# Properties

 Label 9T15 Order $$72$$ n $$9$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $C_3^2:C_8$

# Related objects

## Group action invariants

 Degree $n$ : $9$ Transitive number $t$ : $15$ Group : $C_3^2:C_8$ CHM label : $E(9):8$ Parity: $-1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,6,4,5,2,3,8,7), (1,2,9)(3,4,5)(6,7,8), (1,4,7)(2,5,8)(3,6,9) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 47$

Degree 3: None

## Low degree siblings

12T46, 18T28, 24T81, 36T49

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

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 39]
 Character table:  2 3 3 3 3 3 3 3 3 . 3 2 . . . . . . . 2 1a 8a 8b 4a 4b 8c 8d 2a 3a 2P 1a 4b 4a 2a 2a 4a 4b 1a 3a 3P 1a 8c 8d 4b 4a 8a 8b 2a 1a 5P 1a 8d 8c 4a 4b 8b 8a 2a 3a 7P 1a 8b 8a 4b 4a 8d 8c 2a 3a X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 -1 1 1 X.3 1 A -A -1 -1 -A A 1 1 X.4 1 -A A -1 -1 A -A 1 1 X.5 1 B /B A -A -/B -B -1 1 X.6 1 -/B -B -A A B /B -1 1 X.7 1 /B B -A A -B -/B -1 1 X.8 1 -B -/B A -A /B B -1 1 X.9 8 . . . . . . . -1 A = -E(4) = -Sqrt(-1) = -i B = -E(8)