# Properties

 Label 10T33 Degree $10$ Order $800$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $F_5 \wr C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 33);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $33$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $F_5 \wr C_2$ CHM label: $[F(5)^{2}]2$ 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), (1,6)(2,7)(3,8)(4,9)(5,10), (2,4,8,6) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_4\wr C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 5: None

## Low degree siblings

20T155, 20T161, 20T167, 20T169, 25T50, 40T874, 40T875, 40T876, 40T877, 40T878, 40T879, 40T880, 40T881, 40T882, 40T883

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 1, 1, 1, 1, 1$ $8$ $5$ $( 2, 4, 6, 8,10)$ $5, 5$ $16$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ $2, 2, 2, 2, 1, 1$ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$ $2, 2, 2, 2, 2$ $20$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ $10$ $80$ $10$ $( 1, 8, 3,10, 5, 2, 7, 4, 9, 6)$ $4, 4, 1, 1$ $50$ $4$ $( 3, 5, 9, 7)( 4, 8,10, 6)$ $2, 2, 1, 1, 1, 1, 1, 1$ $10$ $2$ $( 4,10)( 6, 8)$ $5, 2, 2, 1$ $40$ $10$ $( 1, 3, 5, 7, 9)( 4,10)( 6, 8)$ $4, 4, 2$ $100$ $4$ $( 1, 6, 3, 8)( 2, 7)( 4, 5,10, 9)$ $4, 4, 1, 1$ $25$ $4$ $( 3, 5, 9, 7)( 4, 6,10, 8)$ $4, 4, 1, 1$ $25$ $4$ $( 3, 7, 9, 5)( 4, 8,10, 6)$ $4, 1, 1, 1, 1, 1, 1$ $10$ $4$ $( 4, 6,10, 8)$ $5, 4, 1$ $40$ $20$ $( 1, 3, 5, 7, 9)( 4, 6,10, 8)$ $4, 2, 2, 1, 1$ $50$ $4$ $( 3, 9)( 4, 8,10, 6)( 5, 7)$ $8, 2$ $100$ $8$ $( 1, 6, 5,10, 3, 8, 9, 4)( 2, 7)$ $4, 1, 1, 1, 1, 1, 1$ $10$ $4$ $( 4, 8,10, 6)$ $5, 4, 1$ $40$ $20$ $( 1, 3, 5, 7, 9)( 4, 8,10, 6)$ $4, 2, 2, 1, 1$ $50$ $4$ $( 3, 9)( 4, 6,10, 8)( 5, 7)$ $8, 2$ $100$ $8$ $( 1, 6, 9, 4, 3, 8, 5,10)( 2, 7)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $800=2^{5} \cdot 5^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 800.1191 magma: IdentifyGroup(G);
 Character table:  2 5 2 1 5 3 1 4 4 2 3 5 5 4 2 4 3 4 2 4 3 5 2 2 2 . 1 1 . 1 1 . . . 1 1 . . 1 1 . . 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4c 4d 4e 20a 4f 8a 4g 20b 4h 8b 2P 1a 5a 5b 1a 1a 5b 2a 1a 5a 2a 2a 2a 2c 10b 2c 4c 2c 10b 2c 4d 3P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4d 4c 4g 20b 4h 8b 4e 20a 4f 8a 5P 1a 1a 1a 2a 2b 2b 4a 2c 2c 4b 4c 4d 4e 4e 4f 8a 4g 4g 4h 8b 7P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4d 4c 4g 20b 4h 8b 4e 20a 4f 8a 11P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4d 4c 4g 20b 4h 8b 4e 20a 4f 8a 13P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4c 4d 4e 20a 4f 8a 4g 20b 4h 8b 17P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4c 4d 4e 20a 4f 8a 4g 20b 4h 8b 19P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4d 4c 4g 20b 4h 8b 4e 20a 4f 8a 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 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 B B B -B -B -B -B B X.6 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 -B -B -B B B B B -B X.7 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 B B B B -B -B -B -B X.8 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -B -B -B -B B B B B X.9 2 2 2 2 . . -2 -2 -2 . 2 2 . . . . . . . . X.10 2 2 2 2 . . -2 2 2 . -2 -2 . . . . . . . . X.11 2 2 2 -2 . . . . . . A -A C C -C . /C /C -/C . X.12 2 2 2 -2 . . . . . . -A A /C /C -/C . C C -C . X.13 2 2 2 -2 . . . . . . -A A -/C -/C /C . -C -C C . X.14 2 2 2 -2 . . . . . . A -A -C -C C . -/C -/C /C . X.15 8 3 -2 . . . . 4 -1 . . . -4 1 . . -4 1 . . X.16 8 3 -2 . . . . 4 -1 . . . 4 -1 . . 4 -1 . . X.17 8 3 -2 . . . . -4 1 . . . D -B . . -D B . . X.18 8 3 -2 . . . . -4 1 . . . -D B . . D -B . . X.19 16 -4 1 . -4 1 . . . . . . . . . . . . . . X.20 16 -4 1 . 4 -1 . . . . . . . . . . . . . . A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(4) = -Sqrt(-1) = -i C = 1-E(4) = 1-Sqrt(-1) = 1-i D = -4*E(4) = -4*Sqrt(-1) = -4i 

magma: CharacterTable(G);