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Properties

Label 10T23
10T23 1 3 1->3 9 1->9 2 4 2->4 8 2->8 5 3->5 7 3->7 6 4->6 4->6 5->7 10 5->10 6->8 7->9 8->10 9->1 10->2
Degree $10$
Order $320$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times (C_2^4 : D_5)$

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Copy content magma:G := TransitiveGroup(10, 23);
 

Group invariants

Abstract group:  $C_2\times (C_2^4 : D_5)$
Copy content magma:IdentifyGroup(G);
 
Order:  $320=2^{6} \cdot 5$
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 $n$:  $10$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $23$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[2^{5}]D(5)$
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:  $(5,10)$, $(1,9)(2,8)(3,7)(4,6)$, $(1,3,5,7,9)(2,4,6,8,10)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$10$:  $D_{5}$
$20$:  $D_{10}$
$160$:  $(C_2^4 : C_5) : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $D_{5}$

Low degree siblings

10T23 x 5, 20T71 x 6, 20T73 x 6, 20T76 x 6, 20T81 x 3, 20T85 x 6, 20T87 x 6, 32T9313 x 2, 40T204 x 3, 40T270 x 12, 40T271 x 12, 40T272 x 3, 40T273 x 2, 40T284 x 6, 40T286 x 6, 40T288 x 3, 40T293 x 3, 40T295 x 6

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{10}$ $1$ $1$ $0$ $()$
2A $2^{5}$ $1$ $2$ $5$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
2B $2^{3},1^{4}$ $5$ $2$ $3$ $( 2, 7)( 4, 9)( 5,10)$
2C $2^{3},1^{4}$ $5$ $2$ $3$ $( 3, 8)( 4, 9)( 5,10)$
2D $2,1^{8}$ $5$ $2$ $1$ $(3,8)$
2E $2^{4},1^{2}$ $5$ $2$ $4$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$
2F $2^{2},1^{6}$ $5$ $2$ $2$ $(1,6)(4,9)$
2G $2^{2},1^{6}$ $5$ $2$ $2$ $(2,7)(3,8)$
2H $2^{4},1^{2}$ $20$ $2$ $4$ $( 1, 2)( 3, 5)( 6, 7)( 8,10)$
2I $2^{5}$ $20$ $2$ $5$ $( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9)$
4A $4^{2},2$ $20$ $4$ $7$ $( 1, 6)( 2, 5, 7,10)( 3, 4, 8, 9)$
4B $4,2^{3}$ $20$ $4$ $6$ $( 1, 4, 6, 9)( 2, 3)( 5,10)( 7, 8)$
4C $4,2^{3}$ $20$ $4$ $6$ $( 1, 9)( 2, 8, 7, 3)( 4, 6)( 5,10)$
4D $4^{2},1^{2}$ $20$ $4$ $6$ $( 1, 2, 6, 7)( 3,10, 8, 5)$
4E $4,2^{2},1^{2}$ $20$ $4$ $5$ $( 2,10)( 3, 9, 8, 4)( 5, 7)$
4F $4,2^{2},1^{2}$ $20$ $4$ $5$ $( 1, 8, 6, 3)( 4,10)( 5, 9)$
5A1 $5^{2}$ $32$ $5$ $8$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)$
5A2 $5^{2}$ $32$ $5$ $8$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$
10A1 $10$ $32$ $10$ $9$ $( 1,10, 4, 8, 2, 6, 5, 9, 3, 7)$
10A3 $10$ $32$ $10$ $9$ $( 1, 8, 5, 7, 4, 6, 3,10, 2, 9)$

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 5A1 5A2 10A1 10A3
Size 1 1 5 5 5 5 5 5 20 20 20 20 20 20 20 20 32 32 32 32
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2E 2F 2G 2E 2G 2F 5A2 5A1 5A1 5A2
5 P 1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 1A 1A 2A 2A
Type
320.1636.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
320.1636.1b R 1 −1 −1 −1 1 1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 −1 −1
320.1636.1c R 1 −1 −1 −1 1 1 1 −1 1 −1 −1 −1 1 1 1 −1 1 1 −1 −1
320.1636.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
320.1636.2a1 R 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
320.1636.2a2 R 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
320.1636.2b1 R 2 −2 −2 −2 2 2 2 −2 0 0 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
320.1636.2b2 R 2 −2 −2 −2 2 2 2 −2 0 0 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
320.1636.5a R 5 5 −3 1 1 1 −3 1 1 1 1 −1 −1 1 −1 −1 0 0 0 0
320.1636.5b R 5 5 1 −3 1 −3 1 1 1 1 −1 1 −1 −1 1 −1 0 0 0 0
320.1636.5c R 5 5 1 1 −3 1 1 −3 1 1 −1 −1 1 −1 −1 1 0 0 0 0
320.1636.5d R 5 5 −3 1 1 1 −3 1 −1 −1 −1 1 1 −1 1 1 0 0 0 0
320.1636.5e R 5 5 1 −3 1 −3 1 1 −1 −1 1 −1 1 1 −1 1 0 0 0 0
320.1636.5f R 5 5 1 1 −3 1 1 −3 −1 −1 1 1 −1 1 1 −1 0 0 0 0
320.1636.5g R 5 −5 −1 −1 −3 1 1 3 −1 1 −1 −1 −1 1 1 1 0 0 0 0
320.1636.5h R 5 −5 −1 −1 −3 1 1 3 1 −1 1 1 1 −1 −1 −1 0 0 0 0
320.1636.5i R 5 −5 −1 3 1 −3 1 −1 −1 1 −1 1 1 1 −1 −1 0 0 0 0
320.1636.5j R 5 −5 −1 3 1 −3 1 −1 1 −1 1 −1 −1 −1 1 1 0 0 0 0
320.1636.5k R 5 −5 3 −1 1 1 −3 −1 −1 1 1 −1 1 −1 1 −1 0 0 0 0
320.1636.5l R 5 −5 3 −1 1 1 −3 −1 1 −1 −1 1 −1 1 −1 1 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{10} - t x^{6} + \left(3 t + 25\right) x^{4} + \left(-4 t + 300\right)$ Copy content Toggle raw display