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Properties

Label 32T1412
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4.C_4^2$

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Copy content magma:G := TransitiveGroup(32, 1412);
 

Group invariants

Abstract group:  $D_4.C_4^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $128=2^{7}$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1412$
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:  $(1,12,4,9)(2,11,3,10)(5,13,8,16)(6,14,7,15)(17,26,20,27)(18,25,19,28)(21,32,24,29)(22,31,23,30)$, $(1,30,14,25,3,32,16,27)(2,29,13,26,4,31,15,28)(5,17,12,22,7,19,10,24)(6,18,11,21,8,20,9,23)$, $(1,32)(2,31)(3,30)(4,29)(5,17)(6,18)(7,19)(8,20)(9,23)(10,24)(11,21)(12,22)(13,28)(14,27)(15,26)(16,25)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 12, $C_2^2$ x 7
$8$:  $D_{4}$ x 4, $C_4\times C_2$ x 18, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$ x 3, $C_4^2$ x 4
$32$:  $C_4^2:C_2$, $C_4 \times D_4$ x 4, $C_2 \times (C_2^2:C_4)$, 32T36
$64$:  16T106 x 2, 32T198

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 2

Degree 8: $C_4\times C_2$, $D_4\times C_2$, $Q_8:C_2$

Degree 16: $C_4 \times D_4$, 16T107 x 2

Low degree siblings

32T1412 x 3

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 $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2C $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2D $2^{16}$ $2$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2E $2^{8},1^{16}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2F $2^{16}$ $4$ $2$ $16$ $( 1,28)( 2,27)( 3,26)( 4,25)( 5,21)( 6,22)( 7,23)( 8,24)( 9,17)(10,18)(11,19)(12,20)(13,30)(14,29)(15,32)(16,31)$
2G $2^{16}$ $4$ $2$ $16$ $( 1,25)( 2,26)( 3,27)( 4,28)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,31)(14,32)(15,29)(16,30)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1,13, 3,15)( 2,14, 4,16)( 5,11, 7, 9)( 6,12, 8,10)(17,21,19,23)(18,22,20,24)(25,31,27,29)(26,32,28,30)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1,15, 3,13)( 2,16, 4,14)( 5, 9, 7,11)( 6,10, 8,12)(17,23,19,21)(18,24,20,22)(25,29,27,31)(26,30,28,32)$
4B1 $4^{8}$ $1$ $4$ $24$ $( 1,14, 3,16)( 2,13, 4,15)( 5,12, 7,10)( 6,11, 8, 9)(17,22,19,24)(18,21,20,23)(25,32,27,30)(26,31,28,29)$
4B-1 $4^{8}$ $1$ $4$ $24$ $( 1,16, 3,14)( 2,15, 4,13)( 5,10, 7,12)( 6, 9, 8,11)(17,24,19,22)(18,23,20,21)(25,30,27,32)(26,29,28,31)$
4C $4^{8}$ $2$ $4$ $24$ $( 1,16, 3,14)( 2,15, 4,13)( 5,10, 7,12)( 6, 9, 8,11)(17,22,19,24)(18,21,20,23)(25,32,27,30)(26,31,28,29)$
4D $4^{8}$ $2$ $4$ $24$ $( 1,13, 3,15)( 2,14, 4,16)( 5,11, 7, 9)( 6,12, 8,10)(17,23,19,21)(18,24,20,22)(25,29,27,31)(26,30,28,32)$
4E1 $4^{8}$ $2$ $4$ $24$ $( 1, 9, 4,12)( 2,10, 3,11)( 5,16, 8,13)( 6,15, 7,14)(17,25,20,28)(18,26,19,27)(21,31,24,30)(22,32,23,29)$
4E-1 $4^{8}$ $2$ $4$ $24$ $( 1,12, 4, 9)( 2,11, 3,10)( 5,13, 8,16)( 6,14, 7,15)(17,28,20,25)(18,27,19,26)(21,30,24,31)(22,29,23,32)$
4F1 $4^{8}$ $2$ $4$ $24$ $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,13,10,14)(11,15,12,16)(17,32,18,31)(19,30,20,29)(21,28,22,27)(23,26,24,25)$
4F-1 $4^{8}$ $2$ $4$ $24$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,16,10,15)(11,14,12,13)(17,29,18,30)(19,31,20,32)(21,25,22,26)(23,27,24,28)$
4G1 $4^{8}$ $2$ $4$ $24$ $( 1,12, 4, 9)( 2,11, 3,10)( 5,13, 8,16)( 6,14, 7,15)(17,26,20,27)(18,25,19,28)(21,32,24,29)(22,31,23,30)$
4G-1 $4^{8}$ $2$ $4$ $24$ $( 1, 9, 4,12)( 2,10, 3,11)( 5,16, 8,13)( 6,15, 7,14)(17,27,20,26)(18,28,19,25)(21,29,24,32)(22,30,23,31)$
