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Properties

Label 32T216
Degree $32$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2.C_2^2$

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Copy content magma:G := TransitiveGroup(32, 216);
 

Group invariants

Abstract group:  $C_4^2.C_2^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $64=2^{6}$
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:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $216$
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:  $(1,8,19,24)(2,7,20,23)(3,6,17,22)(4,5,18,21)(9,14,25,32)(10,13,26,31)(11,16,27,29)(12,15,28,30)$, $(1,15,19,30)(2,16,20,29)(3,13,17,31)(4,14,18,32)(5,9,21,25)(6,10,22,26)(7,11,23,27)(8,12,24,28)$, $(1,7,2,8)(3,6,4,5)(9,14,10,13)(11,15,12,16)(17,22,18,21)(19,23,20,24)(25,32,26,31)(27,30,28,29)$, $(1,4)(2,3)(5,23)(6,24)(7,21)(8,22)(9,12)(10,11)(13,30)(14,29)(15,31)(16,32)(17,20)(18,19)(25,28)(26,27)$
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
$8$:  $C_2^3$ x 15
$16$:  $Q_8:C_2$ x 4, $C_2^4$
$32$:  $C_2 \times (C_4\times C_2):C_2$ x 2, $(C_2 \times Q_8):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $Q_8:C_2$ x 4

Degree 16: $C_2 \times (C_4\times C_2):C_2$ x 2, $(C_2 \times Q_8):C_2$

Low degree siblings

32T216

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

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E1 4E-1 4F1 4F-1 4G1 4G-1 4H1 4H-1 4I 4J 4K 4L 4M 4N 4O
Size 1 1 1 1 4 4 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2A 2A 2B 2B 2C 2C 2A 2A 2C 2C 2A 2A 2B 2B 2B 2C 2A 2B 2A
Type
64.237.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.237.1b R 1 1 1 1 −1 −1 −1 1 1 −1 −1 −1 1 1 1 1 1 1 −1 1 1 −1 1 −1 −1
64.237.1c R 1 1 1 1 −1 −1 −1 1 1 −1 1 1 1 1 −1 −1 1 1 1 −1 −1 1 −1 1 −1
64.237.1d R 1 1 1 1 −1 −1 1 −1 1 1 −1 −1 1 −1 −1 −1 −1 −1 1 1 −1 1 1 −1 1
64.237.1e R 1 1 1 1 −1 −1 1 −1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1 1 −1 −1 1 1
64.237.1f R 1 1 1 1 −1 1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 −1 1 −1 −1 −1 1 1 1
64.237.1g R 1 1 1 1 −1 1 −1 −1 −1 −1 1 1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 −1 1
64.237.1h R 1 1 1 1 −1 1 1 1 −1 1 −1 −1 −1 −1 −1 −1 −1 1 −1 −1 1 1 1 1 −1
64.237.1i R 1 1 1 1 −1 1 1 1 −1 1 1 1 −1 −1 1 1 −1 1 1 1 −1 −1 −1 −1 −1
64.237.1j R 1 1 1 1 1 −1 −1 1 −1 −1 −1 −1 −1 −1 1 1 −1 1 −1 1 −1 1 −1 1 1
64.237.1k R 1 1 1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 −1 −1 −1 1 1 −1 1 −1 1 −1 1
64.237.1l R 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 −1 −1 1 −1 1 1 1 −1 −1 1 −1
64.237.1m R 1 1 1 1 1 −1 1 −1 −1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
64.237.1n R 1 1 1 1 1 1 −1 −1 1 −1 −1 −1 1 −1 1 1 −1 −1 1 −1 1 1 −1 −1 −1
64.237.1o R 1 1 1 1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 −1 −1 −1 1 −1 −1 1 1 −1
64.237.1p R 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1 1
64.237.2a1 C 2 −2 −2 2 0 0 0 −2i −2 0 0 0 2 2i 0 0 −2i 2i 0 0 0 0 0 0 0
64.237.2a2 C 2 −2 −2 2 0 0 0 2i −2 0 0 0 2 −2i 0 0 2i −2i 0 0 0 0 0 0 0
64.237.2b1 C 2 −2 −2 2 0 0 0 −2i 2 0 0 0 −2 −2i 0 0 2i 2i 0 0 0 0 0 0 0
64.237.2b2 C 2 −2 −2 2 0 0 0 2i 2 0 0 0 −2 2i 0 0 −2i −2i 0 0 0 0 0 0 0
64.237.2c1 C 2 −2 2 −2 0 0 −2 0 0 2 −2i 2i 0 0 2i −2i 0 0 0 0 0 0 0 0 0
64.237.2c2 C 2 −2 2 −2 0 0 −2 0 0 2 2i −2i 0 0 −2i 2i 0 0 0 0 0 0 0 0 0
64.237.2d1 C 2 −2 2 −2 0 0 2 0 0 −2 −2i 2i 0 0 −2i 2i 0 0 0 0 0 0 0 0 0
64.237.2d2 C 2 −2 2 −2 0 0 2 0 0 −2 2i −2i 0 0 2i −2i 0 0 0 0 0 0 0 0 0
64.237.4a S 4 4 −4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed