# Properties

 Label 10T25 Degree $10$ Order $320$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $(C_2^4 : C_5):C_4$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 25);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $25$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $(C_2^4 : C_5):C_4$ CHM label: $1/2[2^{5}]F(5)$ 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: (1,7,9,3)(2,4,8,6)(5,10), (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)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 5: $F_5$

## Low degree siblings

10T24, 16T711, 20T77, 20T78, 20T79, 20T80, 20T83, 20T88, 32T9312, 40T206, 40T207, 40T296, 40T297, 40T298, 40T299, 40T300, 40T301, 40T302, 40T303

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1, 1, 1$ $10$ $2$ $( 4, 9)( 5,10)$ $8, 1, 1$ $40$ $8$ $( 2, 3, 5, 9, 7, 8,10, 4)$ $8, 1, 1$ $40$ $8$ $( 2, 4, 5, 8, 7, 9,10, 3)$ $2, 2, 2, 2, 1, 1$ $20$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)$ $4, 4, 1, 1$ $20$ $4$ $( 2, 5, 7,10)( 3, 4, 8, 9)$ $2, 2, 2, 2, 1, 1$ $5$ $2$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$ $4, 2, 2, 2$ $40$ $4$ $( 1, 2)( 3, 5, 8,10)( 4, 9)( 6, 7)$ $5, 5$ $64$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$ $4, 4, 2$ $40$ $4$ $( 1, 2, 4, 3)( 5,10)( 6, 7, 9, 8)$ $4, 4, 2$ $40$ $4$ $( 1, 2, 5, 4)( 3, 8)( 6, 7,10, 9)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $320=2^{6} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 320.1635 magma: IdentifyGroup(G);
 Character table:  2 6 5 3 3 4 4 6 3 . 3 3 5 1 . . . . . . . 1 . . 1a 2a 8a 8b 2b 4a 2c 4b 5a 4c 4d 2P 1a 1a 4a 4a 1a 2c 1a 2a 5a 2b 2b 3P 1a 2a 8b 8a 2b 4a 2c 4b 5a 4d 4c 5P 1a 2a 8a 8b 2b 4a 2c 4b 1a 4c 4d 7P 1a 2a 8b 8a 2b 4a 2c 4b 5a 4d 4c X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 1 1 -1 -1 X.3 1 1 A -A -1 -1 1 -1 1 A -A X.4 1 1 -A A -1 -1 1 -1 1 -A A X.5 4 4 . . . . 4 . -1 . . X.6 5 1 -1 -1 1 1 -3 -1 . 1 1 X.7 5 1 1 1 1 1 -3 -1 . -1 -1 X.8 5 1 A -A -1 -1 -3 1 . -A A X.9 5 1 -A A -1 -1 -3 1 . A -A X.10 10 -2 . . -2 2 2 . . . . X.11 10 -2 . . 2 -2 2 . . . . A = -E(4) = -Sqrt(-1) = -i 

magma: CharacterTable(G);