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## G = C4×D35order 280 = 23·5·7

### Direct product of C4 and D35

Aliases: C4×D35, C282D5, C202D7, C1402C2, C2.1D70, D70.2C2, C10.9D14, C14.9D10, Dic355C2, C70.9C22, C53(C4×D7), C72(C4×D5), C357(C2×C4), SmallGroup(280,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — C4×D35
 Chief series C1 — C7 — C35 — C70 — D70 — C4×D35
 Lower central C35 — C4×D35
 Upper central C1 — C4

Generators and relations for C4×D35
G = < a,b,c | a4=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C4×D35
On 140 points
Generators in S140
(1 123 40 103)(2 124 41 104)(3 125 42 105)(4 126 43 71)(5 127 44 72)(6 128 45 73)(7 129 46 74)(8 130 47 75)(9 131 48 76)(10 132 49 77)(11 133 50 78)(12 134 51 79)(13 135 52 80)(14 136 53 81)(15 137 54 82)(16 138 55 83)(17 139 56 84)(18 140 57 85)(19 106 58 86)(20 107 59 87)(21 108 60 88)(22 109 61 89)(23 110 62 90)(24 111 63 91)(25 112 64 92)(26 113 65 93)(27 114 66 94)(28 115 67 95)(29 116 68 96)(30 117 69 97)(31 118 70 98)(32 119 36 99)(33 120 37 100)(34 121 38 101)(35 122 39 102)
(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 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)
(1 39)(2 38)(3 37)(4 36)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 60)(16 59)(17 58)(18 57)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(71 119)(72 118)(73 117)(74 116)(75 115)(76 114)(77 113)(78 112)(79 111)(80 110)(81 109)(82 108)(83 107)(84 106)(85 140)(86 139)(87 138)(88 137)(89 136)(90 135)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)(97 128)(98 127)(99 126)(100 125)(101 124)(102 123)(103 122)(104 121)(105 120)

G:=sub<Sym(140)| (1,123,40,103)(2,124,41,104)(3,125,42,105)(4,126,43,71)(5,127,44,72)(6,128,45,73)(7,129,46,74)(8,130,47,75)(9,131,48,76)(10,132,49,77)(11,133,50,78)(12,134,51,79)(13,135,52,80)(14,136,53,81)(15,137,54,82)(16,138,55,83)(17,139,56,84)(18,140,57,85)(19,106,58,86)(20,107,59,87)(21,108,60,88)(22,109,61,89)(23,110,62,90)(24,111,63,91)(25,112,64,92)(26,113,65,93)(27,114,66,94)(28,115,67,95)(29,116,68,96)(30,117,69,97)(31,118,70,98)(32,119,36,99)(33,120,37,100)(34,121,38,101)(35,122,39,102), (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)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,39)(2,38)(3,37)(4,36)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(71,119)(72,118)(73,117)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,124)(102,123)(103,122)(104,121)(105,120)>;

G:=Group( (1,123,40,103)(2,124,41,104)(3,125,42,105)(4,126,43,71)(5,127,44,72)(6,128,45,73)(7,129,46,74)(8,130,47,75)(9,131,48,76)(10,132,49,77)(11,133,50,78)(12,134,51,79)(13,135,52,80)(14,136,53,81)(15,137,54,82)(16,138,55,83)(17,139,56,84)(18,140,57,85)(19,106,58,86)(20,107,59,87)(21,108,60,88)(22,109,61,89)(23,110,62,90)(24,111,63,91)(25,112,64,92)(26,113,65,93)(27,114,66,94)(28,115,67,95)(29,116,68,96)(30,117,69,97)(31,118,70,98)(32,119,36,99)(33,120,37,100)(34,121,38,101)(35,122,39,102), (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)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,39)(2,38)(3,37)(4,36)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(71,119)(72,118)(73,117)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,124)(102,123)(103,122)(104,121)(105,120) );

G=PermutationGroup([(1,123,40,103),(2,124,41,104),(3,125,42,105),(4,126,43,71),(5,127,44,72),(6,128,45,73),(7,129,46,74),(8,130,47,75),(9,131,48,76),(10,132,49,77),(11,133,50,78),(12,134,51,79),(13,135,52,80),(14,136,53,81),(15,137,54,82),(16,138,55,83),(17,139,56,84),(18,140,57,85),(19,106,58,86),(20,107,59,87),(21,108,60,88),(22,109,61,89),(23,110,62,90),(24,111,63,91),(25,112,64,92),(26,113,65,93),(27,114,66,94),(28,115,67,95),(29,116,68,96),(30,117,69,97),(31,118,70,98),(32,119,36,99),(33,120,37,100),(34,121,38,101),(35,122,39,102)], [(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),(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)], [(1,39),(2,38),(3,37),(4,36),(5,70),(6,69),(7,68),(8,67),(9,66),(10,65),(11,64),(12,63),(13,62),(14,61),(15,60),(16,59),(17,58),(18,57),(19,56),(20,55),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(71,119),(72,118),(73,117),(74,116),(75,115),(76,114),(77,113),(78,112),(79,111),(80,110),(81,109),(82,108),(83,107),(84,106),(85,140),(86,139),(87,138),(88,137),(89,136),(90,135),(91,134),(92,133),(93,132),(94,131),(95,130),(96,129),(97,128),(98,127),(99,126),(100,125),(101,124),(102,123),(103,122),(104,121),(105,120)])

76 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 7A 7B 7C 10A 10B 14A 14B 14C 20A 20B 20C 20D 28A ··· 28F 35A ··· 35L 70A ··· 70L 140A ··· 140X order 1 2 2 2 4 4 4 4 5 5 7 7 7 10 10 14 14 14 20 20 20 20 28 ··· 28 35 ··· 35 70 ··· 70 140 ··· 140 size 1 1 35 35 1 1 35 35 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

76 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C4 D5 D7 D10 D14 C4×D5 C4×D7 D35 D70 C4×D35 kernel C4×D35 Dic35 C140 D70 D35 C28 C20 C14 C10 C7 C5 C4 C2 C1 # reps 1 1 1 1 4 2 3 2 3 4 6 12 12 24

Matrix representation of C4×D35 in GL3(𝔽281) generated by

 228 0 0 0 280 0 0 0 280
,
 1 0 0 0 243 232 0 49 4
,
 1 0 0 0 243 232 0 104 38
G:=sub<GL(3,GF(281))| [228,0,0,0,280,0,0,0,280],[1,0,0,0,243,49,0,232,4],[1,0,0,0,243,104,0,232,38] >;

C4×D35 in GAP, Magma, Sage, TeX

C_4\times D_{35}
% in TeX

G:=Group("C4xD35");
// GroupNames label

G:=SmallGroup(280,25);
// by ID

G=gap.SmallGroup(280,25);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-7,26,643,6004]);
// Polycyclic

G:=Group<a,b,c|a^4=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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