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Properties

Label 10T33
10T33 1 6 1->6 2 4 2->4 2->4 7 2->7 3 8 3->8 4->6 4->8 9 4->9 5 10 5->10 6->2 6->8 8->6 8->10 10->2
Degree $10$
Order $800$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_5 \wr C_2$

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Copy content magma:G := TransitiveGroup(10, 33);
 

Group invariants

Abstract group:  $F_5 \wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $800=2^{5} \cdot 5^{2}$
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 $t$:  $33$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[F(5)^{2}]2$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(2,4,6,8,10)$, $(1,6)(2,7)(3,8)(4,9)(5,10)$, $(2,4,8,6)$
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 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

20T155, 20T161, 20T167, 20T169, 25T50, 40T874, 40T875, 40T876, 40T877, 40T878, 40T879, 40T880, 40T881, 40T882, 40T883

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 $2^{2},1^{6}$ $10$ $2$ $2$ $(1,9)(3,7)$
2B $2^{5}$ $20$ $2$ $5$ $( 1, 6)( 2, 9)( 3,10)( 4, 5)( 7, 8)$
2C $2^{4},1^{2}$ $25$ $2$ $4$ $( 1, 9)( 2,10)( 3, 7)( 4, 8)$
4A1 $4,1^{6}$ $10$ $4$ $3$ $(1,3,7,5)$
4A-1 $4,1^{6}$ $10$ $4$ $3$ $(1,5,7,3)$
4B1 $4^{2},1^{2}$ $25$ $4$ $6$ $( 1, 3, 9, 7)( 2, 4,10, 8)$
4B-1 $4^{2},1^{2}$ $25$ $4$ $6$ $( 1, 7, 9, 3)( 2, 8,10, 4)$
4C $4^{2},1^{2}$ $50$ $4$ $6$ $( 1, 9, 3, 5)( 2, 8,10, 4)$
4D1 $4,2^{2},1^{2}$ $50$ $4$ $5$ $( 1, 7, 9, 3)( 4,10)( 6, 8)$
4D-1 $4,2^{2},1^{2}$ $50$ $4$ $5$ $( 1, 3, 9, 7)( 4,10)( 6, 8)$
4E $4^{2},2$ $100$ $4$ $7$ $( 1,10, 7, 6)( 2, 5, 4, 3)( 8, 9)$
5A $5,1^{5}$ $8$ $5$ $4$ $( 2, 8, 4,10, 6)$
5B $5^{2}$ $16$ $5$ $8$ $( 1, 9, 7, 5, 3)( 2, 8, 4,10, 6)$
8A1 $8,2$ $100$ $8$ $8$ $( 1, 2, 3, 4, 9,10, 7, 8)( 5, 6)$
8A-1 $8,2$ $100$ $8$ $8$ $( 1, 8, 7,10, 9, 4, 3, 2)( 5, 6)$
10A $5,2^{2},1$ $40$ $10$ $6$ $( 1, 7)( 2,10, 8, 6, 4)( 3, 5)$
10B $10$ $80$ $10$ $9$ $( 1, 4, 9,10, 7, 6, 5, 2, 3, 8)$
20A1 $5,4,1$ $40$ $20$ $7$ $( 1, 3, 7, 5)( 2, 6,10, 4, 8)$
20A-1 $5,4,1$ $40$ $20$ $7$ $( 1, 5, 7, 3)( 2, 8, 4,10, 6)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 4C 4D1 4D-1 4E 5A 5B 8A1 8A-1 10A 10B 20A1 20A-1
Size 1 10 20 25 10 10 25 25 50 50 50 100 8 16 100 100 40 80 40 40
2 P 1A 1A 1A 1A 2A 2A 2C 2C 2C 2A 2A 2C 5A 5B 4B1 4B-1 5A 5B 10A 10A
5 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 4C 4D1 4D-1 4E 1A 1A 8A1 8A-1 2A 2B 4A1 4A-1
Type
800.1191.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
800.1191.1b R 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 1 1 −1 −1 −1
800.1191.1c R 1 1 −1 1 1 1 1 1 1 1 1 −1 1 1 −1 −1 1 −1 1 1
800.1191.1d R 1 1 1 1 −1 −1 1 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1
800.1191.1e1 C 1 −1 −1 1 −i i −1 −1 1 i −i 1 1 1 −i i −1 −1 −i i
800.1191.1e2 C 1 −1 −1 1 i −i −1 −1 1 −i i 1 1 1 i −i −1 −1 i −i
800.1191.1f1 C 1 −1 1 1 −i i −1 −1 1 i −i −1 1 1 i −i −1 1 −i i
800.1191.1f2 C 1 −1 1 1 i −i −1 −1 1 −i i −1 1 1 −i i −1 1 i −i
800.1191.2a R 2 −2 0 2 0 0 2 2 −2 0 0 0 2 2 0 0 −2 0 0 0
800.1191.2b R 2 2 0 2 0 0 −2 −2 −2 0 0 0 2 2 0 0 2 0 0 0
800.1191.2c1 C 2 0 0 −2 −1−i −1+i −2i 2i 0 1−i 1+i 0 2 2 0 0 0 0 −1−i −1+i
800.1191.2c2 C 2 0 0 −2 −1+i −1−i 2i −2i 0 1+i 1−i 0 2 2 0 0 0 0 −1+i −1−i
800.1191.2d1 C 2 0 0 −2 1−i 1+i 2i −2i 0 −1−i −1+i 0 2 2 0 0 0 0 1−i 1+i
800.1191.2d2 C 2 0 0 −2 1+i 1−i −2i 2i 0 −1+i −1−i 0 2 2 0 0 0 0 1+i 1−i
800.1191.8a R 8 4 0 0 4 4 0 0 0 0 0 0 3 −2 0 0 −1 0 −1 −1
800.1191.8b R 8 4 0 0 −4 −4 0 0 0 0 0 0 3 −2 0 0 −1 0 1 1
800.1191.8c1 C 8 −4 0 0 −4i 4i 0 0 0 0 0 0 3 −2 0 0 1 0 i −i
800.1191.8c2 C 8 −4 0 0 4i −4i 0 0 0 0 0 0 3 −2 0 0 1 0 −i i
800.1191.16a R 16 0 4 0 0 0 0 0 0 0 0 0 −4 1 0 0 0 −1 0 0
800.1191.16b R 16 0 −4 0 0 0 0 0 0 0 0 0 −4 1 0 0 0 1 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{10} + 20 x^{8} - 5 x^{7} + 70 x^{6} + \left(2 t - 50\right) x^{5} + \left(25 t + 100\right) x^{4} + \left(20 t + 75\right) x^{3} + \left(-5 t + 225\right) x^{2} - 30 t x + t^{2}$ Copy content Toggle raw display