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## G = C7×D35order 490 = 2·5·72

### Direct product of C7 and D35

Aliases: C7×D35, C353D7, C351C14, C721D5, C5⋊(C7×D7), C7⋊(C7×D5), (C7×C35)⋊2C2, SmallGroup(490,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — C7×D35
 Chief series C1 — C7 — C35 — C7×C35 — C7×D35
 Lower central C35 — C7×D35
 Upper central C1 — C7

Generators and relations for C7×D35
G = < a,b,c | a7=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C7×D35
On 70 points
Generators in S70
(1 31 26 21 16 11 6)(2 32 27 22 17 12 7)(3 33 28 23 18 13 8)(4 34 29 24 19 14 9)(5 35 30 25 20 15 10)(36 41 46 51 56 61 66)(37 42 47 52 57 62 67)(38 43 48 53 58 63 68)(39 44 49 54 59 64 69)(40 45 50 55 60 65 70)
(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)
(1 36)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)

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

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

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

133 conjugacy classes

 class 1 2 5A 5B 7A ··· 7F 7G ··· 7AA 14A ··· 14F 35A ··· 35CR order 1 2 5 5 7 ··· 7 7 ··· 7 14 ··· 14 35 ··· 35 size 1 35 2 2 1 ··· 1 2 ··· 2 35 ··· 35 2 ··· 2

133 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C7 C14 D5 D7 C7×D5 D35 C7×D7 C7×D35 kernel C7×D35 C7×C35 D35 C35 C72 C35 C7 C7 C5 C1 # reps 1 1 6 6 2 3 12 12 18 72

Matrix representation of C7×D35 in GL2(𝔽71) generated by

 20 0 0 20
,
 10 0 56 64
,
 9 67 20 62
G:=sub<GL(2,GF(71))| [20,0,0,20],[10,56,0,64],[9,20,67,62] >;

C7×D35 in GAP, Magma, Sage, TeX

C_7\times D_{35}
% in TeX

G:=Group("C7xD35");
// GroupNames label

G:=SmallGroup(490,8);
// by ID

G=gap.SmallGroup(490,8);
# by ID

G:=PCGroup([4,-2,-7,-5,-7,674,6723]);
// Polycyclic

G:=Group<a,b,c|a^7=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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