↑

Properties

Label 32T11
32T11 1 7 1->7 10 1->10 18 1->18 23 1->23 2 8 2->8 9 2->9 17 2->17 24 2->24 3 6 3->6 12 3->12 19 3->19 22 3->22 4 5 4->5 11 4->11 20 4->20 21 4->21 5->17 5->24 32 5->32 6->18 6->23 31 6->31 7->19 7->22 29 7->29 8->20 8->21 30 8->30 13 9->13 27 9->27 9->30 14 10->14 28 10->28 10->29 16 11->16 26 11->26 11->32 15 12->15 25 12->25 12->31 13->7 13->23 13->24 13->32 14->8 14->23 14->24 14->31 15->5 15->21 15->22 15->29 16->6 16->21 16->22 16->30 17->12 17->27 17->28 18->11 18->27 18->28 19->9 19->25 19->26 20->10 20->25 20->26 21->31 22->32 23->30 24->29 25->2 26->1 27->3 28->4 29->25 30->26 31->28 32->27
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2\times D_4$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(32, 11);
 

Group invariants

Abstract group:  $C_2^2\times D_4$
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 $t$:  $11$
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:  $(1,18)(2,17)(3,19)(4,20)(5,24)(6,23)(7,22)(8,21)(9,27)(10,28)(11,26)(12,25)(13,32)(14,31)(15,29)(16,30)$, $(1,7)(2,8)(3,6)(4,5)(9,30)(10,29)(11,32)(12,31)(13,23)(14,24)(15,21)(16,22)(17,28)(18,27)(19,26)(20,25)$, $(1,23,30,26)(2,24,29,25)(3,22,32,27)(4,21,31,28)(5,17,12,15)(6,18,11,16)(7,19,9,13)(8,20,10,14)$, $(1,10)(2,9)(3,12)(4,11)(5,32)(6,31)(7,29)(8,30)(13,24)(14,23)(15,22)(16,21)(17,27)(18,28)(19,25)(20,26)$
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
$8$:  $D_{4}$ x 4, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 6, $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35, $D_{4}$ x 8

Degree 8: $C_2^3$ x 15, $D_4$ x 4, $D_4\times C_2$ x 24

Degree 16: $C_2^4$, $D_4\times C_2$ x 6, $C_2^2 \times D_4$ x 8

Low degree siblings

16T25 x 8

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

Malle's constant $a(G)$:     $1/16$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D
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 1A 1A 1A 1A 1A 1A 1A 1A 2G 2G 2G 2G
Type
32.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.46.1b R 1 −1 −1 −1 −1 1 1 1 −1 −1 −1 1 1 1 1 −1 1 −1 −1 1
32.46.1c R 1 −1 −1 −1 −1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1
32.46.1d R 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 1 1 −1
32.46.1e R 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 1
32.46.1f R 1 −1 −1 1 1 −1 1 −1 −1 −1 1 1 −1 −1 1 1 1 1 −1 −1
32.46.1g R 1 −1 −1 1 1 −1 1 −1 −1 1 −1 1 −1 1 −1 1 −1 −1 1 1
32.46.1h R 1 −1 −1 1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 −1 −1 1 1
32.46.1i R 1 −1 −1 1 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1
32.46.1j R 1 1 1 −1 −1 −1 1 −1 −1 −1 1 −1 1 1 −1 1 1 −1 1 −1
32.46.1k R 1 1 1 −1 −1 −1 1 −1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 1
32.46.1l R 1 1 1 −1 −1 −1 1 −1 1 −1 1 1 −1 1 −1 −1 −1 1 −1 1
32.46.1m R 1 1 1 −1 −1 −1 1 −1 1 1 −1 1 −1 −1 1 −1 1 −1 1 −1
32.46.1n R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
32.46.1o R 1 1 1 1 1 1 1 1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1
32.46.1p R 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 −1
32.46.2a R 2 −2 2 −2 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
32.46.2b R 2 −2 2 2 −2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
32.46.2c R 2 2 −2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
32.46.2d R 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