# Properties

 Label 28T20 Degree $28$ Order $112$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times F_8$

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## Group action invariants

 Degree $n$: $28$ Transitive number $t$: $20$ Group: $C_2\times F_8$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,15)(2,16)(3,17)(4,18)(5,6)(7,21)(8,22)(9,10)(11,12)(13,14)(19,20)(23,24)(25,26)(27,28), (1,5,9,14,18,22,25)(2,6,10,13,17,21,26)(3,7,12,16,20,24,27)(4,8,11,15,19,23,28)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$7$:  $C_7$
$14$:  $C_{14}$
$56$:  $C_2^3:C_7$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: None

Degree 7: $C_7$

Degree 14: $C_{14}$, 14T6, 14T9

## Low degree siblings

14T9, 16T196, 28T19 x 3

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

## Group invariants

 Order: $112=2^{4} \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [112, 41]
 Character table:  2 4 4 4 1 1 1 1 1 1 1 1 1 1 1 1 4 7 1 . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 7a 14a 7b 14b 7c 14c 7d 14d 14e 7e 14f 7f 2c 2P 1a 1a 1a 7b 7b 7d 7d 7f 7f 7a 7a 7c 7c 7e 7e 1a 3P 1a 2a 2b 7c 14c 7f 14f 7b 14b 7e 14e 14a 7a 14d 7d 2c 5P 1a 2a 2b 7e 14e 7c 14c 7a 14a 7f 14f 14d 7d 14b 7b 2c 7P 1a 2a 2b 1a 2c 1a 2c 1a 2c 1a 2c 2c 1a 2c 1a 2c 11P 1a 2a 2b 7d 14d 7a 14a 7e 14e 7b 14b 14f 7f 14c 7c 2c 13P 1a 2a 2b 7f 14f 7e 14e 7d 14d 7c 14c 14b 7b 14a 7a 2c X.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 X.3 1 1 -1 A -A B -B C -C /C -/C -/B /B -/A /A -1 X.4 1 1 -1 B -B /C -/C /A -/A A -A -C C -/B /B -1 X.5 1 1 -1 C -C /A -/A B -B /B -/B -A A -/C /C -1 X.6 1 1 -1 /C -/C A -A /B -/B B -B -/A /A -C C -1 X.7 1 1 -1 /B -/B C -C A -A /A -/A -/C /C -B B -1 X.8 1 1 -1 /A -/A /B -/B /C -/C C -C -B B -A A -1 X.9 1 1 1 A A B B C C /C /C /B /B /A /A 1 X.10 1 1 1 B B /C /C /A /A A A C C /B /B 1 X.11 1 1 1 C C /A /A B B /B /B A A /C /C 1 X.12 1 1 1 /C /C A A /B /B B B /A /A C C 1 X.13 1 1 1 /B /B C C A A /A /A /C /C B B 1 X.14 1 1 1 /A /A /B /B /C /C C C B B A A 1 X.15 7 -1 1 . . . . . . . . . . . . -7 X.16 7 -1 -1 . . . . . . . . . . . . 7 A = E(7)^6 B = E(7)^5 C = E(7)^4