Properties

Label 10T33
Degree $10$
Order $800$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_5 \wr C_2$

Related objects

Downloads

Learn more

Show commands: Magma

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

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_5 \wr C_2$
CHM label:   $[F(5)^{2}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
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)
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 $2^{2},1^{6}$ $10$ $2$ $2$ $( 4,10)( 6, 8)$
2B $2^{5}$ $20$ $2$ $5$ $( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)$
2C $2^{4},1^{2}$ $25$ $2$ $4$ $( 2,10)( 3, 9)( 4, 8)( 5, 7)$
4A1 $4,1^{6}$ $10$ $4$ $3$ $( 2,10, 6, 8)$
4A-1 $4,1^{6}$ $10$ $4$ $3$ $(1,9,3,5)$
4B1 $4^{2},1^{2}$ $25$ $4$ $6$ $( 2, 4,10, 8)( 3, 7, 9, 5)$
4B-1 $4^{2},1^{2}$ $25$ $4$ $6$ $( 2, 8,10, 4)( 3, 5, 9, 7)$
4C $4^{2},1^{2}$ $50$ $4$ $6$ $( 2, 8, 6,10)( 3, 5, 9, 7)$
4D1 $4,2^{2},1^{2}$ $50$ $4$ $5$ $(1,7,9,3)(2,8)(4,6)$
4D-1 $4,2^{2},1^{2}$ $50$ $4$ $5$ $( 1, 7)( 3, 5)( 4, 8,10, 6)$
4E $4^{2},2$ $100$ $4$ $7$ $( 1, 4, 7, 2)( 3,10, 5, 6)( 8, 9)$
5A $5,1^{5}$ $8$ $5$ $4$ $( 2, 4, 6, 8,10)$
5B $5^{2}$ $16$ $5$ $8$ $( 1, 7, 3, 9, 5)( 2,10, 8, 6, 4)$
8A1 $8,2$ $100$ $8$ $8$ $( 1, 6)( 2, 3, 4, 7,10, 9, 8, 5)$
8A-1 $8,2$ $100$ $8$ $8$ $( 1, 6)( 2, 7, 8, 3,10, 5, 4, 9)$
10A $5,2^{2},1$ $40$ $10$ $6$ $( 1, 7, 3, 9, 5)( 2, 4)( 6,10)$
10B $10$ $80$ $10$ $9$ $( 1, 4, 5,10, 9, 6, 3, 2, 7, 8)$
20A1 $5,4,1$ $40$ $20$ $7$ $( 1, 7, 3, 9, 5)( 2, 6, 4,10)$
20A-1 $5,4,1$ $40$ $20$ $7$ $( 1, 7, 5, 9)( 2,10, 8, 6, 4)$

Malle's constant $a(G)$:     $1/2$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $800=2^{5} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  800.1191
magma: IdentifyGroup(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 1i 1+i 2i 2i 0 1+i 1i 0 2 2 0 0 0 0 1i 1+i
800.1191.2c2 C 2 0 0 2 1+i 1i 2i 2i 0 1i 1+i 0 2 2 0 0 0 0 1+i 1i
800.1191.2d1 C 2 0 0 2 1i 1+i 2i 2i 0 1+i 1i 0 2 2 0 0 0 0 1i 1+i
800.1191.2d2 C 2 0 0 2 1+i 1i 2i 2i 0 1i 1+i 0 2 2 0 0 0 0 1+i 1i
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

magma: CharacterTable(G);