# Properties

 Label 10T35 Degree $10$ Order $1440$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $(A_6 : C_2) : C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 35);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $35$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $(A_6 : C_2) : C_2$ CHM label: $L(10).2^{2}=P|L(2,9)$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2)(4,7)(5,8)(9,10), (1,2,10)(3,4,5)(6,7,8), (1,7,3,4,2,5,6,8), (3,6)(4,7)(5,8) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 5: None

## Low degree siblings

12T220, 20T201, 20T204, 20T208, 24T2960, 30T264, 36T2341, 40T1198, 40T1199, 40T1201, 45T187

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1$ $45$ $2$ $( 1, 5)( 2, 9)( 4, 6)( 8,10)$ $4, 4, 1, 1$ $180$ $4$ $( 1, 4, 5, 6)( 2,10, 9, 8)$ $2, 2, 2, 2, 2$ $36$ $2$ $( 1, 9)( 2, 5)( 3, 7)( 4, 8)( 6,10)$ $4, 4, 2$ $90$ $4$ $( 1,10, 5, 8)( 2, 4, 9, 6)( 3, 7)$ $2, 2, 2, 1, 1, 1, 1$ $30$ $2$ $( 1, 5)( 3, 7)( 8,10)$ $3, 3, 3, 1$ $80$ $3$ $( 1, 3, 8)( 2, 4, 6)( 5, 7,10)$ $6, 3, 1$ $240$ $6$ $( 1, 2, 8, 6, 3, 4)( 5, 7,10)$ $4, 4, 1, 1$ $90$ $4$ $( 1, 5, 7, 3)( 2, 8, 6,10)$ $8, 1, 1$ $180$ $8$ $( 1, 6, 3, 8, 7, 2, 5,10)$ $8, 2$ $180$ $8$ $( 1, 2, 5, 8, 7, 6, 3,10)( 4, 9)$ $5, 5$ $144$ $5$ $( 1, 5, 8, 9, 7)( 2, 6, 4, 3,10)$ $10$ $144$ $10$ $( 1, 4, 9, 2, 5, 3, 7, 6, 8,10)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $1440=2^{5} \cdot 3^{2} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 1440.5841 magma: IdentifyGroup(G);
 Character table:  2 5 1 4 1 3 1 1 5 4 3 4 3 3 3 2 2 1 1 . . . . . . . . . 5 1 . . . 1 1 1 . . . . . . 1a 3a 2a 6a 2b 5a 10a 2c 4a 8a 4b 4c 8b 2P 1a 3a 1a 3a 1a 5a 5a 1a 2c 4a 2c 2c 4a 3P 1a 1a 2a 2a 2b 5a 10a 2c 4a 8a 4b 4c 8b 5P 1a 3a 2a 6a 2b 1a 2b 2c 4a 8a 4b 4c 8b 7P 1a 3a 2a 6a 2b 5a 10a 2c 4a 8a 4b 4c 8b X.1 1 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 1 1 X.3 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 X.4 1 1 1 1 -1 1 -1 1 1 -1 1 -1 -1 X.5 9 . -3 . -1 -1 -1 1 1 1 1 -1 1 X.6 9 . -3 . 1 -1 1 1 1 -1 1 1 -1 X.7 9 . 3 . -1 -1 -1 1 1 1 -1 1 -1 X.8 9 . 3 . 1 -1 1 1 1 -1 -1 -1 1 X.9 10 1 2 -1 . . . 2 -2 . 2 . . X.10 10 1 -2 1 . . . 2 -2 . -2 . . X.11 16 -2 . . -4 1 1 . . . . . . X.12 16 -2 . . 4 1 -1 . . . . . . X.13 20 2 . . . . . -4 . . . . . 

magma: CharacterTable(G);