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Properties

Label 32T3213
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, 3213);
 

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 $t$:  $3213$
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:  $(1,24,3,21)(2,23,4,22)(5,31,7,30)(6,32,8,29)(9,20,12,17)(10,19,11,18)(13,27,16,26)(14,28,15,25)$, $(1,12,3,9)(2,11,4,10)(5,14,7,15)(6,13,8,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$, $(1,30)(2,29)(3,31)(4,32)(5,24,7,21)(6,23,8,22)(9,25,12,28)(10,26,11,27)(13,19)(14,20)(15,17)(16,18)$, $(1,21,11,19)(2,22,12,20)(3,24,10,18)(4,23,9,17)(5,31,16,26)(6,32,15,25)(7,30,13,27)(8,29,14,28)$
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: $C_2^3$, $D_4\times C_2$ x 6

Degree 16: $C_2^2 \times D_4$, 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 x 2, 32T3210 x 2, 32T3211 x 2, 32T3212 x 2, 32T3213, 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 $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2B $2^{16}$ $2$ $2$ $16$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5, 9)( 6,10)( 7,12)( 8,11)(17,31)(18,32)(19,29)(20,30)(21,28)(22,27)(23,26)(24,25)$
2C $2^{16}$ $2$ $2$ $16$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5,12)( 6,11)( 7, 9)( 8,10)(17,31)(18,32)(19,29)(20,30)(21,28)(22,27)(23,26)(24,25)$
2D $2^{8},1^{16}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)$
2E $2^{16}$ $4$ $2$ $16$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5, 9)( 6,10)( 7,12)( 8,11)(17,30)(18,29)(19,32)(20,31)(21,28)(22,27)(23,26)(24,25)$
2F $2^{16}$ $4$ $2$ $16$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,15)(12,16)(17,27)(18,28)(19,25)(20,26)(21,32)(22,31)(23,30)(24,29)$
2G $2^{8},1^{16}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)(13,16)(14,15)(17,20)(18,19)(29,32)(30,31)$
2H $2^{16}$ $4$ $2$ $16$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,16)( 8,15)(17,23)(18,24)(19,21)(20,22)(25,32)(26,31)(27,30)(28,29)$
2I $2^{16}$ $8$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,31)(30,32)$
2J $2^{16}$ $8$ $2$ $16$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,15)( 8,16)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28)(29,31)(30,32)$
2K $2^{16}$ $8$ $2$ $16$ $( 1,18)( 2,17)( 3,19)( 4,20)( 5,27)( 6,28)( 7,26)( 8,25)( 9,23)(10,24)(11,21)(12,22)(13,30)(14,29)(15,32)(16,31)$
2L $2^{16}$ $8$ $2$ $16$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,16)(12,15)(17,29)(18,30)(19,31)(20,32)(21,27)(22,28)(23,25)(24,26)$
2M $2^{12},1^{8}$ $8$ $2$ $12$ $( 5, 7)( 6, 8)(13,16)(14,15)(17,23)(18,24)(19,21)(20,22)(25,29)(26,30)(27,31)(28,32)$
2N $2^{16}$ $8$ $2$ $16$ $( 1,32)( 2,31)( 3,29)( 4,30)( 5,22)( 6,21)( 7,23)( 8,24)( 9,26)(10,25)(11,28)(12,27)(13,20)(14,19)(15,18)(16,17)$
