metabelian, supersoluble, monomial, A-group
Aliases: C5⋊D35, C35⋊1D5, C52⋊2D7, C7⋊(C5⋊D5), (C5×C35)⋊1C2, SmallGroup(350,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C35 — C5⋊D35 |
Generators and relations for C5⋊D35
G = < a,b,c | a5=b35=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C5⋊D35
►On 175 points
Generators in S175
(1 141 140 96 60)(2 142 106 97 61)(3 143 107 98 62)(4 144 108 99 63)(5 145 109 100 64)(6 146 110 101 65)(7 147 111 102 66)(8 148 112 103 67)(9 149 113 104 68)(10 150 114 105 69)(11 151 115 71 70)(12 152 116 72 36)(13 153 117 73 37)(14 154 118 74 38)(15 155 119 75 39)(16 156 120 76 40)(17 157 121 77 41)(18 158 122 78 42)(19 159 123 79 43)(20 160 124 80 44)(21 161 125 81 45)(22 162 126 82 46)(23 163 127 83 47)(24 164 128 84 48)(25 165 129 85 49)(26 166 130 86 50)(27 167 131 87 51)(28 168 132 88 52)(29 169 133 89 53)(30 170 134 90 54)(31 171 135 91 55)(32 172 136 92 56)(33 173 137 93 57)(34 174 138 94 58)(35 175 139 95 59)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(36 164)(37 163)(38 162)(39 161)(40 160)(41 159)(42 158)(43 157)(44 156)(45 155)(46 154)(47 153)(48 152)(49 151)(50 150)(51 149)(52 148)(53 147)(54 146)(55 145)(56 144)(57 143)(58 142)(59 141)(60 175)(61 174)(62 173)(63 172)(64 171)(65 170)(66 169)(67 168)(68 167)(69 166)(70 165)(71 129)(72 128)(73 127)(74 126)(75 125)(76 124)(77 123)(78 122)(79 121)(80 120)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 140)(96 139)(97 138)(98 137)(99 136)(100 135)(101 134)(102 133)(103 132)(104 131)(105 130)
G:=sub<Sym(175)| (1,141,140,96,60)(2,142,106,97,61)(3,143,107,98,62)(4,144,108,99,63)(5,145,109,100,64)(6,146,110,101,65)(7,147,111,102,66)(8,148,112,103,67)(9,149,113,104,68)(10,150,114,105,69)(11,151,115,71,70)(12,152,116,72,36)(13,153,117,73,37)(14,154,118,74,38)(15,155,119,75,39)(16,156,120,76,40)(17,157,121,77,41)(18,158,122,78,42)(19,159,123,79,43)(20,160,124,80,44)(21,161,125,81,45)(22,162,126,82,46)(23,163,127,83,47)(24,164,128,84,48)(25,165,129,85,49)(26,166,130,86,50)(27,167,131,87,51)(28,168,132,88,52)(29,169,133,89,53)(30,170,134,90,54)(31,171,135,91,55)(32,172,136,92,56)(33,173,137,93,57)(34,174,138,94,58)(35,175,139,95,59), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,164)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,147)(54,146)(55,145)(56,144)(57,143)(58,142)(59,141)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,165)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,122)(79,121)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)(105,130)>;
G:=Group( (1,141,140,96,60)(2,142,106,97,61)(3,143,107,98,62)(4,144,108,99,63)(5,145,109,100,64)(6,146,110,101,65)(7,147,111,102,66)(8,148,112,103,67)(9,149,113,104,68)(10,150,114,105,69)(11,151,115,71,70)(12,152,116,72,36)(13,153,117,73,37)(14,154,118,74,38)(15,155,119,75,39)(16,156,120,76,40)(17,157,121,77,41)(18,158,122,78,42)(19,159,123,79,43)(20,160,124,80,44)(21,161,125,81,45)(22,162,126,82,46)(23,163,127,83,47)(24,164,128,84,48)(25,165,129,85,49)(26,166,130,86,50)(27