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Properties

Label 20T256
20T256 1 8 1->8 10 1->10 2 7 2->7 9 2->9 3 5 3->5 11 3->11 4 6 4->6 12 4->12 5->9 14 5->14 6->10 13 6->13 7->11 16 7->16 8->12 15 8->15 9->13 18 9->18 10->14 17 10->17 11->15 19 11->19 12->16 20 12->20 13->4 13->19 14->3 14->20 15->2 15->18 16->1 16->17 17->2 17->7 18->1 18->8 19->3 19->5 20->4 20->6
Degree $20$
Order $2560$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^8:C_{10}$

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Copy content magma:G := TransitiveGroup(20, 256);
 

Group invariants

Abstract group:  $C_2^8:C_{10}$
Copy content magma:IdentifyGroup(G);
 
Order:  $2560=2^{9} \cdot 5$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $20$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $256$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,8,12,16,17,2,7,11,15,18)(3,5,9,13,19)(4,6,10,14,20)$, $(1,10,17,7,16)(2,9,18,8,15)(3,11,19,5,14)(4,12,20,6,13)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$
$80$:  $C_2^4 : C_5$
$160$:  $C_2 \times (C_2^4 : C_5)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_2^4 : C_5$

Low degree siblings

20T256 x 23, 20T262 x 24, 32T205514 x 8, 40T1883 x 48, 40T1975 x 24, 40T2019 x 12, 40T2128 x 48, 40T2166 x 24, 40T2206 x 24, 40T2209 x 48, 40T2210 x 48, 40T2211 x 48, 40T2212 x 24, 40T2240 x 12, 40T2291 x 8, 40T2292 x 24, 40T2293 x 48, 40T2294 x 48, 40T2295 x 48, 40T2296 x 48, 40T2297 x 96, 40T2298 x 96, 40T2299 x 96, 40T2300 x 96, 40T2301 x 96, 40T2302 x 96, 40T2303 x 96

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{8},1^{4}$ $5$ $2$ $8$ $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B $2^{4},1^{12}$ $5$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2C $2^{4},1^{12}$ $5$ $2$ $4$ $( 5, 6)( 7, 8)(13,14)(15,16)$
2D $2^{4},1^{12}$ $10$ $2$ $4$ $(13,15)(14,16)(17,19)(18,20)$
2E $2^{6},1^{8}$ $10$ $2$ $6$ $( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)$
2F $2^{4},1^{12}$ $10$ $2$ $4$ $( 9,11)(10,12)(17,20)(18,19)$
2G $2^{6},1^{8}$ $10$ $2$ $6$ $( 9,11)(10,12)(13,14)(15,16)(17,19)(18,20)$
2H $2^{6},1^{8}$ $10$ $2$ $6$ $( 9,11)(10,12)(13,15)(14,16)(17,18)(19,20)$
2I $2^{6},1^{8}$ $10$ $2$ $6$ $( 5, 6)( 7, 8)(13,15)(14,16)(17,20)(18,19)$
2J $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,19)(18,20)$
2K $2^{6},1^{8}$ $10$ $2$ $6$ $( 5, 6)( 7, 8)( 9,11)(10,12)(17,19)(18,20)$
2L $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 6)( 7, 8)( 9,11)(10,12)(13,14)(15,16)(17,20)(18,19)$
2M $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 6)( 7, 8)( 9,11)(10,12)(13,16)(14,15)(17,18)(19,20)$
2N $2^{6},1^{8}$ $10$ $2$ $6$ $( 5, 7)( 6, 8)(13,16)(14,15)(17,18)(19,20)$
2O $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 7)( 6, 8)( 9,10)(11,12)(13,14)(15,16)(17,19)(18,20)$
2P $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 7)( 6, 8)( 9,10)(11,12)(13,15)(14,16)(17,18)(19,20)$
2Q $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)$
2R $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)$
2S $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 7)( 6, 8)( 9,12)(10,11)(13,14)(15,16)(17,18)(19,20)$
2T $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)$
2U $2^{8},1^{4}$ $10$ $2$ $8$ $( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)$
2V $2^{10}$ $10$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)$
2W $2^{10}$ $10$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,14)(15,16)(17,19)(18,20)$
2X $2^{10}$ $10$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)$
2Y $2^{10}$ $10$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)$
2Z $2^{10}$ $10$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)$
2AA $2^{10}$ $10$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)$
2AB $2^{5},1^{10}$ $16$ $2$ $5$ $( 1, 2)( 5, 6)( 9,10)(13,14)(19,20)$
4A $4^{4},2,1^{2}$ $80$ $4$ $13$ $( 1, 3, 2, 4)( 5, 6)( 9,11,10,12)(13,15,14,16)(17,19,18,20)$
4B $4^{2},2^{3},1^{6}$ $80$ $4$ $9$ $( 1, 4, 2, 3)( 5, 8, 6, 7)(11,12)(13,14)(17,18)$
4C $4^{2},2^{3},1^{6}$ $80$ $4$ $9$ $( 1, 2)( 5, 7, 6, 8)(11,12)(13,16,14,15)(19,20)$
5A1 $5^{4}$ $256$ $5$ $16$ $( 1,13, 6,20,10)( 2,14, 5,19, 9)( 3,15, 8,18,11)( 4,16, 7,17,12)$
5A-1 $5^{4}$ $256$ $5$ $16$ $( 1,10,20, 6,13)( 2, 9,19, 5,14)( 3,11,18, 8,15)( 4,12,17, 7,16)$
5A2 $5^{4}$ $256$ $5$ $16$ $( 1, 6,10,13,20)( 2, 5, 9,14,19)( 3, 8,11,15,18)( 4, 7,12,16,17)$
5A-2 $5^{4}$ $256$ $5$ $16$ $( 1,20,13,10, 6)( 2,19,14, 9, 5)( 3,18,15,11, 8)( 4,17,16,12, 7)$
10A1 $10,5^{2}$ $256$ $10$ $17$ $( 1,19,13, 9, 6, 2,20,14,10, 5)( 3,18,15,11, 8)( 4,17,16,12, 7)$
10A-1 $10,5^{2}$ $256$ $10$ $17$ $( 1, 5,10,14,20, 2, 6, 9,13,19)( 3, 8,11,15,18)( 4, 7,12,16,17)$
10A3 $10,5^{2}$ $256$ $10$ $17$ $( 1, 9,20, 5,13, 2,10,19, 6,14)( 3,11,18, 8,15)( 4,12,17, 7,16)$
10A-3 $10,5^{2}$ $256$ $10$ $17$ $( 1,14, 6,19,10, 2,13, 5,20, 9)( 3,15, 8,18,11)( 4,16, 7,17,12)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

40 x 40 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed