Properties

Label 35T16
Order \(420\)
n \(35\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5.F_7$

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Group action invariants

Degree $n$ :  $35$
Transitive number $t$ :  $16$
Group :  $D_5.F_7$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,33,30,22,19,11,8,5,32,29,21,18,15,7,4,31,28,25,17,14,6,3,35,27,24,16,13,10,2,34,26,23,20,12,9), (1,16,21,11,31,26)(2,18,25,14,32,28,5,19,22,13,35,29)(3,20,24,12,33,30,4,17,23,15,34,27)(7,8,10,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $C_{12}$
20:  $F_5$
42:  $F_7$
60:  $F_5\times C_3$
84:  28T12

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Degree 7: $F_7$

Low degree siblings

There are no siblings 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, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27) (18,33,28)(19,34,29)(20,35,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$
$ 12, 12, 6, 4, 1 $ $35$ $12$ $( 2, 3, 5, 4)( 6,16,11,31,21,26)( 7,18,15,34,22,28,10,19,12,33,25,29) ( 8,20,14,32,23,30, 9,17,13,35,24,27)$
$ 12, 12, 6, 4, 1 $ $35$ $12$ $( 2, 3, 5, 4)( 6,26,21,31,11,16)( 7,28,25,34,12,18,10,29,22,33,15,19) ( 8,30,24,32,13,20, 9,27,23,35,14,17)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $35$ $4$ $( 2, 3, 5, 4)( 6,31)( 7,33,10,34)( 8,35, 9,32)(11,26)(12,28,15,29) (13,30,14,27)(16,21)(17,23,20,24)(18,25,19,22)$
$ 12, 12, 6, 4, 1 $ $35$ $12$ $( 2, 4, 5, 3)( 6,16,11,31,21,26)( 7,19,15,33,22,29,10,18,12,34,25,28) ( 8,17,14,35,23,27, 9,20,13,32,24,30)$
$ 12, 12, 6, 4, 1 $ $35$ $12$ $( 2, 4, 5, 3)( 6,26,21,31,11,16)( 7,29,25,33,12,19,10,28,22,34,15,18) ( 8,27,24,35,13,17, 9,30,23,32,14,20)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $35$ $4$ $( 2, 4, 5, 3)( 6,31)( 7,34,10,33)( 8,32, 9,35)(11,26)(12,29,15,28) (13,27,14,30)(16,21)(17,24,20,23)(18,22,19,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $35$ $6$ $( 2, 5)( 3, 4)( 6,11,21)( 7,15,22,10,12,25)( 8,14,23, 9,13,24)(16,31,26) (17,35,27,20,32,30)(18,34,28,19,33,29)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $35$ $6$ $( 2, 5)( 3, 4)( 6,21,11)( 7,25,12,10,22,15)( 8,24,13, 9,23,14)(16,26,31) (17,30,32,20,27,35)(18,29,33,19,28,34)$
$ 5, 5, 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
$ 15, 15, 5 $ $28$ $15$ $( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$
$ 15, 15, 5 $ $28$ $15$ $( 1, 2, 3, 4, 5)( 6,22,13, 9,25,11, 7,23,14,10,21,12, 8,24,15)(16,27,33,19,30, 31,17,28,34,20,26,32,18,29,35)$
$ 7, 7, 7, 7, 7 $ $6$ $7$ $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
$ 14, 14, 7 $ $30$ $14$ $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$
$ 35 $ $12$ $35$ $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$
$ 35 $ $12$ $35$ $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$

Group invariants

Order:  $420=2^{2} \cdot 3 \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [420, 15]
Character table:   
      2  2  2  2   2   2  2   2   2  2  2   2   2  .   .   .  1   1   .   .
      3  1  1  1   1   1  1   1   1  1  1   1   1  1   1   1  .   .   .   .
      5  1  1  1   .   .  .   .   .  .  .   .   .  1   1   1  1   .   1   1
      7  1  .  .   .   .  .   .   .  .  1   .   .  1   .   .  1   1   1   1

