Properties

 Label 8T36 Order $$168$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $C_2^3:(C_7: C_3)$

Related objects

Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $36$ Group : $C_2^3:(C_7: C_3)$ CHM label : $E(8):F_{21}$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,3)(2,8)(4,6)(5,7), (1,2,6,3,4,5,7), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (1,5)(2,6)(3,7)(4,8) $|\Aut(F/K)|$: $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
21:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 4: None

Low degree siblings

14T11, 24T283, 28T27, 42T26

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 $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1, 1$ $28$ $3$ $(3,6,7)(4,8,5)$ $3, 3, 1, 1$ $28$ $3$ $(3,7,6)(4,5,8)$ $7, 1$ $24$ $7$ $(2,3,6,5,7,8,4)$ $7, 1$ $24$ $7$ $(2,4,8,7,5,6,3)$ $6, 2$ $28$ $6$ $(1,2)(3,4,6,8,7,5)$ $6, 2$ $28$ $6$ $(1,2)(3,5,7,8,6,4)$ $2, 2, 2, 2$ $7$ $2$ $(1,2)(3,8)(4,7)(5,6)$

Group invariants

 Order: $168=2^{3} \cdot 3 \cdot 7$ Cyclic: No Abelian: No Solvable: Yes GAP id: [168, 43]
 Character table:  2 3 1 1 . . 1 1 3 3 1 1 1 . . 1 1 1 7 1 . . 1 1 . . . 1a 3a 3b 7a 7b 6a 6b 2a 2P 1a 3b 3a 7a 7b 3a 3b 1a 3P 1a 1a 1a 7b 7a 2a 2a 2a 5P 1a 3b 3a 7b 7a 6b 6a 2a 7P 1a 3a 3b 1a 1a 6a 6b 2a X.1 1 1 1 1 1 1 1 1 X.2 1 A /A 1 1 /A A 1 X.3 1 /A A 1 1 A /A 1 X.4 3 . . B /B . . 3 X.5 3 . . /B B . . 3 X.6 7 1 1 . . -1 -1 -1 X.7 7 A /A . . -/A -A -1 X.8 7 /A A . . -A -/A -1 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7