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## G = C71⋊C7order 497 = 7·71

### The semidirect product of C71 and C7 acting faithfully

Aliases: C71⋊C7, SmallGroup(497,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C71 — C71⋊C7
 Chief series C1 — C71 — C71⋊C7
 Lower central C71 — C71⋊C7
 Upper central C1

Generators and relations for C71⋊C7
G = < a,b | a71=b7=1, bab-1=a30 >

Character table of C71⋊C7

 class 1 7A 7B 7C 7D 7E 7F 71A 71B 71C 71D 71E 71F 71G 71H 71I 71J size 1 71 71 71 71 71 71 7 7 7 7 7 7 7 7 7 7 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ74 ζ76 ζ72 ζ75 ζ7 ζ73 1 1 1 1 1 1 1 1 1 1 linear of order 7 ρ3 1 ζ72 ζ73 ζ7 ζ76 ζ74 ζ75 1 1 1 1 1 1 1 1 1 1 linear of order 7 ρ4 1 ζ73 ζ7 ζ75 ζ72 ζ76 ζ74 1 1 1 1 1 1 1 1 1 1 linear of order 7 ρ5 1 ζ76 ζ72 ζ73 ζ74 ζ75 ζ7 1 1 1 1 1 1 1 1 1 1 linear of order 7 ρ6 1 ζ75 ζ74 ζ76 ζ7 ζ73 ζ72 1 1 1 1 1 1 1 1 1 1 linear of order 7 ρ7 1 ζ7 ζ75 ζ74 ζ73 ζ72 ζ76 1 1 1 1 1 1 1 1 1 1 linear of order 7 ρ8 7 0 0 0 0 0 0 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 complex faithful ρ9 7 0 0 0 0 0 0 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 complex faithful ρ10 7 0 0 0 0 0 0 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 complex faithful ρ11 7 0 0 0 0 0 0 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 complex faithful ρ12 7 0 0 0 0 0 0 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 complex faithful ρ13 7 0 0 0 0 0 0 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 complex faithful ρ14 7 0 0 0 0 0 0 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 complex faithful ρ15 7 0 0 0 0 0 0 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 complex faithful ρ16 7 0 0 0 0 0 0 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 complex faithful ρ17 7 0 0 0 0 0 0 ζ7170+ζ7151+ζ7141+ζ7139+ζ7134+ζ7126+ζ7123 ζ7166+ζ7163+ζ7159+ζ7153+ζ7144+ζ7142+ζ7128 ζ7169+ζ7168+ζ7152+ζ7146+ζ7131+ζ7111+ζ717 ζ7161+ζ7156+ζ7155+ζ7147+ζ7135+ζ7117+ζ7113 ζ7164+ζ7160+ζ7140+ζ7125+ζ7119+ζ713+ζ712 ζ7157+ζ7150+ζ7149+ζ7138+ζ719+ζ716+ζ714 ζ7143+ζ7129+ζ7127+ζ7118+ζ7112+ζ718+ζ715 ζ7158+ζ7154+ζ7136+ζ7124+ζ7116+ζ7115+ζ7110 ζ7167+ζ7165+ζ7162+ζ7133+ζ7122+ζ7121+ζ7114 ζ7148+ζ7145+ζ7137+ζ7132+ζ7130+ζ7120+ζ71 complex faithful

Smallest permutation representation of C71⋊C7
On 71 points: primitive
Generators in S71
```(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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71)
(2 46 38 33 21 49 31)(3 20 4 65 41 26 61)(5 39 7 58 10 51 50)(6 13 44 19 30 28 9)(8 32 47 12 70 53 69)(11 25 16 37 59 55 17)(14 18 56 62 48 57 36)(15 63 22 23 68 34 66)(24 42 71 27 35 40 52)(29 54 43 45 64 67 60)```

`G:=sub<Sym(71)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71), (2,46,38,33,21,49,31)(3,20,4,65,41,26,61)(5,39,7,58,10,51,50)(6,13,44,19,30,28,9)(8,32,47,12,70,53,69)(11,25,16,37,59,55,17)(14,18,56,62,48,57,36)(15,63,22,23,68,34,66)(24,42,71,27,35,40,52)(29,54,43,45,64,67,60)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71), (2,46,38,33,21,49,31)(3,20,4,65,41,26,61)(5,39,7,58,10,51,50)(6,13,44,19,30,28,9)(8,32,47,12,70,53,69)(11,25,16,37,59,55,17)(14,18,56,62,48,57,36)(15,63,22,23,68,34,66)(24,42,71,27,35,40,52)(29,54,43,45,64,67,60) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71)], [(2,46,38,33,21,49,31),(3,20,4,65,41,26,61),(5,39,7,58,10,51,50),(6,13,44,19,30,28,9),(8,32,47,12,70,53,69),(11,25,16,37,59,55,17),(14,18,56,62,48,57,36),(15,63,22,23,68,34,66),(24,42,71,27,35,40,52),(29,54,43,45,64,67,60)]])`

Matrix representation of C71⋊C7 in GL7(𝔽6959)

 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 918 1422 1623 1011 4788 1992
,
 1 0 0 0 0 0 0 1801 4396 2987 2575 4866 2466 2647 3677 897 2942 1850 3150 6355 3225 3172 6045 2076 6690 4614 1063 1445 3374 5673 686 4677 3044 2036 2980 1992 5399 1229 5462 4384 4877 6222 4245 1543 6585 3350 659 820 5886

`G:=sub<GL(7,GF(6959))| [0,0,0,0,0,0,1,1,0,0,0,0,0,918,0,1,0,0,0,0,1422,0,0,1,0,0,0,1623,0,0,0,1,0,0,1011,0,0,0,0,1,0,4788,0,0,0,0,0,1,1992],[1,1801,3677,3172,3374,1992,4245,0,4396,897,6045,5673,5399,1543,0,2987,2942,2076,686,1229,6585,0,2575,1850,6690,4677,5462,3350,0,4866,3150,4614,3044,4384,659,0,2466,6355,1063,2036,4877,820,0,2647,3225,1445,2980,6222,5886] >;`

C71⋊C7 in GAP, Magma, Sage, TeX

`C_{71}\rtimes C_7`
`% in TeX`

`G:=Group("C71:C7");`
`// GroupNames label`

`G:=SmallGroup(497,1);`
`// by ID`

`G=gap.SmallGroup(497,1);`
`# by ID`

`G:=PCGroup([2,-7,-71,1261]);`
`// Polycyclic`

`G:=Group<a,b|a^71=b^7=1,b*a*b^-1=a^30>;`
`// generators/relations`

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