# Properties

 Label 7T5 Order $$168$$ n $$7$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\GL(3,2)$

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## Group action invariants

 Degree $n$ : $7$ Transitive number $t$ : $5$ Group : $\GL(3,2)$ CHM label : $L(7) = L(3,2)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2)(3,6), (1,2,3,4,5,6,7) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

7T5, 8T37, 14T10 x 2, 21T14, 24T284, 28T32, 42T37, 42T38 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1$ $21$ $2$ $(3,5)(6,7)$ $4, 2, 1$ $42$ $4$ $(2,3,4,7)(5,6)$ $3, 3, 1$ $56$ $3$ $(2,3,5)(4,7,6)$ $7$ $24$ $7$ $(1,2,3,4,5,6,7)$ $7$ $24$ $7$ $(1,2,3,7,6,4,5)$

## Group invariants

 Order: $168=2^{3} \cdot 3 \cdot 7$ Cyclic: No Abelian: No Solvable: No GAP id: [168, 42]
 Character table:  2 3 3 2 . . . 3 1 . . 1 . . 7 1 . . . 1 1 1a 2a 4a 3a 7a 7b 2P 1a 1a 2a 3a 7a 7b 3P 1a 2a 4a 1a 7b 7a 5P 1a 2a 4a 3a 7b 7a 7P 1a 2a 4a 3a 1a 1a X.1 1 1 1 1 1 1 X.2 3 -1 1 . A /A X.3 3 -1 1 . /A A X.4 6 2 . . -1 -1 X.5 7 -1 -1 1 . . X.6 8 . . -1 1 1 A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7