4A $4^{4},2^{8}$ $4$ $4$ $20$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,14, 7,15)( 6,13, 8,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
4B $4^{8}$ $4$ $4$ $24$ $( 1, 8, 3, 6)( 2, 7, 4, 5)( 9,13,12,16)(10,14,11,15)(17,26,20,27)(18,25,19,28)(21,32,24,29)(22,31,23,30)$
4C $4^{4},2^{8}$ $4$ $4$ $20$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,15, 7,14)( 6,16, 8,13)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
4D $4^{8}$ $4$ $4$ $24$ $( 1,11, 3,10)( 2,12, 4, 9)( 5,13, 7,16)( 6,14, 8,15)(17,22,20,23)(18,21,19,24)(25,32,28,29)(26,31,27,30)$
4E $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 7, 3, 5)( 2, 8, 4, 6)( 9,14,12,15)(10,13,11,16)(17,32)(18,31)(19,30)(20,29)(21,27)(22,28)(23,25)(24,26)$
4F $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,14, 7,15)( 6,13, 8,16)(17,24)(18,23)(19,22)(20,21)(25,30)(26,29)(27,32)(28,31)$
4G $4^{8}$ $8$ $4$ $24$ $( 1,16, 3,13)( 2,15, 4,14)( 5,10, 7,11)( 6, 9, 8,12)(17,32,20,29)(18,31,19,30)(21,26,24,27)(22,25,23,28)$
4H $4^{4},2^{4},1^{8}$ $8$ $4$ $16$ $( 1,10, 3,11)( 2, 9, 4,12)( 5,13, 7,16)( 6,14, 8,15)(21,24)(22,23)(29,32)(30,31)$
4I $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15)(10,16)(11,13)(12,14)(17,25,20,28)(18,26,19,27)(21,30,24,31)(22,29,23,32)$
4J $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,16)(10,15)(11,14)(12,13)(17,31,20,30)(18,32,19,29)(21,25,24,28)(22,26,23,27)$
4K $4^{8}$ $8$ $4$ $24$ $( 1,32, 3,29)( 2,31, 4,30)( 5,22, 7,23)( 6,21, 8,24)( 9,26,12,27)(10,25,11,28)(13,20,16,17)(14,19,15,18)$
4L $4^{8}$ $8$ $4$ $24$ $( 1,18, 3,19)( 2,17, 4,20)( 5,27, 7,26)( 6,28, 8,25)( 9,23,12,22)(10,24,11,21)(13,30,16,31)(14,29,15,32)$
4M $4^{8}$ $8$ $4$ $24$ $( 1,14, 3,15)( 2,13, 4,16)( 5,12, 7, 9)( 6,11, 8,10)(17,26,20,27)(18,25,19,28)(21,29,24,32)(22,30,23,31)$
4N $4^{4},2^{8}$ $16$ $4$ $20$ $( 1,20, 3,17)( 2,19, 4,18)( 5,28)( 6,27)( 7,25)( 8,26)( 9,21)(10,22)(11,23)(12,24)(13,29,16,32)(14,30,15,31)$
4O $4^{4},2^{8}$ $16$ $4$ $20$ $( 1,30, 3,31)( 2,29, 4,32)( 5,21)( 6,22)( 7,24)( 8,23)( 9,28)(10,27)(11,26)(12,25)(13,19,16,18)(14,20,15,17)$
4P $4^{8}$ $16$ $4$ $24$ $( 1,19,10,21)( 2,20, 9,22)( 3,18,11,24)( 4,17,12,23)( 5,27,13,30)( 6,28,14,29)( 7,26,16,31)( 8,25,15,32)$
4Q $4^{8}$ $16$ $4$ $24$ $( 1,29,10,28)( 2,30, 9,27)( 3,32,11,25)( 4,31,12,26)( 5,22,13,20)( 6,21,14,19)( 7,23,16,17)( 8,24,15,18)$
8A $8^{4}$ $16$ $8$ $28$ $( 1,17,11,22, 3,20,10,23)( 2,18,12,21, 4,19, 9,24)( 5,28,13,29, 7,25,16,32)( 6,27,14,30, 8,26,15,31)$
8B $8^{4}$ $16$ $8$ $28$ $( 1,31,10,26, 3,30,11,27)( 2,32, 9,25, 4,29,12,28)( 5,21,16,18, 7,24,13,19)( 6,22,15,17, 8,23,14,20)$

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