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Properties

Label 28T20
Order \(112\)
n \(28\)
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)
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)
$|\Aut(F/K)|$:  $4$

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 TypeSizeOrderRepresentative
$ 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