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## G = C7×C5⋊D5order 350 = 2·52·7

### Direct product of C7 and C5⋊D5

Aliases: C7×C5⋊D5, C353D5, C522C14, C5⋊(C7×D5), (C5×C35)⋊5C2, SmallGroup(350,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C7×C5⋊D5
 Chief series C1 — C5 — C52 — C5×C35 — C7×C5⋊D5
 Lower central C52 — C7×C5⋊D5
 Upper central C1 — C7

Generators and relations for C7×C5⋊D5
G = < a,b,c,d | a7=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Smallest permutation representation of C7×C5⋊D5
On 175 points
Generators in S175
(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)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)
(1 115 50 130 157)(2 116 51 131 158)(3 117 52 132 159)(4 118 53 133 160)(5 119 54 127 161)(6 113 55 128 155)(7 114 56 129 156)(8 166 15 125 74)(9 167 16 126 75)(10 168 17 120 76)(11 162 18 121 77)(12 163 19 122 71)(13 164 20 123 72)(14 165 21 124 73)(22 91 93 135 144)(23 85 94 136 145)(24 86 95 137 146)(25 87 96 138 147)(26 88 97 139 141)(27 89 98 140 142)(28 90 92 134 143)(29 150 37 46 99)(30 151 38 47 100)(31 152 39 48 101)(32 153 40 49 102)(33 154 41 43 103)(34 148 42 44 104)(35 149 36 45 105)(57 66 175 80 108)(58 67 169 81 109)(59 68 170 82 110)(60 69 171 83 111)(61 70 172 84 112)(62 64 173 78 106)(63 65 174 79 107)
(1 29 96 125 106)(2 30 97 126 107)(3 31 98 120 108)(4 32 92 121 109)(5 33 93 122 110)(6 34 94 123 111)(7 35 95 124 112)(8 64 50 37 147)(9 65 51 38 141)(10 66 52 39 142)(11 67 53 40 143)(12 68 54 41 144)(13 69 55 42 145)(14 70 56 36 146)(15 78 157 99 87)(16 79 158 100 88)(17 80 159 101 89)(18 81 160 102 90)(19 82 161 103 91)(20 83 155 104 85)(21 84 156 105 86)(22 163 170 127 43)(23 164 171 128 44)(24 165 172 129 45)(25 166 173 130 46)(26 167 174 131 47)(27 168 175 132 48)(28 162 169 133 49)(57 117 152 140 76)(58 118 153 134 77)(59 119 154 135 71)(60 113 148 136 72)(61 114 149 137 73)(62 115 150 138 74)(63 116 151 139 75)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 46)(9 47)(10 48)(11 49)(12 43)(13 44)(14 45)(15 150)(16 151)(17 152)(18 153)(19 154)(20 148)(21 149)(22 144)(23 145)(24 146)(25 147)(26 141)(27 142)(28 143)(29 125)(30 126)(31 120)(32 121)(33 122)(34 123)(35 124)(36 165)(37 166)(38 167)(39 168)(40 162)(41 163)(42 164)(50 173)(51 174)(52 175)(53 169)(54 170)(55 171)(56 172)(57 159)(58 160)(59 161)(60 155)(61 156)(62 157)(63 158)(64 130)(65 131)(66 132)(67 133)(68 127)(69 128)(70 129)(71 103)(72 104)(73 105)(74 99)(75 100)(76 101)(77 102)(78 115)(79 116)(80 117)(81 118)(82 119)(83 113)(84 114)(85 136)(86 137)(87 138)(88 139)(89 140)(90 134)(91 135)

