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## G = D7×C52order 350 = 2·52·7

### Direct product of C52 and D7

Aliases: D7×C52, C352C10, C7⋊(C5×C10), (C5×C35)⋊3C2, SmallGroup(350,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C52
 Chief series C1 — C7 — C35 — C5×C35 — D7×C52
 Lower central C7 — D7×C52
 Upper central C1 — C52

Generators and relations for D7×C52
G = < a,b,c,d | a5=b5=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of D7×C52
On 175 points
Generators in S175
(1 174 132 90 48)(2 175 133 91 49)(3 169 127 85 43)(4 170 128 86 44)(5 171 129 87 45)(6 172 130 88 46)(7 173 131 89 47)(8 141 134 92 50)(9 142 135 93 51)(10 143 136 94 52)(11 144 137 95 53)(12 145 138 96 54)(13 146 139 97 55)(14 147 140 98 56)(15 148 106 99 57)(16 149 107 100 58)(17 150 108 101 59)(18 151 109 102 60)(19 152 110 103 61)(20 153 111 104 62)(21 154 112 105 63)(22 155 113 71 64)(23 156 114 72 65)(24 157 115 73 66)(25 158 116 74 67)(26 159 117 75 68)(27 160 118 76 69)(28 161 119 77 70)(29 162 120 78 36)(30 163 121 79 37)(31 164 122 80 38)(32 165 123 81 39)(33 166 124 82 40)(34 167 125 83 41)(35 168 126 84 42)
(1 34 27 20 13)(2 35 28 21 14)(3 29 22 15 8)(4 30 23 16 9)(5 31 24 17 10)(6 32 25 18 11)(7 33 26 19 12)(36 64 57 50 43)(37 65 58 51 44)(38 66 59 52 45)(39 67 60 53 46)(40 68 61 54 47)(41 69 62 55 48)(42 70 63 56 49)(71 99 92 85 78)(72 100 93 86 79)(73 101 94 87 80)(74 102 95 88 81)(75 103 96 89 82)(76 104 97 90 83)(77 105 98 91 84)(106 134 127 120 113)(107 135 128 121 114)(108 136 129 122 115)(109 137 130 123 116)(110 138 131 124 117)(111 139 132 125 118)(112 140 133 126 119)(141 169 162 155 148)(142 170 163 156 149)(143 171 164 157 150)(144 172 165 158 151)(145 173 166 159 152)(146 174 167 160 153)(147 175 168 161 154)
(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 87)(88 91)(89 90)(92 94)(95 98)(96 97)(99 101)(102 105)(103 104)(106 108)(109 112)(110 111)(113 115)(116 119)(117 118)(120 122)(123 126)(124 125)(127 129)(130 133)(131 132)(134 136)(137 140)(138 139)(141 143)(144 147)(145 146)(148 150)(151 154)(152 153)(155 157)(158 161)(159 160)(162 164)(165 168)(166 167)(169 171)(172 175)(173 174)

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

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

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

125 conjugacy classes

 class 1 2 5A ··· 5X 7A 7B 7C 10A ··· 10X 35A ··· 35BT order 1 2 5 ··· 5 7 7 7 10 ··· 10 35 ··· 35 size 1 7 1 ··· 1 2 2 2 7 ··· 7 2 ··· 2

125 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C5 C10 D7 C5×D7 kernel D7×C52 C5×C35 C5×D7 C35 C52 C5 # reps 1 1 24 24 3 72

Matrix representation of D7×C52 in GL3(𝔽71) generated by

 1 0 0 0 57 0 0 0 57
,
 5 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 1 0 70 14
,
 70 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(71))| [1,0,0,0,57,0,0,0,57],[5,0,0,0,1,0,0,0,1],[1,0,0,0,0,70,0,1,14],[70,0,0,0,0,1,0,1,0] >;

D7×C52 in GAP, Magma, Sage, TeX

D_7\times C_5^2
% in TeX

G:=Group("D7xC5^2");
// GroupNames label

G:=SmallGroup(350,5);
// by ID

G=gap.SmallGroup(350,5);
# by ID

G:=PCGroup([4,-2,-5,-5,-7,4803]);
// Polycyclic

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

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