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Properties

Label 32T3209
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, 3209);
 

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 $t$:  $3209$
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:  $(1,14,4,15)(2,13,3,16)(5,10,7,11)(6,9,8,12)(17,29,20,32)(18,30,19,31)(21,26,23,28)(22,25,24,27)$, $(1,9,4,12)(2,10,3,11)(5,13,7,16)(6,14,8,15)(17,19)(18,20)(21,22)(23,24)(25,28)(26,27)(29,30)(31,32)$, $(1,28,10,29,4,26,11,32)(2,27,9,30,3,25,12,31)(5,17,15,21,7,20,14,23)(6,18,16,22,8,19,13,24)$, $(1,3)(2,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,24)(18,23)(19,21)(20,22)(25,32)(26,31)(27,29)(28,30)$
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 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, 16T657 x 2

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, 32T3210 x 2, 32T3211 x 2, 32T3212 x 2, 32T3213 x 2, 32T4769 x 4, 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 $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)$
2B $2^{16}$ $2$ $2$ $16$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,10)( 6, 9)( 7,11)( 8,12)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)$
2C $2^{16}$ $2$ $2$ $16$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,10)( 6, 9)( 7,11)( 8,12)(17,29)(18,30)(19,31)(20,32)(21,26)(22,25)(23,28)(24,27)$
2D $2^{8},1^{16}$ $2$ $2$ $8$ $(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)$
2E $2^{16}$ $4$ $2$ $16$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,11)( 6,12)( 7,10)( 8, 9)(17,32)(18,31)(19,30)(20,29)(21,26)(22,25)(23,28)(24,27)$
2F $2^{16}$ $4$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,16)(10,15)(11,14)(12,13)(17,26)(18,25)(19,27)(20,28)(21,29)(22,30)(23,32)(24,31)$
2G $2^{8},1^{16}$ $4$ $2$ $8$ $( 5, 7)( 6, 8)( 9,12)(10,11)(21,23)(22,24)(25,27)(26,28)$
2H $2^{16}$ $4$ $2$ $16$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,15)( 8,16)(17,23)(18,24)(19,22)(20,21)(25,31)(26,32)(27,30)(28,29)$
2I $2^{8},1^{16}$ $8$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)(17,20)(18,19)(21,23)(22,24)$
2J $2^{16}$ $8$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,22)(18,21)(19,23)(20,24)(25,29)(26,30)(27,32)(28,31)$
2K $2^{16}$ $8$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,27)( 6,28)( 7,25)( 8,26)( 9,23)(10,24)(11,22)(12,21)(13,29)(14,30)(15,31)(16,32)$
2L $2^{16}$ $8$ $2$ $16$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5, 9)( 6,10)( 7,12)( 8,11)(17,27)(18,28)(19,26)(20,25)(21,31)(22,32)(23,30)(24,29)$
2M $2^{16}$ $8$ $2$ $16$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,13)( 6,14)( 7,16)( 8,15)(17,19)(18,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)$
2N $2^{16}$ $8$ $2$ $16$ $( 1,30)( 2,29)( 3,32)( 4,31)( 5,24)( 6,23)( 7,22)( 8,21)( 9,28)(10,27)(11,25)(12,26)(13,17)(14,18)(15,19)(16,20)$
4A $4^{4},2^{8}$ $4$ $4$ $20$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,22,20,24)(18,21,19,23)(25,32,27,29)(26,31,28,30)$
4B $4^{8}$ $4$ $4$ $24$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,13,12,16)(10,14,11,15)(17,26,20,28)(18,25,19,27)(21,32,23,29)(22,31,24,30)$
4C $4^{4},2^{8}$ $4$ $4$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,24,20,22)(18,23,19,21)(25,29,27,32)(26,30,28,31)$
4D $4^{8}$ $4$ $4$ $24$ $( 1,10, 4,11)( 2, 9, 3,12)( 5,15, 7,14)( 6,16, 8,13)(17,23,20,21)(18,24,19,22)(25,30,27,31)(26,29,28,32)$
4E $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7, 9)( 8,10)(17,27,20,25)(18,28,19,26)(21,30,23,31)(22,29,24,32)$
4F $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,15)( 6,16)( 7,14)( 8,13)(17,21,20,23)(18,22,19,24)(25,30,27,31)(26,29,28,32)$
4G $4^{8}$ $8$ $4$ $24$ $( 1,15, 4,14)( 2,16, 3,13)( 5,11, 7,10)( 6,12, 8, 9)(17,29,20,32)(18,30,19,31)(21,26,23,28)(22,25,24,27)$
4H $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,15)(14,16)(17,24,20,22)(18,23,19,21)(25,32,27,29)(26,31,28,30)$
4I $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,13,12,16)(10,14,11,15)(17,28)(18,27)(19,25)(20,26)(21,29)(22,30)(23,32)(24,31)$
4J $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,13, 4,16)( 2,14, 3,15)( 5, 9, 7,12)( 6,10, 8,11)(17,25)(18,26)(19,28)(20,27)(21,31)(22,32)(23,30)(24,29)$
4K $4^{8}$ $8$ $4$ $24$ $( 1,31, 4,30)( 2,32, 3,29)( 5,22, 7,24)( 6,21, 8,23)( 9,26,12,28)(10,25,11,27)(13,20,16,17)(14,19,15,18)$
4L $4^{8}$ $8$ $4$ $24$ $( 1,19, 4,18)( 2,20, 3,17)( 5,25, 7,27)( 6,26, 8,28)( 9,21,12,23)(10,22,11,24)(13,32,16,29)(14,31,15,30)$
4M $4^{8}$ $8$ $4$ $24$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,15,12,14)(10,16,11,13)(17,30,20,31)(18,29,19,32)(21,25,23,27)(22,26,24,28)$
4N $4^{4},2^{8}$ $16$ $4$ $20$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,25, 7,27)( 6,26, 8,28)( 9,23,12,21)(10,24,11,22)(13,29)(14,30)(15,31)(16,32)$
4O $4^{4},2^{8}$ $16$ $4$ $20$ $( 1,30)( 2,29)( 3,32)( 4,31)( 5,24, 7,22)( 6,23, 8,21)( 9,26,12,28)(10,25,11,27)(13,20)(14,19)(15,18)(16,17)$
4P $4^{8}$ $16$ $4$ $24$ $( 1,23,10,17)( 2,24, 9,18)( 3,22,12,19)( 4,21,11,20)( 5,32,14,26)( 6,31,13,25)( 7,29,15,28)( 8,30,16,27)$
4Q $4^{8}$ $16$ $4$ $24$ $( 1,26,10,32)( 2,25, 9,31)( 3,27,12,30)( 4,28,11,29)( 5,17,14,23)( 6,18,13,24)( 7,20,15,21)( 8,19,16,22)$
8A $8^{4}$ $16$ $8$ $28$ $( 1,21,10,17, 4,23,11,20)( 2,22, 9,18, 3,24,12,19)( 5,29,15,28, 7,32,14,26)( 6,30,16,27, 8,31,13,25)$
8B $8^{4}$ $16$ $8$ $28$ $( 1,28,11,29, 4,26,10,32)( 2,27,12,30, 3,25, 9,31)( 5,20,14,23, 7,17,15,21)( 6,19,13,24, 8,18,16,22)$

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