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Properties

Label 32T64
32T64 1 26 1->26 31 1->31 2 25 2->25 32 2->32 3 27 3->27 29 3->29 4 28 4->28 30 4->30 5 17 5->17 21 5->21 6 18 6->18 22 6->22 7 20 7->20 23 7->23 8 19 8->19 24 8->24 9 9->17 9->24 10 10->18 10->23 11 11->19 11->21 12 12->20 12->22 13 13->27 13->32 14 14->28 14->31 15 15->26 15->30 16 16->25 16->29 17->10 17->14 18->9 18->13 19->12 19->15 20->11 20->16 21->4 21->7 22->3 22->8 23->2 23->5 24->1 24->6 25->1 25->6 26->2 26->5 27->4 27->7 28->3 28->8 29->10 29->14 30->9 30->13 31->11 31->16 32->12 32->15
Degree $32$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2:C_4$

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Copy content magma:G := TransitiveGroup(32, 64);
 

Group invariants

Abstract group:  $C_4^2:C_4$
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 $t$:  $64$
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:  $(1,26,5,21,4,28,8,24)(2,25,6,22,3,27,7,23)(9,17,14,31,11,19,15,30)(10,18,13,32,12,20,16,29)$, $(1,31,16,25)(2,32,15,26)(3,29,14,28)(4,30,13,27)(5,17,10,23)(6,18,9,24)(7,20,11,21)(8,19,12,22)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 6, $C_2^2$
$8$:  $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$
$16$:  $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3
$32$:  32T41

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: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2

