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Properties

Label 32T1293
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^5:C_4$

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Copy content magma:G := TransitiveGroup(32, 1293);
 

Group invariants

Abstract group:  $C_2^5:C_4$
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 $t$:  $1293$
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:  $(1,10)(2,9)(3,12)(4,11)(5,32)(6,31)(7,30)(8,29)(13,22)(14,21)(15,24)(16,23)(17,28)(18,27)(19,26)(20,25)$, $(1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(13,15)(14,16)(29,31)(30,32)$, $(1,21,9,29)(2,22,10,30)(3,23,11,31)(4,24,12,32)(5,25,15,19)(6,26,16,20)(7,27,13,17)(8,28,14,18)$, $(1,5,9,15)(2,6,10,16)(3,7,11,13)(4,8,12,14)(17,23,27,31)(18,24,28,32)(19,21,25,29)(20,22,26,30)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_4$ x 8, $C_2^2$ x 35
$8$:  $D_{4}$ x 8, $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 12, $C_2^2:C_4$ x 16, $C_4\times C_2^2$ x 14, $C_2^4$
$32$:  $C_2^3 : C_4 $ x 4, $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
$64$:  16T76 x 6, 32T262

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 4

Degree 8: $C_4\times C_2$, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_2^3: C_4$ x 4

Degree 16: $C_2 \times (C_2^2:C_4)$, 16T93 x 4, 16T94 x 2

Low degree siblings

32T1107 x 8, 32T1144 x 3, 32T1145 x 6, 32T1282, 32T1283 x 6, 32T1293 x 11, 32T1295 x 4

