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## G = D7×C35order 490 = 2·5·72

### Direct product of C35 and D7

Aliases: D7×C35, C7⋊C70, C352C14, C721C10, (C7×C35)⋊3C2, SmallGroup(490,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C35
 Chief series C1 — C7 — C72 — C7×C35 — D7×C35
 Lower central C7 — D7×C35
 Upper central C1 — C35

Generators and relations for D7×C35
G = < a,b,c | a35=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D7×C35
On 70 points
Generators in S70
(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 11 21 31 6 16 26)(2 12 22 32 7 17 27)(3 13 23 33 8 18 28)(4 14 24 34 9 19 29)(5 15 25 35 10 20 30)(36 61 51 41 66 56 46)(37 62 52 42 67 57 47)(38 63 53 43 68 58 48)(39 64 54 44 69 59 49)(40 65 55 45 70 60 50)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)

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

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

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

175 conjugacy classes

 class 1 2 5A 5B 5C 5D 7A ··· 7F 7G ··· 7AA 10A 10B 10C 10D 14A ··· 14F 35A ··· 35X 35Y ··· 35DD 70A ··· 70X order 1 2 5 5 5 5 7 ··· 7 7 ··· 7 10 10 10 10 14 ··· 14 35 ··· 35 35 ··· 35 70 ··· 70 size 1 7 1 1 1 1 1 ··· 1 2 ··· 2 7 7 7 7 7 ··· 7 1 ··· 1 2 ··· 2 7 ··· 7

175 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C5 C7 C10 C14 C35 C70 D7 C5×D7 C7×D7 D7×C35 kernel D7×C35 C7×C35 C7×D7 C5×D7 C72 C35 D7 C7 C35 C7 C5 C1 # reps 1 1 4 6 4 6 24 24 3 12 18 72

Matrix representation of D7×C35 in GL3(𝔽71) generated by

 16 0 0 0 48 0 0 0 48
,
 1 0 0 0 45 26 0 0 30
,
 70 0 0 0 10 20 0 27 61
G:=sub<GL(3,GF(71))| [16,0,0,0,48,0,0,0,48],[1,0,0,0,45,0,0,26,30],[70,0,0,0,10,27,0,20,61] >;

D7×C35 in GAP, Magma, Sage, TeX

D_7\times C_{35}
% in TeX

G:=Group("D7xC35");
// GroupNames label

G:=SmallGroup(490,5);
// by ID

G=gap.SmallGroup(490,5);
# by ID

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

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

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