        1a 3a 3b 12a 12b 4a 12c 12d 4b 2a  6a  6b 5a 15a 15b 7a 14a 35a 35b
     2P 1a 3b 3a  6a  6b 2a  6a  6b 2a 1a  3b  3a 5a 15b 15a 7a  7a 35b 35a
     3P 1a 1a 1a  4b  4b 4b  4a  4a 4a 2a  2a  2a 5a  5a  5a 7a 14a 35a 35b
     5P 1a 3b 3a 12b 12a 4a 12d 12c 4b 2a  6b  6a 1a  3b  3a 7a 14a  7a  7a
     7P 1a 3a 3b 12c 12d 4b 12a 12b 4a 2a  6a  6b 5a 15a 15b 1a  2a  5a  5a
    11P 1a 3b 3a 12d 12c 4b 12b 12a 4a 2a  6b  6a 5a 15b 15a 7a 14a 35a 35b
    13P 1a 3a 3b 12a 12b 4a 12c 12d 4b 2a  6a  6b 5a 15a 15b 7a 14a 35a 35b
    17P 1a 3b 3a 12b 12a 4a 12d 12c 4b 2a  6b  6a 5a 15b 15a 7a 14a 35a 35b
    19P 1a 3a 3b 12c 12d 4b 12a 12b 4a 2a  6a  6b 5a 15a 15b 7a 14a 35b 35a
    23P 1a 3b 3a 12d 12c 4b 12b 12a 4a 2a  6b  6a 5a 15b 15a 7a 14a 35b 35a
    29P 1a 3b 3a 12b 12a 4a 12d 12c 4b 2a  6b  6a 5a 15b 15a 7a 14a 35a 35b
    31P 1a 3a 3b 12c 12d 4b 12a 12b 4a 2a  6a  6b 5a 15a 15b 7a 14a 35b 35a

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   C   C  C  -C  -C -C -1  -1  -1  1   1   1  1  -1   1   1
X.4      1  1  1  -C  -C -C   C   C  C -1  -1  -1  1   1   1  1  -1   1   1
X.5      1  A /A -/A  -A -1 -/A  -A -1  1   A  /A  1   A  /A  1   1   1   1
X.6      1 /A  A  -A -/A -1  -A -/A -1  1  /A   A  1  /A   A  1   1   1   1
X.7      1  A /A  /A   A  1  /A   A  1  1   A  /A  1   A  /A  1   1   1   1
X.8      1 /A  A   A  /A  1   A  /A  1  1  /A   A  1  /A   A  1   1   1   1
X.9      1  A /A   D -/D  C  -D  /D -C -1  -A -/A  1   A  /A  1  -1   1   1
X.10     1  A /A  -D  /D -C   D -/D  C -1  -A -/A  1   A  /A  1  -1   1   1
X.11     1 /A  A -/D   D  C  /D  -D -C -1 -/A  -A  1  /A   A  1  -1   1   1
X.12     1 /A  A  /D  -D -C -/D   D  C -1 -/A  -A  1  /A   A  1  -1   1   1
X.13     4  4  4   .   .  .   .   .  .  .   .   . -1  -1  -1  4   .  -1  -1
X.14     4  B /B   .   .  .   .   .  .  .   .   . -1  -A -/A  4   .  -1  -1
X.15     4 /B  B   .   .  .   .   .  .  .   .   . -1 -/A  -A  4   .  -1  -1
X.16     6  .  .   .   .  .   .   .  . -6   .   .  6   .   . -1   1  -1  -1
X.17     6  .  .   .   .  .   .   .  .  6   .   .  6   .   . -1  -1  -1  -1
X.18    12  .  .   .   .  .   .   .  .  .   .   . -3   .   . -2   .   E  /E
X.19    12  .  .   .   .  .   .   .  .  .   .   . -3   .   . -2   .  /E   E

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3
C = -E(4)
  = -Sqrt(-1) = -i
D = -E(12)^7
E = E(35)^2+E(35)^6+E(35)^8+E(35)^18+E(35)^19+E(35)^22+E(35)^23+E(35)^24+E(35)^26+E(35)^31+E(35)^32+E(35)^34
  = (1-Sqrt(-35))/2 = -b35