4H1 $4^{8}$ $2$ $4$ $24$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,16,10,15)(11,14,12,13)(17,31,18,32)(19,29,20,30)(21,27,22,28)(23,25,24,26)$
4H-1 $4^{8}$ $2$ $4$ $24$ $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,13,10,14)(11,15,12,16)(17,30,18,29)(19,32,20,31)(21,26,22,25)(23,28,24,27)$
4I $4^{8}$ $4$ $4$ $24$ $( 1,25, 3,27)( 2,26, 4,28)( 5,24, 7,22)( 6,23, 8,21)( 9,20,11,18)(10,19,12,17)(13,31,15,29)(14,32,16,30)$
4J $4^{8}$ $4$ $4$ $24$ $( 1,28, 3,26)( 2,27, 4,25)( 5,21, 7,23)( 6,22, 8,24)( 9,17,11,19)(10,18,12,20)(13,30,15,32)(14,29,16,31)$
4K1 $4^{8}$ $4$ $4$ $24$ $( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,14,12,15)(10,13,11,16)(17,28,18,27)(19,26,20,25)(21,30,22,29)(23,32,24,31)$
4K-1 $4^{8}$ $4$ $4$ $24$ $( 1, 8, 4, 5)( 2, 7, 3, 6)( 9,15,12,14)(10,16,11,13)(17,27,18,28)(19,25,20,26)(21,29,22,30)(23,31,24,32)$
4L1 $4^{8}$ $4$ $4$ $24$ $( 1,17, 4,20)( 2,18, 3,19)( 5,31, 8,30)( 6,32, 7,29)( 9,25,12,28)(10,26,11,27)(13,21,16,24)(14,22,15,23)$
4L-1 $4^{8}$ $4$ $4$ $24$ $( 1,20, 4,17)( 2,19, 3,18)( 5,30, 8,31)( 6,29, 7,32)( 9,28,12,25)(10,27,11,26)(13,24,16,21)(14,23,15,22)$
4M1 $4^{8}$ $4$ $4$ $24$ $( 1, 8, 4, 5)( 2, 7, 3, 6)( 9,15,12,14)(10,16,11,13)(17,25,18,26)(19,27,20,28)(21,31,22,32)(23,29,24,30)$
4M-1 $4^{8}$ $4$ $4$ $24$ $( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,14,12,15)(10,13,11,16)(17,26,18,25)(19,28,20,27)(21,32,22,31)(23,30,24,29)$
4N1 $4^{4},2^{4},1^{8}$ $4$ $4$ $16$ $( 1,16, 3,14)( 2,15, 4,13)( 5,12, 7,10)( 6,11, 8, 9)(25,27)(26,28)(29,31)(30,32)$
4N-1 $4^{4},2^{4},1^{8}$ $4$ $4$ $16$ $( 1,16, 3,14)( 2,15, 4,13)( 5,12, 7,10)( 6,11, 8, 9)(17,19)(18,20)(21,23)(22,24)$
4O1 $4^{8}$ $4$ $4$ $24$ $( 1,20, 2,19)( 3,18, 4,17)( 5,30, 6,29)( 7,32, 8,31)( 9,28,10,27)(11,26,12,25)(13,24,14,23)(15,22,16,21)$
4O-1 $4^{8}$ $4$ $4$ $24$ $( 1,17, 2,18)( 3,19, 4,20)( 5,31, 6,32)( 7,29, 8,30)( 9,25,10,26)(11,27,12,28)(13,21,14,22)(15,23,16,24)$
4P1 $4^{4},2^{8}$ $4$ $4$ $20$ $( 1,13, 3,15)( 2,14, 4,16)( 5, 9, 7,11)( 6,10, 8,12)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)$
4P-1 $4^{4},2^{8}$ $4$ $4$ $20$ $( 1,13, 3,15)( 2,14, 4,16)( 5, 9, 7,11)( 6,10, 8,12)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)$
8A1 $8^{4}$ $4$ $8$ $28$ $( 1,21,13,19, 3,23,15,17)( 2,22,14,20, 4,24,16,18)( 5,25,11,31, 7,27, 9,29)( 6,26,12,32, 8,28,10,30)$
8A-1 $8^{4}$ $4$ $8$ $28$ $( 1,24,15,20, 3,22,13,18)( 2,23,16,19, 4,21,14,17)( 5,28, 9,32, 7,26,11,30)( 6,27,10,31, 8,25,12,29)$
8B1 $8^{4}$ $4$ $8$ $28$ $( 1,24,13,18, 3,22,15,20)( 2,23,14,17, 4,21,16,19)( 5,28,11,30, 7,26, 9,32)( 6,27,12,29, 8,25,10,31)$
8B-1 $8^{4}$ $4$ $8$ $28$ $( 1,21,15,17, 3,23,13,19)( 2,22,16,18, 4,24,14,20)( 5,25, 9,29, 7,27,11,31)( 6,26,10,30, 8,28,12,32)$
8C1 $8^{4}$ $4$ $8$ $28$ $( 1,31,14,28, 3,29,16,26)( 2,32,13,27, 4,30,15,25)( 5,20,12,23, 7,18,10,21)( 6,19,11,24, 8,17, 9,22)$
8C-1 $8^{4}$ $4$ $8$ $28$ $( 1,31,16,26, 3,29,14,28)( 2,32,15,25, 4,30,13,27)( 5,20,10,21, 7,18,12,23)( 6,19, 9,22, 8,17,11,24)$
8D1 $8^{4}$ $4$ $8$ $28$ $( 1,30,14,25, 3,32,16,27)( 2,29,13,26, 4,31,15,28)( 5,17,12,22, 7,19,10,24)( 6,18,11,21, 8,20, 9,23)$
8D-1 $8^{4}$ $4$ $8$ $28$ $( 1,30,16,27, 3,32,14,25)( 2,29,15,28, 4,31,13,26)( 5,17,10,24, 7,19,12,22)( 6,18, 9,23, 8,20,11,21)$

Malle's constant $a(G)$:     $1/8$

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Character table

44 x 44 character table

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Regular extensions

Data not computed