Degree 16: $C_2^2 : C_4$, 16T74 x 2, 16T123

Low degree siblings

16T74 x 4, 16T123, 32T64, 32T131

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 $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)$
2B $2^{16}$ $2$ $2$ $16$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,12)( 6,11)( 7, 9)( 8,10)(17,23)(18,24)(19,22)(20,21)(25,31)(26,32)(27,30)(28,29)$
2C $2^{16}$ $2$ $2$ $16$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,10)( 6, 9)( 7,11)( 8,12)(17,23)(18,24)(19,22)(20,21)(25,31)(26,32)(27,30)(28,29)$
2D $2^{8},1^{16}$ $2$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1,12, 4,10)( 2,11, 3, 9)( 5,16, 8,13)( 6,15, 7,14)(17,25,19,27)(18,26,20,28)(21,29,24,32)(22,30,23,31)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,13, 8,16)( 6,14, 7,15)(17,27,19,25)(18,28,20,26)(21,32,24,29)(22,31,23,30)$
4B $4^{8}$ $2$ $4$ $24$ $( 1, 8, 4, 5)( 2, 7, 3, 6)( 9,15,11,14)(10,16,12,13)(17,31,19,30)(18,32,20,29)(21,28,24,26)(22,27,23,25)$
4C $4^{8}$ $2$ $4$ $24$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,13, 8,16)( 6,14, 7,15)(17,25,19,27)(18,26,20,28)(21,29,24,32)(22,30,23,31)$
4D $4^{8}$ $2$ $4$ $24$ $( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,14,11,15)(10,13,12,16)(17,31,19,30)(18,32,20,29)(21,28,24,26)(22,27,23,25)$
4E1 $4^{8}$ $4$ $4$ $24$ $( 1,31,16,25)( 2,32,15,26)( 3,29,14,28)( 4,30,13,27)( 5,17,10,23)( 6,18, 9,24)( 7,20,11,21)( 8,19,12,22)$
4E-1 $4^{8}$ $4$ $4$ $24$ $( 1,25,16,31)( 2,26,15,32)( 3,28,14,29)( 4,27,13,30)( 5,23,10,17)( 6,24, 9,18)( 7,21,11,20)( 8,22,12,19)$
4F1 $4^{4},2^{8}$ $4$ $4$ $20$ $( 1,15, 4,14)( 2,16, 3,13)( 5,11, 8, 9)( 6,12, 7,10)(17,29)(18,30)(19,32)(20,31)(21,27)(22,28)(23,26)(24,25)$
4F-1 $4^{4},2^{8}$ $4$ $4$ $20$ $( 1,14, 4,15)( 2,13, 3,16)( 5, 9, 8,11)( 6,10, 7,12)(17,29)(18,30)(19,32)(20,31)(21,27)(22,28)(23,26)(24,25)$
4G1 $4^{8}$ $4$ $4$ $24$ $( 1,30,13,25)( 2,29,14,26)( 3,32,15,28)( 4,31,16,27)( 5,19,12,23)( 6,20,11,24)( 7,18, 9,21)( 8,17,10,22)$
4G-1 $4^{8}$ $4$ $4$ $24$ $( 1,27,13,31)( 2,28,14,32)( 3,26,15,29)( 4,25,16,30)( 5,22,12,17)( 6,21,11,18)( 7,24, 9,20)( 8,23,10,19)$
4H1 $4^{4},2^{8}$ $4$ $4$ $20$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,15)(14,16)(17,28,19,26)(18,27,20,25)(21,30,24,31)(22,29,23,32)$
4H-1 $4^{4},2^{8}$ $4$ $4$ $20$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,28,19,26)(18,27,20,25)(21,30,24,31)(22,29,23,32)$
8A1 $8^{4}$ $4$ $8$ $28$ $( 1,26, 5,21, 4,28, 8,24)( 2,25, 6,22, 3,27, 7,23)( 9,17,14,31,11,19,15,30)(10,18,13,32,12,20,16,29)$
8A-1 $8^{4}$ $4$ $8$ $28$ $( 1,29, 5,18, 4,32, 8,20)( 2,30, 6,17, 3,31, 7,19)( 9,22,14,27,11,23,15,25)(10,21,13,28,12,24,16,26)$
8B1 $8^{4}$ $4$ $8$ $28$ $( 1,32, 8,18, 4,29, 5,20)( 2,31, 7,17, 3,30, 6,19)( 9,23,15,27,11,22,14,25)(10,24,16,28,12,21,13,26)$
8B-1 $8^{4}$ $4$ $8$ $28$ $( 1,28, 8,21, 4,26, 5,24)( 2,27, 7,22, 3,25, 6,23)( 9,19,15,31,11,17,14,30)(10,20,16,32,12,18,13,29)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 4A1 4A-1 4B 4C 4D 4E1 4E-1 4F1 4F-1 4G1 4G-1 4H1 4H-1 8A1 8A-1 8B1 8B-1
Size 1 1 2 2 2 1 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2C 2C 2D 2D 2C 2C 2D 2D 4D 4D 4D 4D
Type
64.18.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.18.1b R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
64.18.1c R 1 1 1 1 1 1 1 1 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 −1 −1
64.18.1d R 1 1 1 1 1 1 1 1 1 1 1 −1 1 1 −1 −1 −1 1 −1 −1 −1 −1
64.18.1e1 C 1 1 −1 −1 1 1 1 −1 −1 1 −i −i i −i i i −i i −1 1 −1 1
64.18.1e2 C 1 1 −1 −1 1 1 1 −1 −1 1 i i −i i −i −i i −i −1 1 −1 1
64.18.1f1 C 1 1 −1 −1 1 1 1 −1 −1 1 −i i i −i −i −i i i 1 −1 1 −1
64.18.1f2 C 1 1 −1 −1 1 1 1 −1 −1 1 i −i −i i i i −i −i 1 −1 1 −1
64.18.1g1 C 1 1 −1 1 −1 1 1 −1 1 −1 −1 −i −1 1 −i i i 1 i i −i −i
64.18.1g2 C 1 1 −1 1 −1 1 1 −1 1 −1 −1 i −1 1 i −i −i 1 −i −i i i
64.18.1h1 C 1 1 −1 1 −1 1 1 −1 1 −1 1 −i 1 −1 −i i i −1 −i −i i i
64.18.1h2 C 1 1 −1 1 −1 1 1 −1 1 −1 1 i 1 −1 i −i −i −1 i i −i −i
64.18.1i1 C 1 1 1 −1 −1 1 1 1 −1 −1 −i −1 i i 1 −1 1 −i i −i −i i
64.18.1i2 C 1 1 1 −1 −1 1 1 1 −1 −1 i −1 −i −i 1 −1 1 i −i i i −i
64.18.1j1 C 1 1 1 −1 −1 1 1 1 −1 −1 −i 1 i i −1 1 −1 −i −i i i −i
64.18.1j2 C 1 1 1 −1 −1 1 1 1 −1 −1 i 1 −i −i −1 1 −1 i i −i −i i
64.18.2a R 2 2 −2 2 −2 −2 −2 2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
64.18.2b R 2 2 2 −2 −2 −2 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
64.18.2c R 2 2 2 2 2 −2 −2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
64.18.2d S 2 2 −2 −2 2 −2 −2 2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
64.18.4a1 C 4 −4 0 0 0 −4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
64.18.4a2 C 4 −4 0 0 0 4i −4i 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