,167,131,87,51)(28,168,132,88,52)(29,169,133,89,53)(30,170,134,90,54)(31,171,135,91,55)(32,172,136,92,56)(33,173,137,93,57)(34,174,138,94,58)(35,175,139,95,59), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,164)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,147)(54,146)(55,145)(56,144)(57,143)(58,142)(59,141)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,165)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,122)(79,121)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)(105,130) );
G=PermutationGroup([[(1,141,140,96,60),(2,142,106,97,61),(3,143,107,98,62),(4,144,108,99,63),(5,145,109,100,64),(6,146,110,101,65),(7,147,111,102,66),(8,148,112,103,67),(9,149,113,104,68),(10,150,114,105,69),(11,151,115,71,70),(12,152,116,72,36),(13,153,117,73,37),(14,154,118,74,38),(15,155,119,75,39),(16,156,120,76,40),(17,157,121,77,41),(18,158,122,78,42),(19,159,123,79,43),(20,160,124,80,44),(21,161,125,81,45),(22,162,126,82,46),(23,163,127,83,47),(24,164,128,84,48),(25,165,129,85,49),(26,166,130,86,50),(27,167,131,87,51),(28,168,132,88,52),(29,169,133,89,53),(30,170,134,90,54),(31,171,135,91,55),(32,172,136,92,56),(33,173,137,93,57),(34,174,138,94,58),(35,175,139,95,59)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(36,164),(37,163),(38,162),(39,161),(40,160),(41,159),(42,158),(43,157),(44,156),(45,155),(46,154),(47,153),(48,152),(49,151),(50,150),(51,149),(52,148),(53,147),(54,146),(55,145),(56,144),(57,143),(58,142),(59,141),(60,175),(61,174),(62,173),(63,172),(64,171),(65,170),(66,169),(67,168),(68,167),(69,166),(70,165),(71,129),(72,128),(73,127),(74,126),(75,125),(76,124),(77,123),(78,122),(79,121),(80,120),(81,119),(82,118),(83,117),(84,116),(85,115),(86,114),(87,113),(88,112),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,140),(96,139),(97,138),(98,137),(99,136),(100,135),(101,134),(102,133),(103,132),(104,131),(105,130)]])
89 conjugacy classes
| class | 1 | 2 | 5A | ··· | 5L | 7A | 7B | 7C | 35A | ··· | 35BT |
| order | 1 | 2 | 5 | ··· | 5 | 7 | 7 | 7 | 35 | ··· | 35 |
| size | 1 | 175 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
89 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D5 | D7 | D35 |
| kernel | C5⋊D35 | C5×C35 | C35 | C52 | C5 |
| # reps | 1 | 1 | 12 | 3 | 72 |
Matrix representation of C5⋊D35 ►in GL4(𝔽71) generated by
| 19 | 6 | 0 | 0 |
| 65 | 43 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 47 | 53 | 0 | 0 |
| 18 | 46 | 0 | 0 |
| 0 | 0 | 47 | 53 |
| 0 | 0 | 18 | 46 |
| 47 | 53 | 0 | 0 |
| 28 | 24 | 0 | 0 |
| 0 | 0 | 11 | 46 |
| 0 | 0 | 19 | 60 |
G:=sub<GL(4,GF(71))| [19,65,0,0,6,43,0,0,0,0,1,0,0,0,0,1],[47,18,0,0,53,46,0,0,0,0,47,18,0,0,53,46],[47,28,0,0,53,24,0,0,0,0,11,19,0,0,46,60] >;
C5⋊D35 in GAP, Magma, Sage, TeX
C_5\rtimes D_{35} % in TeX
G:=Group("C5:D35"); // GroupNames label
G:=SmallGroup(350,9);
// by ID
G=gap.SmallGroup(350,9);
# by ID
G:=PCGroup([4,-2,-5,-5,-7,65,482,4803]);
// Polycyclic
G:=Group<a,b,c|a^5=b^35=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export