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Properties

Label 32T6557
32T6557 1 2 1->2 13 1->13 17 1->17 14 2->14 18 2->18 3 11 3->11 24 3->24 4 12 4->12 23 4->23 5 7 5->7 5->11 6 8 6->8 6->12 7->6 7->8 7->18 8->5 8->17 9 21 9->21 25 9->25 10 22 10->22 26 10->26 11->12 29 11->29 30 12->30 13->14 20 13->20 13->24 19 14->19 14->23 15 15->1 15->22 16 16->2 16->21 32 17->32 31 18->31 19->16 19->20 27 19->27 20->15 28 20->28 21->4 21->15 21->22 22->3 22->16 23->9 23->13 24->10 24->14 25->6 25->26 26->5 27->7 28->8 29->26 29->30 29->31 30->25 30->32 31->27 31->30 32->28 32->29
Degree $32$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4.(C_2\times D_4)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(32, 6557);
 

Group invariants

Abstract group:  $C_2^4.(C_2\times D_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $256=2^{8}$
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:  $4$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6557$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,13,20,15)(2,14,19,16)(3,24,10,22)(4,23,9,21)(5,11,29,26)(6,12,30,25)(7,18,31,27)(8,17,32,28)$, $(1,2)(5,7,6,8)(11,12)(13,24,14,23)(15,22,16,21)(19,20)(25,26)(29,31,30,32)$, $(1,17)(2,18)(3,11)(4,12)(7,8)(9,25)(10,26)(13,14)(19,27)(20,28)(21,22)(29,30)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 12, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$ x 4, $C_2 \times (C_2^2:C_4)$ x 3
$64$:  16T79, 16T101 x 2
$128$:  32T1350

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 6

Degree 8: $C_2^2:C_4$ x 3, $C_2^2 \wr C_2$ x 4

Degree 16: 16T79

Low degree siblings

32T6555 x 2, 32T6557, 32T7453 x 4, 32T7789, 32T8577 x 4, 32T8780 x 4, 32T9262 x 2

