# Properties

 Label 10T8 Degree $10$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^4 : C_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 8);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $8$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_2^4 : C_5$ CHM label: $[2^{4}]5$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 5: $C_5$

## Low degree siblings

10T8 x 2, 16T178, 20T17 x 6, 20T23, 40T57 x 3

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 4, 9)( 5,10)$ $2, 2, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 3, 8)( 5,10)$ $2, 2, 2, 2, 1, 1$ $5$ $2$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$ $5, 5$ $16$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$ $5, 5$ $16$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)$ $5, 5$ $16$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)$ $5, 5$ $16$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $80=2^{4} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 80.49 magma: IdentifyGroup(G);
 Character table:  2 4 4 4 4 . . . . 5 1 . . . 1 1 1 1 1a 2a 2b 2c 5a 5b 5c 5d 2P 1a 1a 1a 1a 5b 5d 5a 5c 3P 1a 2a 2b 2c 5c 5a 5d 5b 5P 1a 2a 2b 2c 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 A B /B /A X.3 1 1 1 1 B /A A /B X.4 1 1 1 1 /B A /A B X.5 1 1 1 1 /A /B B A X.6 5 -3 1 1 . . . . X.7 5 1 -3 1 . . . . X.8 5 1 1 -3 . . . . A = E(5)^4 B = E(5)^3 

magma: CharacterTable(G);