Properties

Label 28T44
Order \(336\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times F_8:C_3$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $44$
Group :  $C_2\times F_8:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,18,26,20,13,22,15,24,4,12,6,27,8)(2,9,17,25,19,14,21,16,23,3,11,5,28,7), (1,13,9,15,27,23)(2,14,10,16,28,24)(3,17)(4,18)(5,8,12,19,22,26)(6,7,11,20,21,25)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
21:  $C_7:C_3$
42:  $(C_7:C_3) \times C_2$
168:  $C_2^3:(C_7: C_3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$, 14T11, 14T18

Low degree siblings

14T18, 16T712, 42T67

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 $ $7$ $2$ $( 5,20)( 6,19)( 9,24)(10,23)(11,26)(12,25)(13,28)(14,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $28$ $3$ $( 3, 5, 9)( 4, 6,10)( 7,14,25)( 8,13,26)(11,21,28)(12,22,27)(17,19,23) (18,20,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $28$ $3$ $( 3, 9, 5)( 4,10, 6)( 7,25,14)( 8,26,13)(11,28,21)(12,27,22)(17,23,19) (18,24,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,21)( 8,22)( 9,23)(10,24)(11,12)(13,27)(14,28)(15,16) (17,18)(19,20)(25,26)$
$ 6, 6, 6, 6, 2, 2 $ $28$ $6$ $( 1, 2)( 3, 6, 9,17,20,23)( 4, 5,10,18,19,24)( 7,28,25, 8,27,26) (11,22,13,12,21,14)(15,16)$
$ 6, 6, 6, 6, 2, 2 $ $28$ $6$ $( 1, 2)( 3,10,20,17,24, 6)( 4, 9,19,18,23, 5)( 7,11,27, 8,12,28) (13,22,26,14,21,25)(15,16)$
$ 7, 7, 7, 7 $ $24$ $7$ $( 1, 3, 5, 7,24,12,14)( 2, 4, 6, 8,23,11,13)( 9,25,27,16,18,20,22) (10,26,28,15,17,19,21)$
$ 6, 6, 3, 3, 3, 3, 2, 2 $ $28$ $6$ $( 1, 3, 7)( 2, 4, 8)( 5,25,24,20,12, 9)( 6,26,23,19,11,10)(13,28)(14,27) (15,17,21)(16,18,22)$
$ 6, 6, 3, 3, 3, 3, 2, 2 $ $28$ $6$ $( 1, 3,12,16,18,25)( 2, 4,11,15,17,26)( 5,20)( 6,19)( 7,14, 9)( 8,13,10) (21,28,23)(22,27,24)$
$ 14, 14 $ $24$ $14$ $( 1, 4, 5, 8, 9,11,14,15,18,19,22,23,25,28)( 2, 3, 6, 7,10,12,13,16,17,20,21, 24,26,27)$
$ 6, 6, 6, 6, 2, 2 $ $28$ $6$ $( 1, 4, 7,15,18,21)( 2, 3, 8,16,17,22)( 5,11,24,19,25,10)( 6,12,23,20,26, 9) (13,27)(14,28)$
$ 6, 6, 6, 6, 2, 2 $ $28$ $6$ $( 1, 4,12,15,18,26)( 2, 3,11,16,17,25)( 5,19)( 6,20)( 7,28, 9,21,14,23) ( 8,27,10,22,13,24)$
$ 7, 7, 7, 7 $ $24$ $7$ $( 1, 7,27,20,25, 3, 9)( 2, 8,28,19,26, 4,10)( 5,12,18,24,16,22,14) ( 6,11,17,23,15,21,13)$
$ 14, 14 $ $24$ $14$ $( 1, 8,14,19,25, 4, 9,15,22,28, 5,11,18,23)( 2, 7,13,20,26, 3,10,16,21,27, 6, 12,17,24)$
$ 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:  $336=2^{4} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [336, 210]
Character table:   
      2  4  4  2  2  4   2   2  1   2   2   1   2   2  1   1  4
      3  1  1  1  1  1   1   1  .   1   1   .   1   1  .   .  1
      7  1  .  .  .  .   .   .  1   .   .   1   .   .  1   1  1

        1a 2a 3a 3b 2b  6a  6b 7a  6c  6d 14a  6e  6f 7b 14b 2c
     2P 1a 1a 3b 3a 1a  3b  3a 7a  3b  3a  7a  3b  3a 7b  7b 1a
     3P 1a 2a 1a 1a 2b  2b  2b 7b  2a  2a 14b  2c  2c 7a 14a 2c
     5P 1a 2a 3b 3a 2b  6b  6a 7b  6d  6c 14b  6f  6e 7a 14a 2c
     7P 1a 2a 3a 3b 2b  6a  6b 1a  6c  6d  2c  6e  6f 1a  2c 2c
    11P 1a 2a 3b 3a 2b  6b  6a 7a  6d  6c 14a  6f  6e 7b 14b 2c
    13P 1a 2a 3a 3b 2b  6a  6b 7b  6c  6d 14b  6e  6f 7a 14a 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  A /A -1  -A -/A  1   A  /A  -1  -A -/A  1  -1 -1
X.4      1  1 /A  A -1 -/A  -A  1  /A   A  -1 -/A  -A  1  -1 -1
X.5      1  1  A /A  1   A  /A  1   A  /A   1   A  /A  1   1  1
X.6      1  1 /A  A  1  /A   A  1  /A   A   1  /A   A  1   1  1
X.7      3  3  .  . -3   .   .  B   .   .  -B   .   . /B -/B -3
X.8      3  3  .  . -3   .   . /B   .   . -/B   .   .  B  -B -3
X.9      3  3  .  .  3   .   .  B   .   .   B   .   . /B  /B  3
X.10     3  3  .  .  3   .   . /B   .   .  /B   .   .  B   B  3
X.11     7 -1  1  1 -1  -1  -1  .  -1  -1   .   1   1  .   .  7
X.12     7 -1  1  1  1   1   1  .  -1  -1   .  -1  -1  .   . -7
X.13     7 -1  A /A -1  -A -/A  .  -A -/A   .   A  /A  .   .  7
X.14     7 -1 /A  A -1 -/A  -A  . -/A  -A   .  /A   A  .   .  7
X.15     7 -1  A /A  1   A  /A  .  -A -/A   .  -A -/A  .   . -7
X.16     7 -1 /A  A  1  /A   A  . -/A  -A   . -/A  -A  .   . -7

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7