G:=sub<Sym(175)| (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)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175), (1,115,50,130,157)(2,116,51,131,158)(3,117,52,132,159)(4,118,53,133,160)(5,119,54,127,161)(6,113,55,128,155)(7,114,56,129,156)(8,166,15,125,74)(9,167,16,126,75)(10,168,17,120,76)(11,162,18,121,77)(12,163,19,122,71)(13,164,20,123,72)(14,165,21,124,73)(22,91,93,135,144)(23,85,94,136,145)(24,86,95,137,146)(25,87,96,138,147)(26,88,97,139,141)(27,89,98,140,142)(28,90,92,134,143)(29,150,37,46,99)(30,151,38,47,100)(31,152,39,48,101)(32,153,40,49,102)(33,154,41,43,103)(34,148,42,44,104)(35,149,36,45,105)(57,66,175,80,108)(58,67,169,81,109)(59,68,170,82,110)(60,69,171,83,111)(61,70,172,84,112)(62,64,173,78,106)(63,65,174,79,107), (1,29,96,125,106)(2,30,97,126,107)(3,31,98,120,108)(4,32,92,121,109)(5,33,93,122,110)(6,34,94,123,111)(7,35,95,124,112)(8,64,50,37,147)(9,65,51,38,141)(10,66,52,39,142)(11,67,53,40,143)(12,68,54,41,144)(13,69,55,42,145)(14,70,56,36,146)(15,78,157,99,87)(16,79,158,100,88)(17,80,159,101,89)(18,81,160,102,90)(19,82,161,103,91)(20,83,155,104,85)(21,84,156,105,86)(22,163,170,127,43)(23,164,171,128,44)(24,165,172,129,45)(25,166,173,130,46)(26,167,174,131,47)(27,168,175,132,48)(28,162,169,133,49)(57,117,152,140,76)(58,118,153,134,77)(59,119,154,135,71)(60,113,148,136,72)(61,114,149,137,73)(62,115,150,138,74)(63,116,151,139,75), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,46)(9,47)(10,48)(11,49)(12,43)(13,44)(14,45)(15,150)(16,151)(17,152)(18,153)(19,154)(20,148)(21,149)(22,144)(23,145)(24,146)(25,147)(26,141)(27,142)(28,143)(29,125)(30,126)(31,120)(32,121)(33,122)(34,123)(35,124)(36,165)(37,166)(38,167)(39,168)(40,162)(41,163)(42,164)(50,173)(51,174)(52,175)(53,169)(54,170)(55,171)(56,172)(57,159)(58,160)(59,161)(60,155)(61,156)(62,157)(63,158)(64,130)(65,131)(66,132)(67,133)(68,127)(69,128)(70,129)(71,103)(72,104)(73,105)(74,99)(75,100)(76,101)(77,102)(78,115)(79,116)(80,117)(81,118)(82,119)(83,113)(84,114)(85,136)(86,137)(87,138)(88,139)(89,140)(90,134)(91,135)>;

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)(71,72,73,74,75,76,77)(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)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175), (1,115,50,130,157)(2,116,51,131,158)(3,117,52,132,159)(4,118,53,133,160)(5,119,54,127,161)(6,113,55,128,155)(7,114,56,129,156)(8,166,15,125,74)(9,167,16,126,75)(10,168,17,120,76)(11,162,18,121,77)(12,163,19,122,71)(13,164,20,123,72)(14,165,21,124,73)(22,91,93,135,144)(23,85,94,136,145)(24,86,95,137,146)(25,87,96,138,147)(26,88,97,139,141)(27,89,98,140,142)(28,90,92,134,143)(29,150,37,46,99)(30,151,38,47,100)(31,152,39,48,101)(32,153,40,49,102)(33,154,41,43,103)(34,148,42,44,104)(35,149,36,45,105)(57,66,175,80,108)(58,67,169,81,109)(59,68,170,82,110)(60,69,171,83,111)(61,70,172,84,112)(62,64,173,78,106)(63,65,174,79,107), (1,29,96,125,106)(2,30,97,126,107)(3,31,98,120,108)(4,32,92,121,109)(5,33,93,122,110)(6,34,94,123,111)(7,35,95,124,112)(8,64,50,37,147)(9,65,51,38,141)(10,66,52,39,142)(11,67,53,40,143)(12,68,54,41,144)(13,69,55,42,145)(14,70,56,36,146)(15,78,157,99,87)(16,79,158,100,88)(17,80,159,101,89)(18,81,160,102,90)(19,82,161,103,91)(20,83,155,104,85)(21,84,156,105,86)(22,163,170,127,43)(23,164,171,128,44)(24,165,172,129,45)(25,166,173,130,46)(26,167,174,131,47)(27,168,175,132,48)(28,162,169,133,49)(57,117,152,140,76)(58,118,153,134,77)(59,119,154,135,71)(60,113,148,136,72)(61,114,149,137,73)(62,115,150,138,74)(63,116,151,139,75), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,46)(9,47)(10,48)(11,49)(12,43)(13,44)(14,45)(15,150)(16,151)(17,152)(18,153)(19,154)(20,148)(21,149)(22,144)(23,145)(24,146)(25,147)(26,141)(27,142)(28,143)(29,125)(30,126)(31,120)(32,121)(33,122)(34,123)(35,124)(36,165)(37,166)(38,167)(39,168)(40,162)(41,163)(42,164)(50,173)(51,174)(52,175)(53,169)(54,170)(55,171)(56,172)(57,159)(58,160)(59,161)(60,155)(61,156)(62,157)(63,158)(64,130)(65,131)(66,132)(67,133)(68,127)(69,128)(70,129)(71,103)(72,104)(73,105)(74,99)(75,100)(76,101)(77,102)(78,115)(79,116)(80,117)(81,118)(82,119)(83,113)(84,114)(85,136)(86,137)(87,138)(88,139)(89,140)(90,134)(91,135) );

