T636 x 4, 16T652 x 4, 16T657 x 4, 32T3110 x 4, 32T3111 x 2, 32T3112 x 2, 32T3113 x 4, 32T3114 x 2, 32T3115 x 2, 32T3116 x 2, 32T3185 x 2, 32T3186 x 2, 32T3187 x 4, 32T3188 x 2, 32T3189 x 2, 32T3190 x 2, 32T3209 x 2, 32T3210 x 2, 32T3211 x 2, 32T3212 x 2, 32T3213 x 2, 32T4769 x 3, 32T7442 x 2, 32T7470 x 2

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

Malle's constant $a(G)$:     $1/12$

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Character table

34 x 34 character table

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Regular extensions

Data not computed