# Properties

 Label 28T28 Degree $28$ Order $168$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_7:A_4$

## Group action invariants

 Degree $n$: $28$ Transitive number $t$: $28$ Group: $D_7:A_4$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,24,26,12,19,14)(2,21,28,11,18,16)(3,23,27,10,20,15)(4,22,25,9,17,13)(5,7,8), (1,10,15)(2,11,13)(3,9,14)(4,12,16)(5,25,23)(6,28,21)(7,26,22)(8,27,24)(17,18,19)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$
$42$:  $F_7$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 4: $A_4$

Degree 7: $F_7$

Degree 14: None

## Low degree siblings

42T30, 42T31

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $7$ $2$ $( 5,25)( 6,26)( 7,27)( 8,28)( 9,21)(10,22)(11,23)(12,24)(13,20)(14,19)(15,18) (16,17)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 1$ $28$ $3$ $( 2, 3, 4)( 5,10,18)( 6,12,19)( 7,11,17)( 8, 9,20)(13,28,21)(14,26,24) (15,25,22)(16,27,23)$ $6, 6, 6, 6, 3, 1$ $28$ $6$ $( 2, 3, 4)( 5,22,18,25,10,15)( 6,24,19,26,12,14)( 7,23,17,27,11,16) ( 8,21,20,28, 9,13)$ $6, 6, 6, 6, 3, 1$ $28$ $6$ $( 2, 4, 3)( 5,15,10,25,18,22)( 6,14,12,26,19,24)( 7,16,11,27,17,23) ( 8,13, 9,28,20,21)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 1$ $28$ $3$ $( 2, 4, 3)( 5,18,10)( 6,19,12)( 7,17,11)( 8,20, 9)(13,21,28)(14,24,26) (15,22,25)(16,23,27)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $21$ $2$ $( 1, 2)( 3, 4)( 5,26)( 6,25)( 7,28)( 8,27)( 9,22)(10,21)(11,24)(12,23)(13,19) (14,20)(15,17)(16,18)$ $14, 14$ $6$ $14$ $( 1, 5,12,13,19,23,26, 2, 6,11,14,20,24,25)( 3, 7,10,15,17,21,28, 4, 8, 9,16, 18,22,27)$ $7, 7, 7, 7$ $6$ $7$ $( 1, 6,12,14,19,24,26)( 2, 5,11,13,20,23,25)( 3, 8,10,16,17,22,28) ( 4, 7, 9,15,18,21,27)$ $14, 14$ $6$ $14$ $( 1, 7,12,15,19,21,26, 4, 6, 9,14,18,24,27)( 2, 8,11,16,20,22,25, 3, 5,10,13, 17,23,28)$ $14, 14$ $6$ $14$ $( 1, 8,12,16,19,22,26, 3, 6,10,14,17,24,28)( 2, 7,11,15,20,21,25, 4, 5, 9,13, 18,23,27)$

## Group invariants

 Order: $168=2^{3} \cdot 3 \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [168, 49]
 Character table:  2 3 3 1 1 1 1 3 3 2 2 2 2 3 1 1 1 1 1 1 . . . . . . 7 1 . . . . . 1 . 1 1 1 1 1a 2a 3a 6a 6b 3b 2b 2c 14a 7a 14b 14c 2P 1a 1a 3b 3b 3a 3a 1a 1a 7a 7a 7a 7a 3P 1a 2a 1a 2a 2a 1a 2b 2c 14c 7a 14a 14b 5P 1a 2a 3b 6b 6a 3a 2b 2c 14b 7a 14c 14a 7P 1a 2a 3a 6a 6b 3b 2b 2c 2b 1a 2b 2b 11P 1a 2a 3b 6b 6a 3a 2b 2c 14c 7a 14a 14b 13P 1a 2a 3a 6a 6b 3b 2b 2c 14a 7a 14b 14c X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 1 -1 1 1 1 1 X.3 1 -1 A -A -/A /A 1 -1 1 1 1 1 X.4 1 -1 /A -/A -A A 1 -1 1 1 1 1 X.5 1 1 A A /A /A 1 1 1 1 1 1 X.6 1 1 /A /A A A 1 1 1 1 1 1 X.7 3 -3 . . . . -1 1 -1 3 -1 -1 X.8 3 3 . . . . -1 -1 -1 3 -1 -1 X.9 6 . . . . . 6 . -1 -1 -1 -1 X.10 6 . . . . . -2 . B -1 D C X.11 6 . . . . . -2 . C -1 B D X.12 6 . . . . . -2 . D -1 C B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(7)-E(7)^2-E(7)^3-E(7)^4-E(7)^5+E(7)^6 C = -E(7)-E(7)^2+E(7)^3+E(7)^4-E(7)^5-E(7)^6 D = -E(7)+E(7)^2-E(7)^3-E(7)^4+E(7)^5-E(7)^6