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Properties

Label 32T3186
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, 3186);
 

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 $t$:  $3186$
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:  $(1,30,2,29)(3,31,4,32)(5,28,8,26)(6,27,7,25)(9,22,12,24)(10,21,11,23)(13,17,14,18)(15,20,16,19)$, $(1,24,6,20,4,21,8,17)(2,23,5,19,3,22,7,18)(9,29,14,28,11,31,16,25)(10,30,13,27,12,32,15,26)$, $(1,13)(2,14)(3,16)(4,15)(5,9)(6,10)(7,11)(8,12)(17,29)(18,30)(19,32)(20,31)(21,25)(22,26)(23,27)(24,28)$, $(1,25,5,30)(2,26,6,29)(3,27,8,31)(4,28,7,32)(9,17,15,22)(10,18,16,21)(11,20,13,23)(12,19,14,24)$
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, 16T652 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, 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 4, 32T7442 x 2, 32T7470 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)$
2B $2^{8},1^{16}$ $2$ $2$ $8$ $( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)$
2C $2^{8},1^{16}$ $2$ $2$ $8$ $( 9,11)(10,12)(13,15)(14,16)(25,28)(26,27)(29,31)(30,32)$
2D $2^{8},1^{16}$ $2$ $2$ $8$ $(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)$
2E $2^{16}$ $4$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,27)(26,28)(29,30)(31,32)$
2F $2^{16}$ $4$ $2$ $16$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13)(10,14)(11,15)(12,16)(17,23)(18,24)(19,21)(20,22)(25,30)(26,29)(27,31)(28,32)$
2G $2^{16}$ $4$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32)$
2H $2^{16}$ $4$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,13)(10,14)(11,15)(12,16)(17,22)(18,21)(19,24)(20,23)(25,30)(26,29)(27,31)(28,32)$
2I $2^{16}$ $8$ $2$ $16$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5,11)( 6,12)( 7, 9)( 8,10)(17,31)(18,32)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)$
2J $2^{12},1^{8}$ $8$ $2$ $12$ $( 1, 4)( 2, 3)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,32)(28,31)$
2K $2^{16}$ $8$ $2$ $16$ $( 1,24)( 2,23)( 3,22)( 4,21)( 5,18)( 6,17)( 7,19)( 8,20)( 9,29)(10,30)(11,31)(12,32)(13,26)(14,25)(15,27)(16,28)$
2L $2^{12},1^{8}$ $8$ $2$ $12$ $( 1, 4)( 2, 3)( 9,11)(10,12)(17,24)(18,23)(19,22)(20,21)(25,29)(26,30)(27,32)(28,31)$
2M $2^{16}$ $8$ $2$ $16$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,13)( 6,14)( 7,15)( 8,16)(17,30)(18,29)(19,31)(20,32)(21,26)(22,25)(23,28)(24,27)$
2N $2^{16}$ $8$ $2$ $16$ $( 1,24)( 2,23)( 3,22)( 4,21)( 5,18)( 6,17)( 7,19)( 8,20)( 9,31)(10,32)(11,29)(12,30)(13,27)(14,28)(15,26)(16,25)$
4A $4^{4},2^{8}$ $4$ $4$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,22,20,23)(18,21,19,24)(25,32,28,30)(26,31,27,29)$
4B $4^{8}$ $4$ $4$ $24$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,14,11,16)(10,13,12,15)(17,24,20,21)(18,23,19,22)(25,31,28,29)(26,32,27,30)$
4C $4^{4},2^{8}$ $4$ $4$ $20$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,23,20,22)(18,24,19,21)(25,30,28,32)(26,29,27,31)$
4D $4^{8}$ $4$ $4$ $24$ $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,14,11,16)(10,13,12,15)(17,21,20,24)(18,22,19,23)(25,31,28,29)(26,32,27,30)$
4E $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,16)(14,15)(17,23,20,22)(18,24,19,21)(25,32,28,30)(26,31,27,29)$
4F $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,14)( 6,13)( 7,16)( 8,15)(17,28,20,25)(18,27,19,26)(21,29,24,31)(22,30,23,32)$
4G $4^{8}$ $8$ $4$ $24$ $( 1,13, 4,15)( 2,14, 3,16)( 5, 9, 7,11)( 6,10, 8,12)(17,31,20,29)(18,32,19,30)(21,28,24,25)(22,27,23,26)$
4H $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,16)( 2,15)( 3,13)( 4,14)( 5,12)( 6,11)( 7,10)( 8, 9)(17,27,20,26)(18,28,19,25)(21,30,24,32)(22,29,23,31)$
4I $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,16, 7,14)( 6,15, 8,13)(17,28)(18,27)(19,26)(20,25)(21,29)(22,30)(23,32)(24,31)$
4J $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,14, 4,16)( 2,13, 3,15)( 5,10, 7,12)( 6, 9, 8,11)(17,27)(18,28)(19,25)(20,26)(21,30)(22,29)(23,31)(24,32)$
4K $4^{8}$ $8$ $4$ $24$ $( 1,21, 4,24)( 2,22, 3,23)( 5,19, 7,18)( 6,20, 8,17)( 9,29,11,31)(10,30,12,32)(13,26,15,27)(14,25,16,28)$
4L $4^{8}$ $8$ $4$ $24$ $( 1,21, 4,24)( 2,22, 3,23)( 5,19, 7,18)( 6,20, 8,17)( 9,31,11,29)(10,32,12,30)(13,27,15,26)(14,28,16,25)$
4M $4^{8}$ $8$ $4$ $24$ $( 1, 9, 4,11)( 2,10, 3,12)( 5,15, 7,13)( 6,16, 8,14)(17,30,20,32)(18,29,19,31)(21,26,24,27)(22,25,23,28)$
4N $4^{8}$ $16$ $4$ $24$ $( 1,27, 3,25)( 2,28, 4,26)( 5,31, 6,32)( 7,29, 8,30)( 9,19,10,20)(11,18,12,17)(13,21,16,23)(14,22,15,24)$
4O $4^{8}$ $16$ $4$ $24$ $( 1,27, 3,25)( 2,28, 4,26)( 5,31, 6,32)( 7,29, 8,30)( 9,18,10,17)(11,19,12,20)(13,24,16,22)(14,23,15,21)$
4P $4^{8}$ $16$ $4$ $24$ $( 1,29, 7,26)( 2,30, 8,25)( 3,32, 6,28)( 4,31, 5,27)( 9,24,13,19)(10,23,14,20)(11,21,15,18)(12,22,16,17)$
4Q $4^{8}$ $16$ $4$ $24$ $( 1,29, 7,26)( 2,30, 8,25)( 3,32, 6,28)( 4,31, 5,27)( 9,21,13,18)(10,22,14,17)(11,24,15,19)(12,23,16,20)$
8A $8^{4}$ $16$ $8$ $28$ $( 1,18, 6,22, 4,19, 8,23)( 2,17, 5,21, 3,20, 7,24)( 9,26,14,32,11,27,16,30)(10,25,13,31,12,28,15,29)$
8B $8^{4}$ $16$ $8$ $28$ $( 1,19, 8,22, 4,18, 6,23)( 2,20, 7,21, 3,17, 5,24)( 9,26,16,30,11,27,14,32)(10,25,15,29,12,28,13,31)$

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