# Properties

 Label 20T27 Order $$100$$ n $$20$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5:F_5$

# Related objects

## Group action invariants

 Degree $n$ : $20$ Transitive number $t$ : $27$ Group : $C_5:F_5$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,12,18,7)(2,11,17,8)(3,5,15,14)(4,6,16,13)(9,20,10,19), (3,8,12,16,19)(4,7,11,15,20) $|\Aut(F/K)|$: $10$

## 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 4: $C_4$

Degree 5: None

Degree 10: $C_5^2 : C_4$

## Low degree siblings

10T10 x 2, 20T27, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $5$ $( 3, 8,12,16,19)( 4, 7,11,15,20)$ $5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $5$ $( 3,12,19, 8,16)( 4,11,20, 7,15)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $25$ $2$ $( 1, 2)( 3, 4)( 5,17)( 6,18)( 7,19)( 8,20)( 9,14)(10,13)(11,16)(12,15)$ $4, 4, 4, 4, 4$ $25$ $4$ $( 1, 3, 2, 4)( 5, 7,17,19)( 6, 8,18,20)( 9,11,14,16)(10,12,13,15)$ $4, 4, 4, 4, 4$ $25$ $4$ $( 1, 4, 2, 3)( 5,19,17, 7)( 6,20,18, 8)( 9,16,14,11)(10,15,13,12)$ $5, 5, 5, 5$ $4$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3, 8,12,16,19)( 4, 7,11,15,20)$ $5, 5, 5, 5$ $4$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,12,19, 8,16)( 4,11,20, 7,15)$ $5, 5, 5, 5$ $4$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,16, 8,19,12)( 4,15, 7,20,11)$ $5, 5, 5, 5$ $4$ $5$ $( 1,10,17, 6,14)( 2, 9,18, 5,13)( 3,12,19, 8,16)( 4,11,20, 7,15)$

## Group invariants

 Order: $100=2^{2} \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [100, 12]
 Character table:  2 2 . . 2 2 2 . . . . 5 2 2 2 . . . 2 2 2 2 1a 5a 5b 2a 4a 4b 5c 5d 5e 5f 2P 1a 5b 5a 1a 2a 2a 5f 5d 5e 5c 3P 1a 5b 5a 2a 4b 4a 5f 5d 5e 5c 5P 1a 1a 1a 2a 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