# Properties

 Label 10T10 Degree $10$ Order $100$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5^2 : C_4$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 10);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $10$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5^2 : C_4$ CHM label: $1/2[D(5)^{2}]2$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (2,4,6,8,10), (1,6,9,4)(2,3,8,7)(5,10) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$ x 2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 5: None

## Low degree siblings

10T10, 20T27 x 2, 25T10

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1$ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$ $5, 1, 1, 1, 1, 1$ $4$ $5$ $( 2, 4, 6, 8,10)$ $5, 1, 1, 1, 1, 1$ $4$ $5$ $( 2, 6,10, 4, 8)$ $4, 4, 2$ $25$ $4$ $( 1, 2)( 3, 4, 9,10)( 5, 6, 7, 8)$ $4, 4, 2$ $25$ $4$ $( 1, 2)( 3,10, 9, 4)( 5, 8, 7, 6)$ $5, 5$ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ $5, 5$ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$ $5, 5$ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 8, 4,10, 6)$ $5, 5$ $4$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $100=2^{2} \cdot 5^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 100.12 magma: IdentifyGroup(G);
 Character table:  2 2 2 . . 2 2 . . . . 5 2 . 2 2 . . 2 2 2 2 1a 2a 5a 5b 4a 4b 5c 5d 5e 5f 2P 1a 1a 5b 5a 2a 2a 5f 5d 5e 5c 3P 1a 2a 5b 5a 4b 4a 5f 5d 5e 5c 5P 1a 2a 1a 1a 4a 4b 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 1 1 1 1 X.3 1 -1 1 1 C -C 1 1 1 1 X.4 1 -1 1 1 -C C 1 1 1 1 X.5 4 . -1 -1 . . -1 4 -1 -1 X.6 4 . -1 -1 . . -1 -1 4 -1 X.7 4 . A *A . . B -1 -1 *B X.8 4 . *A A . . *B -1 -1 B X.9 4 . B *B . . *A -1 -1 A X.10 4 . *B B . . A -1 -1 *A A = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 B = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4 = (3+Sqrt(5))/2 = 2+b5 C = -E(4) = -Sqrt(-1) = -i 

magma: CharacterTable(G);