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),(71,72,73,74,75,76,77),(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),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175)], [(1,115,50,130,157),(2,116,51,131,158),(3,117,52,132,159),(4,118,53,133,160),(5,119,54,127,161),(6,113,55,128,155),(7,114,56,129,156),(8,166,15,125,74),(9,167,16,126,75),(10,168,17,120,76),(11,162,18,121,77),(12,163,19,122,71),(13,164,20,123,72),(14,165,21,124,73),(22,91,93,135,144),(23,85,94,136,145),(24,86,95,137,146),(25,87,96,138,147),(26,88,97,139,141),(27,89,98,140,142),(28,90,92,134,143),(29,150,37,46,99),(30,151,38,47,100),(31,152,39,48,101),(32,153,40,49,102),(33,154,41,43,103),(34,148,42,44,104),(35,149,36,45,105),(57,66,175,80,108),(58,67,169,81,109),(59,68,170,82,110),(60,69,171,83,111),(61,70,172,84,112),(62,64,173,78,106),(63,65,174,79,107)], [(1,29,96,125,106),(2,30,97,126,107),(3,31,98,120,108),(4,32,92,121,109),(5,33,93,122,110),(6,34,94,123,111),(7,35,95,124,112),(8,64,50,37,147),(9,65,51,38,141),(10,66,52,39,142),(11,67,53,40,143),(12,68,54,41,144),(13,69,55,42,145),(14,70,56,36,146),(15,78,157,99,87),(16,79,158,100,88),(17,80,159,101,89),(18,81,160,102,90),(19,82,161,103,91),(20,83,155,104,85),(21,84,156,105,86),(22,163,170,127,43),(23,164,171,128,44),(24,165,172,129,45),(25,166,173,130,46),(26,167,174,131,47),(27,168,175,132,48),(28,162,169,133,49),(57,117,152,140,76),(58,118,153,134,77),(59,119,154,135,71),(60,113,148,136,72),(61,114,149,137,73),(62,115,150,138,74),(63,116,151,139,75)], [(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,46),(9,47),(10,48),(11,49),(12,43),(13,44),(14,45),(15,150),(16,151),(17,152),(18,153),(19,154),(20,148),(21,149),(22,144),(23,145),(24,146),(25,147),(26,141),(27,142),(28,143),(29,125),(30,126),(31,120),(32,121),(33,122),(34,123),(35,124),(36,165),(37,166),(38,167),(39,168),(40,162),(41,163),(42,164),(50,173),(51,174),(52,175),(53,169),(54,170),(55,171),(56,172),(57,159),(58,160),(59,161),(60,155),(61,156),(62,157),(63,158),(64,130),(65,131),(66,132),(67,133),(68,127),(69,128),(70,129),(71,103),(72,104),(73,105),(74,99),(75,100),(76,101),(77,102),(78,115),(79,116),(80,117),(81,118),(82,119),(83,113),(84,114),(85,136),(86,137),(87,138),(88,139),(89,140),(90,134),(91,135)]])

98 conjugacy classes

 class 1 2 5A ··· 5L 7A ··· 7F 14A ··· 14F 35A ··· 35BT order 1 2 5 ··· 5 7 ··· 7 14 ··· 14 35 ··· 35 size 1 25 2 ··· 2 1 ··· 1 25 ··· 25 2 ··· 2

98 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C7 C14 D5 C7×D5 kernel C7×C5⋊D5 C5×C35 C5⋊D5 C52 C35 C5 # reps 1 1 6 6 12 72

Matrix representation of C7×C5⋊D5 in GL4(𝔽71) generated by

 37 0 0 0 0 37 0 0 0 0 37 0 0 0 0 37
,
 8 1 0 0 70 0 0 0 0 0 0 1 0 0 70 62
,
 1 0 0 0 0 1 0 0 0 0 9 9 0 0 62 70
,
 1 8 0 0 0 70 0 0 0 0 1 0 0 0 62 70
G:=sub<GL(4,GF(71))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[8,70,0,0,1,0,0,0,0,0,0,70,0,0,1,62],[1,0,0,0,0,1,0,0,0,0,9,62,0,0,9,70],[1,0,0,0,8,70,0,0,0,0,1,62,0,0,0,70] >;

C7×C5⋊D5 in GAP, Magma, Sage, TeX

C_7\times C_5\rtimes D_5
% in TeX

G:=Group("C7xC5:D5");
// GroupNames label

G:=SmallGroup(350,8);
// by ID

G=gap.SmallGroup(350,8);
# by ID

G:=PCGroup([4,-2,-7,-5,-5,674,4483]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^5=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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