# Properties

 Label 8T25 Order $$56$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $C_2^3:C_7$

# Related objects

## Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $25$ Group : $C_2^3:C_7$ CHM label : $E(8):7=F_{56}(8)$ 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), (1,5)(2,6)(3,7)(4,8) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
7:  $C_7$

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 4: None

## Low degree siblings

14T6, 28T11

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

## Group invariants

 Order: $56=2^{3} \cdot 7$ Cyclic: No Abelian: No Solvable: Yes GAP id: [56, 11]
 Character table:  2 3 . . . . . . 3 7 1 1 1 1 1 1 1 . 1a 7a 7b 7c 7d 7e 7f 2a 2P 1a 7d 7f 7b 7e 7a 7c 1a 3P 1a 7c 7e 7d 7b 7f 7a 2a 5P 1a 7f 7d 7a 7c 7b 7e 2a 7P 1a 1a 1a 1a 1a 1a 1a 2a X.1 1 1 1 1 1 1 1 1 X.2 1 A /A C B /C /B 1 X.3 1 B /B /A /C A C 1 X.4 1 C /C B /A /B A 1 X.5 1 /C C /B A B /A 1 X.6 1 /B B A C /A /C 1 X.7 1 /A A /C /B C B 1 X.8 7 . . . . . . -1 A = E(7)^6 B = E(7)^5 C = E(7)^4