# Properties

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

# Learn more

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); 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), (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

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1, 1, 1$ $10$ $2$ $( 4,10)( 6, 8)$ $4, 4, 1, 1$ $25$ $4$ $( 3, 5, 9, 7)( 4, 6,10, 8)$ $4, 4, 1, 1$ $50$ $4$ $( 3, 5, 9, 7)( 4, 8,10, 6)$ $4, 4, 1, 1$ $25$ $4$ $( 3, 7, 9, 5)( 4, 8,10, 6)$ $2, 2, 2, 2, 1, 1$ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$ $5, 1, 1, 1, 1, 1$ $8$ $5$ $( 2, 4, 6, 8,10)$ $5, 2, 2, 1$ $40$ $10$ $( 2, 4, 6, 8,10)( 3, 9)( 5, 7)$ $8, 2$ $50$ $8$ $( 1, 2)( 3, 4, 5, 6, 9,10, 7, 8)$ $8, 2$ $50$ $8$ $( 1, 2)( 3, 4, 7, 8, 9,10, 5, 6)$ $8, 2$ $50$ $8$ $( 1, 2)( 3, 6, 5,10, 9, 8, 7, 4)$ $8, 2$ $50$ $8$ $( 1, 2)( 3, 6, 7, 4, 9, 8, 5,10)$ $5, 5$ $16$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$

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); Label: 400.206 magma: IdentifyGroup(G);
 Character table:  2 4 3 4 3 4 4 1 1 3 3 3 3 . 5 2 1 . . . . 2 1 . . . . 2 1a 2a 4a 4b 4c 2b 5a 10a 8a 8b 8c 8d 5b 2P 1a 1a 2b 2b 2b 1a 5a 5a 4a 4c 4a 4c 5b 3P 1a 2a 4c 4b 4a 2b 5a 10a 8d 8c 8b 8a 5b 5P 1a 2a 4a 4b 4c 2b 1a 2a 8a 8b 8c 8d 1a 7P 1a 2a 4c 4b 4a 2b 5a 10a 8d 8c 8b 8a 5b 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 1 -1 -1 1 -1 1 1 -1 B -B B -B 1 X.6 1 -1 -1 1 -1 1 1 -1 -B B -B B 1 X.7 1 1 -1 -1 -1 1 1 1 B B -B -B 1 X.8 1 1 -1 -1 -1 1 1 1 -B -B B B 1 X.9 2 . A . -A -2 2 . . . . . 2 X.10 2 . -A . A -2 2 . . . . . 2 X.11 8 -4 . . . . 3 1 . . . . -2 X.12 8 4 . . . . 3 -1 . . . . -2 X.13 16 . . . . . -4 . . . . . 1 A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(4) = -Sqrt(-1) = -i 

magma: CharacterTable(G);