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Properties

Label 32T369
Degree $32$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Q_{16}:C_2^2$

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Copy content magma:G := TransitiveGroup(32, 369);
 

Group invariants

Abstract group:  $Q_{16}:C_2^2$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $369$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $16$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,15,3,13)(2,16,4,14)(5,11,7,9)(6,12,8,10)(17,21,19,23)(18,22,20,24)(25,31,27,29)(26,32,28,30)$, $(1,16,3,14)(2,15,4,13)(5,12,7,10)(6,11,8,9)(17,22,19,24)(18,21,20,23)(25,32,27,30)(26,31,28,29)$, $(1,32,7,18,3,30,5,20)(2,31,8,17,4,29,6,19)(9,26,15,22,11,28,13,24)(10,25,16,21,12,27,14,23)$, $(1,19,7,31,3,17,5,29)(2,20,8,32,4,18,6,30)(9,23,15,25,11,21,13,27)(10,24,16,26,12,22,14,28)$
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 4, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 6, $C_2^4$
$32$:  $C_2^2 \times D_4$, 16T32 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

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

Degree 16: $D_4\times C_2$, 16T50 x 2

Low degree siblings

32T278 x 2, 32T369

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

Malle's constant $a(G)$:     $1/8$

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 8A 8B 8C 8D
Size 1 1 1 1 2 2 4 4 2 2 2 2 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2C 2C 2C 2C 2C 2C 2C 2C 2C 2C 4D 4D 4D 4D
Type
64.255.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.255.1b R 1 1 −1 −1 −1 1 −1 1 −1 −1 1 1 −1 1 1 −1 −1 1 −1 1 1 −1
64.255.1c R 1 1 −1 −1 −1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 −1 −1 1
64.255.1d R 1 1 −1 −1 −1 1 1 −1 −1 −1 1 1 −1 1 1 1 −1 −1 1 −1 −1 1
64.255.1e R 1 1 −1 −1 −1 1 1 −1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1
64.255.1f R 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 −1 1 −1 1 1 −1 −1 −1 1 1
64.255.1g R 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 1 −1 1 1 −1 −1 1 1 −1 −1
64.255.1h R 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1
64.255.1i R 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 1 −1 1 −1 −1 1 −1 −1 1 1
64.255.1j R 1 1 1 1 −1 −1 −1 −1 −1 1 −1 1 −1 −1 1 1 1 1 1 −1 1 −1
64.255.1k R 1 1 1 1 −1 −1 −1 −1 −1 1 −1 1 1 1 −1 1 −1 1 −1 1 −1 1
64.255.1l R 1 1 1 1 −1 −1 1 1 −1 1 −1 1 −1 −1 1 −1 1 −1 −1 1 −1 1
64.255.1m R 1 1 1 1 −1 −1 1 1 −1 1 −1 1 1 1 −1 −1 −1 −1 1 −1 1 −1
64.255.1n R 1 1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1
64.255.1o R 1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1
64.255.1p R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 1 −1 1 −1 −1 −1 −1
64.255.2a R 2 2 −2 −2 −2 2 0 0 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0
64.255.2b R 2 2 −2 −2 2 −2 0 0 −2 2 2 −2 0 0 0 0 0 0 0 0 0 0
64.255.2c R 2 2 2 2 −2 −2 0 0 2 −2 2 −2 0 0 0 0 0 0 0 0 0 0
64.255.2d R 2 2 2 2 2 2 0 0 −2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0
64.255.4a S 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
64.255.4b S 4 −4 4 −4 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