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 $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B $2^{16}$ $2$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,20)(10,19)(11,17)(12,18)(13,23)(14,24)(15,21)(16,22)(25,27)(26,28)(29,32)(30,31)$
2C $2^{16}$ $2$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,20)(10,19)(11,17)(12,18)(13,24)(14,23)(15,22)(16,21)(25,27)(26,28)(29,31)(30,32)$
2D $2^{8},1^{16}$ $2$ $2$ $8$ $( 5, 6)( 7, 8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$
2E $2^{16}$ $4$ $2$ $16$ $( 1,20)( 2,19)( 3,10)( 4, 9)( 5,29)( 6,30)( 7,31)( 8,32)(11,26)(12,25)(13,15)(14,16)(17,28)(18,27)(21,23)(22,24)$
2F $2^{16}$ $4$ $2$ $16$ $( 1, 9)( 2,10)( 3,19)( 4,20)( 5,31)( 6,32)( 7,29)( 8,30)(11,28)(12,27)(13,22)(14,21)(15,24)(16,23)(17,26)(18,25)$
2G $2^{16}$ $4$ $2$ $16$ $( 1,28)( 2,27)( 3,25)( 4,26)( 5,24)( 6,23)( 7,13)( 8,14)( 9,12)(10,11)(15,32)(16,31)(17,19)(18,20)(21,29)(22,30)$
2H $2^{16}$ $4$ $2$ $16$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,14)( 6,13)( 7,23)( 8,24)( 9,17)(10,18)(11,20)(12,19)(15,29)(16,30)(21,32)(22,31)$
2I $2^{12},1^{8}$ $8$ $2$ $12$ $( 3, 4)( 5,22)( 6,21)( 7,16)( 8,15)(13,32)(14,31)(17,18)(19,20)(23,30)(24,29)(25,26)$
4A $4^{8}$ $4$ $4$ $24$ $( 1,11, 2,12)( 3,18, 4,17)( 5,16, 6,15)( 7,21, 8,22)( 9,27,10,28)(13,30,14,29)(19,26,20,25)(23,31,24,32)$
4B $4^{8}$ $4$ $4$ $24$ $( 1,17, 2,18)( 3,12, 4,11)( 5,22, 6,21)( 7,15, 8,16)( 9,25,10,26)(13,31,14,32)(19,28,20,27)(23,30,24,29)$
4C1 $4^{4},2^{4},1^{8}$ $4$ $4$ $16$ $( 3, 4)( 5, 7, 6, 8)( 9,10)(13,24,14,23)(15,22,16,21)(17,18)(27,28)(29,31,30,32)$
4C-1 $4^{4},2^{4},1^{8}$ $4$ $4$ $16$ $( 3, 4)( 5, 8, 6, 7)( 9,10)(13,23,14,24)(15,21,16,22)(17,18)(27,28)(29,32,30,31)$
4D1 $4^{4},2^{8}$ $4$ $4$ $20$ $( 1,11, 2,12)( 3,17, 4,18)( 5,21)( 6,22)( 7,15)( 8,16)( 9,28,10,27)(13,32)(14,31)(19,26,20,25)(23,30)(24,29)$
4D-1 $4^{4},2^{8}$ $4$ $4$ $20$ $( 1,11, 2,12)( 3,17, 4,18)( 5,22)( 6,21)( 7,16)( 8,15)( 9,28,10,27)(13,31)(14,32)(19,26,20,25)(23,29)(24,30)$
4E1 $4^{4},2^{4},1^{8}$ $4$ $4$ $16$ $( 5,16, 6,15)( 7,21, 8,22)( 9,10)(13,29,14,30)(19,20)(23,32,24,31)(25,26)(27,28)$
4E-1 $4^{4},2^{4},1^{8}$ $4$ $4$ $16$ $( 1, 2)( 3, 4)( 5,15, 6,16)( 7,22, 8,21)(11,12)(13,30,14,29)(17,18)(23,31,24,32)$
4F $4^{8}$ $8$ $4$ $24$ $( 1,11, 2,12)( 3,17, 4,18)( 5, 7, 6, 8)( 9,27,10,28)(13,23,14,24)(15,22,16,21)(19,25,20,26)(29,32,30,31)$
4G $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,20)( 2,19)( 3, 9)( 4,10)( 5,31, 6,32)( 7,30, 8,29)(11,26)(12,25)(13,22,14,21)(15,24,16,23)(17,27)(18,28)$
4H $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,19, 2,20)( 3,10, 4, 9)( 5,24)( 6,23)( 7,14)( 8,13)(11,25,12,26)(15,32)(16,31)(17,28,18,27)(21,30)(22,29)$
4I $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 4)( 2, 3)( 5,22, 6,21)( 7,15, 8,16)( 9,19)(10,20)(11,17)(12,18)(13,32,14,31)(23,29,24,30)(25,28)(26,27)$
4J $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,25)( 2,26)( 3,27)( 4,28)( 5,24, 6,23)( 7,14, 8,13)( 9,18)(10,17)(11,20)(12,19)(15,32,16,31)(21,30,22,29)$
4K $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,26, 2,25)( 3,28, 4,27)( 5,32)( 6,31)( 7,29)( 8,30)( 9,18,10,17)(11,19,12,20)(13,22)(14,21)(15,23)(16,24)$
4L $4^{8}$ $8$ $4$ $24$ $( 1,19, 2,20)( 3, 9, 4,10)( 5,14, 6,13)( 7,23, 8,24)(11,25,12,26)(15,29,16,30)(17,27,18,28)(21,32,22,31)$
4M $4^{8}$ $8$ $4$ $24$ $( 1,10, 2, 9)( 3,20, 4,19)( 5,24, 6,23)( 7,13, 8,14)(11,27,12,28)(15,32,16,31)(17,25,18,26)(21,29,22,30)$
4N1 $4^{8}$ $16$ $4$ $24$ $( 1,13,20,15)( 2,14,19,16)( 3,24,10,22)( 4,23, 9,21)( 5,11,29,26)( 6,12,30,25)( 7,18,31,27)( 8,17,32,28)$
4N-1 $4^{8}$ $16$ $4$ $24$ $( 1,15,20,13)( 2,16,19,14)( 3,22,10,24)( 4,21, 9,23)( 5,26,29,11)( 6,25,30,12)( 7,27,31,18)( 8,28,32,17)$
4O1 $4^{8}$ $16$ $4$ $24$ $( 1,24, 9,15)( 2,23,10,16)( 3,14,19,21)( 4,13,20,22)( 5,11,31,28)( 6,12,32,27)( 7,17,29,26)( 8,18,30,25)$
4O-1 $4^{8}$ $16$ $4$ $24$ $( 1,22, 9,13)( 2,21,10,14)( 3,16,19,23)( 4,15,20,24)( 5,26,31,17)( 6,25,32,18)( 7,28,29,11)( 8,27,30,12)$
4P1 $4^{8}$ $16$ $4$ $24$ $( 1,32,28,15)( 2,31,27,16)( 3,29,25,21)( 4,30,26,22)( 5,11,24,10)( 6,12,23, 9)( 7,17,13,19)( 8,18,14,20)$
4P-1 $4^{8}$ $16$ $4$ $24$ $( 1, 8,28,13)( 2, 7,27,14)( 3, 5,25,23)( 4, 6,26,24)( 9,30,12,21)(10,29,11,22)(15,19,31,17)(16,20,32,18)$
4Q1 $4^{8}$ $16$ $4$ $24$ $( 1,29,25,15)( 2,30,26,16)( 3,31,28,22)( 4,32,27,21)( 5,11,14,20)( 6,12,13,19)( 7,18,23,10)( 8,17,24, 9)$
4Q-1 $4^{8}$ $16$ $4$ $24$ $( 1, 5,25,13)( 2, 6,26,14)( 3, 7,28,24)( 4, 8,27,23)( 9,32,17,22)(10,31,18,21)(11,16,20,29)(12,15,19,30)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

34 x 34 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed