# Properties

 Label 10T23 Degree $10$ Order $320$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times (C_2^4 : D_5)$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 23);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $23$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_2\times (C_2^4 : D_5)$ CHM label: $[2^{5}]D(5)$ 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: (5,10), (1,9)(2,8)(3,7)(4,6), (1,3,5,7,9)(2,4,6,8,10) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$10$:  $D_{5}$
$20$:  $D_{10}$
$160$:  $(C_2^4 : C_5) : C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 5: $D_{5}$

## Low degree siblings

10T23 x 5, 20T71 x 6, 20T73 x 6, 20T76 x 6, 20T81 x 3, 20T85 x 6, 20T87 x 6, 32T9313 x 2, 40T204 x 3, 40T270 x 12, 40T271 x 12, 40T272 x 3, 40T273 x 2, 40T284 x 6, 40T286 x 6, 40T288 x 3, 40T293 x 3, 40T295 x 6

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 5,10)$ $2, 2, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 4, 9)( 5,10)$ $2, 2, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 3, 8)( 5,10)$ $2, 2, 2, 1, 1, 1, 1$ $5$ $2$ $( 3, 8)( 4, 9)( 5,10)$ $2, 2, 2, 2, 1, 1$ $20$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)$ $4, 2, 2, 1, 1$ $20$ $4$ $( 2, 5, 7,10)( 3, 4)( 8, 9)$ $4, 2, 2, 1, 1$ $20$ $4$ $( 2, 5)( 3, 4, 8, 9)( 7,10)$ $4, 4, 1, 1$ $20$ $4$ $( 2, 5, 7,10)( 3, 4, 8, 9)$ $2, 2, 2, 1, 1, 1, 1$ $5$ $2$ $( 2, 7)( 4, 9)( 5,10)$ $2, 2, 2, 2, 1, 1$ $5$ $2$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$ $2, 2, 2, 2, 2$ $20$ $2$ $( 1, 2)( 3, 5)( 4, 9)( 6, 7)( 8,10)$ $4, 2, 2, 2$ $20$ $4$ $( 1, 2)( 3, 5, 8,10)( 4, 9)( 6, 7)$ $5, 5$ $32$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$ $10$ $32$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$ $4, 2, 2, 2$ $20$ $4$ $( 1, 2, 6, 7)( 3, 5)( 4, 9)( 8,10)$ $4, 4, 2$ $20$ $4$ $( 1, 2, 6, 7)( 3, 5, 8,10)( 4, 9)$ $5, 5$ $32$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)$ $10$ $32$ $10$ $( 1, 3, 5, 7, 9, 6, 8,10, 2, 4)$ $2, 2, 2, 2, 2$ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$

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: 320.1636 magma: IdentifyGroup(G);
 Character table:  2 6 6 6 6 6 4 4 4 4 6 6 4 4 1 1 4 4 1 1 6 5 1 . . . . . . . . . . . . 1 1 . . 1 1 1 1a 2a 2b 2c 2d 2e 4a 4b 4c 2f 2g 2h 4d 5a 10a 4e 4f 5b 10b 2i 2P 1a 1a 1a 1a 1a 1a 2c 2b 2g 1a 1a 1a 2c 5b 5b 2b 2g 5a 5a 1a 3P 1a 2a 2b 2c 2d 2e 4a 4b 4c 2f 2g 2h 4d 5b 10b 4e 4f 5a 10a 2i 5P 1a 2a 2b 2c 2d 2e 4a 4b 4c 2f 2g 2h 4d 1a 2i 4e 4f 1a 2i 2i 7P 1a 2a 2b 2c 2d 2e 4a 4b 4c 2f 2g 2h 4d 5b 10b 4e 4f 5a 10a 2i X.1 1 1 1 1 1 1 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 1 -1 -1 1 1 -1 -1 X.3 1 -1 1 1 -1 1 -1 -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 1 1 -1 -1 1 1 1 X.5 2 -2 2 2 -2 . . . . -2 2 . . A -A . . *A -*A -2 X.6 2 -2 2 2 -2 . . . . -2 2 . . *A -*A . . A -A -2 X.7 2 2 2 2 2 . . . . 2 2 . . A A . . *A *A 2 X.8 2 2 2 2 2 . . . . 2 2 . . *A *A . . A A 2 X.9 5 -3 1 1 1 -1 1 1 -1 1 -3 -1 1 . . 1 -1 . . 5 X.10 5 -3 1 1 1 1 -1 -1 1 1 -3 1 -1 . . -1 1 . . 5 X.11 5 3 1 1 -1 -1 -1 -1 -1 -1 -3 1 1 . . 1 1 . . -5 X.12 5 3 1 1 -1 1 1 1 1 -1 -3 -1 -1 . . -1 -1 . . -5 X.13 5 -1 -3 1 3 -1 -1 1 1 -1 1 1 1 . . -1 -1 . . -5 X.14 5 -1 -3 1 3 1 1 -1 -1 -1 1 -1 -1 . . 1 1 . . -5 X.15 5 -1 1 -3 -1 -1 1 -1 1 3 1 1 -1 . . 1 -1 . . -5 X.16 5 -1 1 -3 -1 1 -1 1 -1 3 1 -1 1 . . -1 1 . . -5 X.17 5 1 -3 1 -3 -1 1 -1 1 1 1 -1 1 . . -1 1 . . 5 X.18 5 1 -3 1 -3 1 -1 1 -1 1 1 1 -1 . . 1 -1 . . 5 X.19 5 1 1 -3 1 -1 -1 1 1 -3 1 -1 -1 . . 1 1 . . 5 X.20 5 1 1 -3 1 1 1 -1 -1 -3 1 1 1 . . -1 -1 . . 5 A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 

magma: CharacterTable(G);