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 $2^{16}$ $1$ $2$ $16$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)(11,27)(12,28)(13,31)(14,32)(15,29)(16,30)$
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 $2^{16}$ $1$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,24)( 6,23)( 7,22)( 8,21)( 9,28)(10,27)(11,26)(12,25)(13,30)(14,29)(15,32)(16,31)$
2D $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2E $2^{16}$ $1$ $2$ $16$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,22)( 6,21)( 7,24)( 8,23)( 9,26)(10,25)(11,28)(12,27)(13,32)(14,31)(15,30)(16,29)$
2F $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2G $2^{16}$ $1$ $2$ $16$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,23)( 6,24)( 7,21)( 8,22)( 9,27)(10,28)(11,25)(12,26)(13,29)(14,30)(15,31)(16,32)$
2H $2^{16}$ $2$ $2$ $16$ $( 1,27)( 2,28)( 3,25)( 4,26)( 5,15)( 6,16)( 7,13)( 8,14)( 9,17)(10,18)(11,19)(12,20)(21,29)(22,30)(23,31)(24,32)$
2I $2^{16}$ $2$ $2$ $16$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,32)( 6,31)( 7,30)( 8,29)(13,22)(14,21)(15,24)(16,23)(17,28)(18,27)(19,26)(20,25)$
2J $2^{16}$ $2$ $2$ $16$ $( 1,26)( 2,25)( 3,28)( 4,27)( 5,14)( 6,13)( 7,16)( 8,15)( 9,20)(10,19)(11,18)(12,17)(21,32)(22,31)(23,30)(24,29)$
2K $2^{16}$ $2$ $2$ $16$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5, 7)( 6, 8)( 9,25)(10,26)(11,27)(12,28)(13,15)(14,16)(21,23)(22,24)(29,31)(30,32)$
2L $2^{8},1^{16}$ $2$ $2$ $8$ $( 5,23)( 6,24)( 7,21)( 8,22)(13,29)(14,30)(15,31)(16,32)$
2M $2^{16}$ $2$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5, 6)( 7, 8)( 9,28)(10,27)(11,26)(12,25)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$
2N $2^{16}$ $2$ $2$ $16$ $( 1, 4)( 2, 3)( 5,22)( 6,21)( 7,24)( 8,23)( 9,12)(10,11)(13,32)(14,31)(15,30)(16,29)(17,20)(18,19)(25,28)(26,27)$
2O $2^{16}$ $2$ $2$ $16$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,13)( 6,14)( 7,15)( 8,16)(17,25)(18,26)(19,27)(20,28)(21,31)(22,32)(23,29)(24,30)$
2P $2^{16}$ $2$ $2$ $16$ $( 1,26)( 2,25)( 3,28)( 4,27)( 5,30)( 6,29)( 7,32)( 8,31)( 9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)$
2Q $2^{16}$ $2$ $2$ $16$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,16)( 6,15)( 7,14)( 8,13)(17,28)(18,27)(19,26)(20,25)(21,30)(22,29)(23,32)(24,31)$
2R $2^{16}$ $2$ $2$ $16$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,29)( 6,30)( 7,31)( 8,32)(13,23)(14,24)(15,21)(16,22)(17,25)(18,26)(19,27)(20,28)$
2S $2^{16}$ $2$ $2$ $16$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,15)( 6,16)( 7,13)( 8,14)(17,27)(18,28)(19,25)(20,26)(21,29)(22,30)(23,31)(24,32)$
2T $2^{16}$ $4$ $2$ $16$ $( 1, 2)( 3, 4)( 5,22)( 6,21)( 7,24)( 8,23)( 9,28)(10,27)(11,26)(12,25)(13,16)(14,15)(17,18)(19,20)(29,32)(30,31)$
2U $2^{16}$ $4$ $2$ $16$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5, 6)( 7, 8)( 9,12)(10,11)(13,30)(14,29)(15,32)(16,31)(21,22)(23,24)(25,28)(26,27)$
2V $2^{12},1^{8}$ $4$ $2$ $12$ $( 1, 3)( 2, 4)( 5,23)( 6,24)( 7,21)( 8,22)( 9,25)(10,26)(11,27)(12,28)(17,19)(18,20)$
2W $2^{12},1^{8}$ $4$ $2$ $12$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5, 7)( 6, 8)(13,31)(14,32)(15,29)(16,30)(21,23)(22,24)$
4A $4^{8}$ $4$ $4$ $24$ $( 1,26,17,12)( 2,25,18,11)( 3,28,19,10)( 4,27,20, 9)( 5,32,23,16)( 6,31,24,15)( 7,30,21,14)( 8,29,22,13)$
4B $4^{8}$ $4$ $4$ $24$ $( 1,10,17,28)( 2, 9,18,27)( 3,12,19,26)( 4,11,20,25)( 5,14,23,30)( 6,13,24,29)( 7,16,21,32)( 8,15,22,31)$
4C $4^{8}$ $4$ $4$ $24$ $( 1,27,17, 9)( 2,28,18,10)( 3,25,19,11)( 4,26,20,12)( 5,29,23,13)( 6,30,24,14)( 7,31,21,15)( 8,32,22,16)$
4D $4^{8}$ $4$ $4$ $24$ $( 1,11,17,25)( 2,12,18,26)( 3, 9,19,27)( 4,10,20,28)( 5,15,23,31)( 6,16,24,32)( 7,13,21,29)( 8,14,22,30)$
4E1 $4^{8}$ $4$ $4$ $24$ $( 1,24,11,14)( 2,23,12,13)( 3,22, 9,16)( 4,21,10,15)( 5,26,29,18)( 6,25,30,17)( 7,28,31,20)( 8,27,32,19)$
4E-1 $4^{8}$ $4$ $4$ $24$ $( 1,14,11,24)( 2,13,12,23)( 3,16, 9,22)( 4,15,10,21)( 5,18,29,26)( 6,17,30,25)( 7,20,31,28)( 8,19,32,27)$
4F1 $4^{8}$ $4$ $4$ $24$ $( 1, 8,11,32)( 2, 7,12,31)( 3, 6, 9,30)( 4, 5,10,29)(13,20,23,28)(14,19,24,27)(15,18,21,26)(16,17,22,25)$
4F-1 $4^{8}$ $4$ $4$ $24$ $( 1,32,11, 8)( 2,31,12, 7)( 3,30, 9, 6)( 4,29,10, 5)(13,28,23,20)(14,27,24,19)(15,26,21,18)(16,25,22,17)$
4G1 $4^{8}$ $4$ $4$ $24$ $( 1,21,11,15)( 2,22,12,16)( 3,23, 9,13)( 4,24,10,14)( 5,27,29,19)( 6,28,30,20)( 7,25,31,17)( 8,26,32,18)$
4G-1 $4^{8}$ $4$ $4$ $24$ $( 1,15,11,21)( 2,16,12,22)( 3,13, 9,23)( 4,14,10,24)( 5,19,29,27)( 6,20,30,28)( 7,17,31,25)( 8,18,32,26)$
4H1 $4^{8}$ $4$ $4$ $24$ $( 1, 5,11,29)( 2, 6,12,30)( 3, 7, 9,31)( 4, 8,10,32)(13,17,23,25)(14,18,24,26)(15,19,21,27)(16,20,22,28)$
4H-1 $4^{8}$ $4$ $4$ $24$ $( 1,29,11, 5)( 2,30,12, 6)( 3,31, 9, 7)( 4,32,10, 8)(13,25,23,17)(14,26,24,18)(15,27,21,19)(16,28,22,20)$
4I1 $4^{8}$ $4$ $4$ $24$ $( 1, 7, 9,13)( 2, 8,10,14)( 3, 5,11,15)( 4, 6,12,16)(17,21,27,29)(18,22,28,30)(19,23,25,31)(20,24,26,32)$
4I-1 $4^{8}$ $4$ $4$ $24$ $( 1,15,27,23)( 2,16,28,24)( 3,13,25,21)( 4,14,26,22)( 5,17,31, 9)( 6,18,32,10)( 7,19,29,11)( 8,20,30,12)$
4J1 $4^{8}$ $4$ $4$ $24$ $( 1,23, 9,31)( 2,24,10,32)( 3,21,11,29)( 4,22,12,30)( 5,27,15,17)( 6,28,16,18)( 7,25,13,19)( 8,26,14,20)$
4J-1 $4^{8}$ $4$ $4$ $24$ $( 1,29,27, 7)( 2,30,28, 8)( 3,31,25, 5)( 4,32,26, 6)( 9,21,17,13)(10,22,18,14)(11,23,19,15)(12,24,20,16)$
4K1 $4^{8}$ $4$ $4$ $24$ $( 1, 6, 9,16)( 2, 5,10,15)( 3, 8,11,14)( 4, 7,12,13)(17,24,27,32)(18,23,28,31)(19,22,25,30)(20,21,26,29)$
4K-1 $4^{8}$ $4$ $4$ $24$ $( 1,14,27,22)( 2,13,28,21)( 3,16,25,24)( 4,15,26,23)( 5,20,31,12)( 6,19,32,11)( 7,18,29,10)( 8,17,30, 9)$
4L1 $4^{8}$ $4$ $4$ $24$ $( 1,22, 9,30)( 2,21,10,29)( 3,24,11,32)( 4,23,12,31)( 5,26,15,20)( 6,25,16,19)( 7,28,13,18)( 8,27,14,17)$
4L-1 $4^{8}$ $4$ $4$ $24$ $( 1,32,27, 6)( 2,31,28, 5)( 3,30,25, 8)( 4,29,26, 7)( 9,24,17,16)(10,23,18,15)(11,22,19,14)(12,21,20,13)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

44 x 44 character table

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Regular extensions

Data not computed