# Properties

 Label 28T21 Degree $28$ Order $168$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^2:F_7$

## Group action invariants

 Degree $n$: $28$ Transitive number $t$: $21$ Group: $C_2^2:F_7$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,12,10)(2,11,9)(3,5,13)(4,6,14)(15,21,18,16,22,17)(19,23,26,20,24,25)(27,28), (1,19,8,24,5,18)(2,20,7,23,6,17)(3,26,12,22,13,27)(4,25,11,21,14,28)(9,16)(10,15)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $D_{4}$
$12$:  $C_6\times C_2$
$24$:  $D_4 \times C_3$
$42$:  $F_7$
$84$:  $F_7 \times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $F_7$

Degree 14: $F_7$

## Low degree siblings

28T25

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

## Group invariants

 Order: $168=2^{3} \cdot 3 \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [168, 11]
 Character table:  2 3 2 2 3 2 3 3 3 3 2 2 2 2 2 2 2 2 2 2 3 1 1 1 1 1 1 1 1 1 . . . . 1 1 1 1 1 1 7 1 1 . . . . 1 . . 1 1 1 1 . . . . . . 1a 2a 6a 3a 6b 3b 2b 6c 6d 14a 7a 14b 14c 6e 12a 12b 6f 2c 4a 2P 1a 1a 3b 3b 3a 3a 1a 3b 3a 7a 7a 7a 7a 3a 6c 6d 3b 1a 2b 3P 1a 2a 2a 1a 2a 1a 2b 2b 2b 14c 7a 14b 14a 2c 4a 4a 2c 2c 4a 5P 1a 2a 6b 3b 6a 3a 2b 6d 6c 14c 7a 14b 14a 6f 12b 12a 6e 2c 4a 7P 1a 2a 6a 3a 6b 3b 2b 6c 6d 2a 1a 2b 2a 6e 12a 12b 6f 2c 4a 11P 1a 2a 6b 3b 6a 3a 2b 6d 6c 14a 7a 14b 14c 6f 12b 12a 6e 2c 4a 13P 1a 2a 6a 3a 6b 3b 2b 6c 6d 14c 7a 14b 14a 6e 12a 12b 6f 2c 4a X.1 1 1 1 1 1 1 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 -1 -1 1 1 -1 -1 1 X.3 1 -1 -1 1 -1 1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 X.4 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 X.5 1 -1 A -A /A -/A 1 -A -/A -1 1 1 -1 /A -/A -A A -1 1 X.6 1 -1 /A -/A A -A 1 -/A -A -1 1 1 -1 A -A -/A /A -1 1 X.7 1 -1 A -A /A -/A 1 -A -/A -1 1 1 -1 -/A /A A -A 1 -1 X.8 1 -1 /A -/A A -A 1 -/A -A -1 1 1 -1 -A A /A -/A 1 -1 X.9 1 1 -/A -/A -A -A 1 -/A -A 1 1 1 1 A A /A /A -1 -1 X.10 1 1 -A -A -/A -/A 1 -A -/A 1 1 1 1 /A /A A A -1 -1 X.11 1 1 -/A -/A -A -A 1 -/A -A 1 1 1 1 -A -A -/A -/A 1 1 X.12 1 1 -A -A -/A -/A 1 -A -/A 1 1 1 1 -/A -/A -A -A 1 1 X.13 2 . . 2 . 2 -2 -2 -2 . 2 -2 . . . . . . . X.14 2 . . B . /B -2 -B -/B . 2 -2 . . . . . . . X.15 2 . . /B . B -2 -/B -B . 2 -2 . . . . . . . X.16 6 -6 . . . . 6 . . 1 -1 -1 1 . . . . . . X.17 6 6 . . . . 6 . . -1 -1 -1 -1 . . . . . . X.18 6 . . . . . -6 . . C -1 1 -C . . . . . . X.19 6 . . . . . -6 . . -C -1 1 C . . . . . . A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 C = -E(7)-E(7)^2+E(7)^3-E(7)^4+E(7)^5+E(7)^6 = -Sqrt(-7) = -i7