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Properties

Label 10T28
Degree $10$
Order $400$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_5^2 : C_8):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(10, 28);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $28$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_5^2 : C_8):C_2$
CHM label:   $1/2[F(5)^{2}]2$
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), (2,8)(4,6), (1,6,7,2,9,4,3,8)(5,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$16$:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

20T104, 20T107, 20T109, 20T115, 25T31, 40T397, 40T398, 40T399, 40T400

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 4,10)( 6, 8)$
$ 4, 4, 1, 1 $ $25$ $4$ $( 3, 5, 9, 7)( 4, 6,10, 8)$
$ 4, 4, 1, 1 $ $50$ $4$ $( 3, 5, 9, 7)( 4, 8,10, 6)$
$ 4, 4, 1, 1 $ $25$ $4$ $( 3, 7, 9, 5)( 4, 8,10, 6)$
$ 2, 2, 2, 2, 1, 1 $ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$
$ 5, 1, 1, 1, 1, 1 $ $8$ $5$ $( 2, 4, 6, 8,10)$
$ 5, 2, 2, 1 $ $40$ $10$ $( 2, 4, 6, 8,10)( 3, 9)( 5, 7)$
$ 8, 2 $ $50$ $8$ $( 1, 2)( 3, 4, 5, 6, 9,10, 7, 8)$
$ 8, 2 $ $50$ $8$ $( 1, 2)( 3, 4, 7, 8, 9,10, 5, 6)$
$ 8, 2 $ $50$ $8$ $( 1, 2)( 3, 6, 5,10, 9, 8, 7, 4)$
$ 8, 2 $ $50$ $8$ $( 1, 2)( 3, 6, 7, 4, 9, 8, 5,10)$
$ 5, 5 $ $16$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $400=2^{4} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  400.206
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 4A1 4A-1 4B 5A 5B 8A1 8A-1 8B1 8B-1 10A
Size 1 10 25 25 25 50 8 16 50 50 50 50 40
2 P 1A 1A 1A 2B 2B 2B 5A 5B 4A-1 4A1 4A1 4A-1 5A
5 P 1A 2A 2B 4A1 4A-1 4B 1A 1A 8A-1 8A1 8B-1 8B1 2A
Type
400.206.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
400.206.1b R 1 −1 1 1 1 −1 1 1 −1 −1 1 1 −1
400.206.1c R 1 −1 1 1 1 −1 1 1 1 1 −1 −1 −1
400.206.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1
400.206.1e1 C 1 −1 1 −1 −1 1 1 1 −i i i −i −1
400.206.1e2 C 1 −1 1 −1 −1 1 1 1 i −i −i i −1
400.206.1f1 C 1 1 1 −1 −1 −1 1 1 −i i −i i 1
400.206.1f2 C 1 1 1 −1 −1 −1 1 1 i −i i −i 1
400.206.2a1 C 2 0 −2 −2i 2i 0 2 2 0 0 0 0 0
400.206.2a2 C 2 0 −2 2i −2i 0 2 2 0 0 0 0 0
400.206.8a R 8 4 0 0 0 0 3 −2 0 0 0 0 −1
400.206.8b R 8 −4 0 0 0 0 3 −2 0 0 0 0 1
400.206.16a R 16 0 0 0 0 0 −4 1 0 0 0 0 0

magma: CharacterTable(G);