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Properties

Label 32T40
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2\times Q_8$

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Copy content magma:G := TransitiveGroup(32, 40);
 

Group invariants

Abstract group:  $C_2^2\times Q_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $32=2^{5}$
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 $t$:  $40$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $32$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,24,2,23)(3,22,4,21)(5,17,6,18)(7,19,8,20)(9,15,10,16)(11,13,12,14)(25,29,26,30)(27,31,28,32)$, $(1,7,2,8)(3,5,4,6)(9,32,10,31)(11,30,12,29)(13,26,14,25)(15,28,16,27)(17,22,18,21)(19,24,20,23)$, $(1,6,2,5)(3,8,4,7)(9,29,10,30)(11,31,12,32)(13,27,14,28)(15,25,16,26)(17,23,18,24)(19,21,20,22)$, $(1,12,2,11)(3,10,4,9)(5,31,6,32)(7,29,8,30)(13,23,14,24)(15,21,16,22)(17,27,18,28)(19,25,20,26)$
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, $Q_8$ x 4
$16$:  $C_2^4$, $Q_8\times C_2$ x 6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35

Degree 8: $C_2^3$ x 15, $Q_8$ x 4

Degree 16: $C_2^4$, $Q_8\times C_2$ x 6

Low degree siblings

There are no siblings 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 $2^{16}$ $1$ $2$ $16$ $( 1,32)( 2,31)( 3,30)( 4,29)( 5,12)( 6,11)( 7,10)( 8, 9)(13,18)(14,17)(15,20)(16,19)(21,26)(22,25)(23,28)(24,27)$
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 $2^{16}$ $1$ $2$ $16$ $( 1,31)( 2,32)( 3,29)( 4,30)( 5,11)( 6,12)( 7, 9)( 8,10)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28)$
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 $2^{16}$ $1$ $2$ $16$ $( 1,29)( 2,30)( 3,31)( 4,32)( 5, 9)( 6,10)( 7,11)( 8,12)(13,19)(14,20)(15,17)(16,18)(21,27)(22,28)(23,25)(24,26)$
2F $2^{16}$ $1$ $2$ $16$ $( 1,30)( 2,29)( 3,32)( 4,31)( 5,10)( 6, 9)( 7,12)( 8,11)(13,20)(14,19)(15,18)(16,17)(21,28)(22,27)(23,26)(24,25)$
2G $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
4A $4^{8}$ $2$ $4$ $24$ $( 1,24, 2,23)( 3,22, 4,21)( 5,17, 6,18)( 7,19, 8,20)( 9,15,10,16)(11,13,12,14)(25,29,26,30)(27,31,28,32)$
4B $4^{8}$ $2$ $4$ $24$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,32, 6,31)( 7,30, 8,29)(13,24,14,23)(15,22,16,21)(17,28,18,27)(19,26,20,25)$
4C $4^{8}$ $2$ $4$ $24$ $( 1,13, 2,14)( 3,15, 4,16)( 5,27, 6,28)( 7,25, 8,26)( 9,21,10,22)(11,23,12,24)(17,32,18,31)(19,30,20,29)$
4D $4^{8}$ $2$ $4$ $24$ $( 1,27, 2,28)( 3,25, 4,26)( 5,14, 6,13)( 7,16, 8,15)( 9,20,10,19)(11,18,12,17)(21,30,22,29)(23,32,24,31)$
4E $4^{8}$ $2$ $4$ $24$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,29,10,30)(11,31,12,32)(13,27,14,28)(15,25,16,26)(17,23,18,24)(19,21,20,22)$
4F $4^{8}$ $2$ $4$ $24$ $( 1,21, 2,22)( 3,23, 4,24)( 5,20, 6,19)( 7,18, 8,17)( 9,14,10,13)(11,16,12,15)(25,32,26,31)(27,30,28,29)$
4G $4^{8}$ $2$ $4$ $24$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,29, 6,30)( 7,31, 8,32)(13,21,14,22)(15,23,16,24)(17,25,18,26)(19,27,20,28)$
4H $4^{8}$ $2$ $4$ $24$ $( 1,18, 2,17)( 3,20, 4,19)( 5,24, 6,23)( 7,22, 8,21)( 9,26,10,25)(11,28,12,27)(13,31,14,32)(15,29,16,30)$
4I $4^{8}$ $2$ $4$ $24$ $( 1,16, 2,15)( 3,14, 4,13)( 5,26, 6,25)( 7,28, 8,27)( 9,24,10,23)(11,22,12,21)(17,29,18,30)(19,31,20,32)$
4J $4^{8}$ $2$ $4$ $24$ $( 1,26, 2,25)( 3,28, 4,27)( 5,15, 6,16)( 7,13, 8,14)( 9,17,10,18)(11,19,12,20)(21,31,22,32)(23,29,24,30)$
4K $4^{8}$ $2$ $4$ $24$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,32,10,31)(11,30,12,29)(13,26,14,25)(15,28,16,27)(17,22,18,21)(19,24,20,23)$
4L $4^{8}$ $2$ $4$ $24$ $( 1,19, 2,20)( 3,17, 4,18)( 5,21, 6,22)( 7,23, 8,24)( 9,27,10,28)(11,25,12,26)(13,30,14,29)(15,32,16,31)$

Malle's constant $a(G)$:     $1/16$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L
Size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2G 2G 2G 2G 2G 2G 2G 2G 2G 2G 2G 2G
Type
32.47.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.47.1b R 1 −1 −1 −1 −1 1 1 1 −1 1 −1 1 1 −1 1 1 −1 1 −1 −1
32.47.1c R 1 −1 −1 −1 −1 1 1 1 −1 1 1 1 −1 −1 −1 −1 1 −1 1 1
32.47.1d R 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 1 1 1 −1 −1 −1 1 1
32.47.1e R 1 −1 −1 −1 −1 1 1 1 1 −1 1 −1 −1 1 −1 1 1 1 −1 −1
32.47.1f R 1 −1 −1 1 1 −1 1 −1 −1 −1 −1 1 −1 1 1 1 1 −1 1 −1
32.47.1g R 1 −1 −1 1 1 −1 1 −1 −1 −1 1 1 1 1 −1 −1 −1 1 −1 1
32.47.1h R 1 −1 −1 1 1 −1 1 −1 1 1 −1 −1 −1 −1 1 −1 1 1 −1 1
32.47.1i R 1 −1 −1 1 1 −1 1 −1 1 1 1 −1 1 −1 −1 1 −1 −1 1 −1
32.47.1j R 1 1 1 −1 −1 −1 1 −1 −1 1 −1 −1 1 1 −1 1 1 −1 −1 1
32.47.1k R 1 1 1 −1 −1 −1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1 1 −1
32.47.1l R 1 1 1 −1 −1 −1 1 −1 1 −1 −1 1 1 −1 −1 −1 1 1 1 −1
32.47.1m R 1 1 1 −1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 −1 1
32.47.1n R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 1 −1 1 1 1
32.47.1o R 1 1 1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 1 −1 −1 −1
32.47.1p R 1 1 1 1 1 1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 −1 −1
32.47.2a S 2 −2 2 −2 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
32.47.2b S 2 −2 2 2 −2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
32.47.2c S 2 2 −2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
32.47.2d S 2 2 −2 2 −2 2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed