direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.8D10, C20.34C24, D20.30C23, Dic10.29C23, C4○D4⋊15D10, C10⋊5(C4○D8), D4⋊D5⋊23C22, (C2×C20).502D4, C20.263(C2×D4), Q8⋊D5⋊21C22, C4.34(C23×D5), (C2×D4).232D10, C4○D20⋊20C22, C5⋊2C8.30C23, D4.D5⋊20C22, (C2×Q8).190D10, C5⋊Q16⋊20C22, (C5×D4).22C23, D4.22(C22×D5), (C5×Q8).22C23, Q8.22(C22×D5), (C2×C20).556C23, (C22×C10).123D4, C10.159(C22×D4), (C22×C4).387D10, C23.45(C5⋊D4), (D4×C10).272C22, (C2×D20).288C22, (Q8×C10).237C22, (C22×C20).291C22, (C2×Dic10).317C22, C5⋊6(C2×C4○D8), (C2×C4○D4)⋊3D5, (C2×D4⋊D5)⋊33C2, (C10×C4○D4)⋊3C2, (C2×Q8⋊D5)⋊33C2, C4.30(C2×C5⋊D4), (C2×C4○D20)⋊30C2, (C2×D4.D5)⋊33C2, (C2×C5⋊Q16)⋊33C2, (C2×C10).76(C2×D4), (C2×C5⋊2C8)⋊42C22, (C22×C5⋊2C8)⋊15C2, (C5×C4○D4)⋊17C22, C2.32(C22×C5⋊D4), (C2×C4).158(C5⋊D4), (C2×C4).636(C22×D5), C22.119(C2×C5⋊D4), SmallGroup(320,1493)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4.8D10
G = < a,b,c,d,e | a2=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d9 >
Subgroups: 830 in 266 conjugacy classes, 111 normal (35 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, C10, C10, C10, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4○D4, C5⋊2C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, C22×C10, C2×C4○D8, C2×C5⋊2C8, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C22×C5⋊2C8, C2×D4⋊D5, C2×D4.D5, C2×Q8⋊D5, C2×C5⋊Q16, D4.8D10, C2×C4○D20, C10×C4○D4, C2×D4.8D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C4○D8, C22×D4, C5⋊D4, C22×D5, C2×C4○D8, C2×C5⋊D4, C23×D5, D4.8D10, C22×C5⋊D4, C2×D4.8D10
►On 160 points
Generators in S160
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 58)(22 59)(23 60)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(61 132)(62 133)(63 134)(64 135)(65 136)(66 137)(67 138)(68 139)(69 140)(70 121)(71 122)(72 123)(73 124)(74 125)(75 126)(76 127)(77 128)(78 129)(79 130)(80 131)(81 150)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)(91 160)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)
(1 98 11 88)(2 99 12 89)(3 100 13 90)(4 81 14 91)(5 82 15 92)(6 83 16 93)(7 84 17 94)(8 85 18 95)(9 86 19 96)(10 87 20 97)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)(41 135 51 125)(42 136 52 126)(43 137 53 127)(44 138 54 128)(45 139 55 129)(46 140 56 130)(47 121 57 131)(48 122 58 132)(49 123 59 133)(50 124 60 134)(101 158 111 148)(102 159 112 149)(103 160 113 150)(104 141 114 151)(105 142 115 152)(106 143 116 153)(107 144 117 154)(108 145 118 155)(109 146 119 156)(110 147 120 157)
(1 38)(2 39)(3 40)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(41 116)(42 117)(43 118)(44 119)(45 120)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 141)(134 142)(135 143)(136 144)(137 145)(138 146)(139 147)(140 148)
(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)
(1 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 64 31 74)(22 73 32 63)(23 62 33 72)(24 71 34 61)(25 80 35 70)(26 69 36 79)(27 78 37 68)(28 67 38 77)(29 76 39 66)(30 65 40 75)(41 122 51 132)(42 131 52 121)(43 140 53 130)(44 129 54 139)(45 138 55 128)(46 127 56 137)(47 136 57 126)(48 125 58 135)(49 134 59 124)(50 123 60 133)(81 84 91 94)(82 93 92 83)(85 100 95 90)(86 89 96 99)(87 98 97 88)(101 118 111 108)(102 107 112 117)(103 116 113 106)(104 105 114 115)(109 110 119 120)(141 152 151 142)(143 150 153 160)(144 159 154 149)(145 148 155 158)(146 157 156 147)
G:=sub<Sym(160)| (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,58)(22,59)(23,60)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,129)(79,130)(80,131)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149), (1,98,11,88)(2,99,12,89)(3,100,13,90)(4,81,14,91)(5,82,15,92)(6,83,16,93)(7,84,17,94)(8,85,18,95)(9,86,19,96)(10,87,20,97)(21,61,31,71)(22,62,32,72)(23,63,33,73)(24,64,34,74)(25,65,35,75)(26,66,36,76)(27,67,37,77)(28,68,38,78)(29,69,39,79)(30,70,40,80)(41,135,51,125)(42,136,52,126)(43,137,53,127)(44,138,54,128)(45,139,55,129)(46,140,56,130)(47,121,57,131)(48,122,58,132)(49,123,59,133)(50,124,60,134)(101,158,111,148)(102,159,112,149)(103,160,113,150)(104,141,114,151)(105,142,115,152)(106,143,116,153)(107,144,117,154)(108,145,118,155)(109,146,119,156)(110,147,120,157), (1,38)(2,39)(3,40)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(41,116)(42,117)(43,118)(44,119)(45,120)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,64,31,74)(22,73,32,63)(23,62,33,72)(24,71,34,61)(25,80,35,70)(26,69,36,79)(27,78,37,68)(28,67,38,77)(29,76,39,66)(30,65,40,75)(41,122,51,132)(42,131,52,121)(43,140,53,130)(44,129,54,139)(45,138,55,128)(46,127,56,137)(47,136,57,126)(48,125,58,135)(49,134,59,124)(50,123,60,133)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(101,118,111,108)(102,107,112,117)(103,116,113,106)(104,105,114,115)(109,110,119,120)(141,152,151,142)(143,150,153,160)(144,159,154,149)(145,148,155,158)(146,157,156,147)>;
G:=Group( (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,58)(22,59)(23,60)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,129)(79,130)(80,131)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149), (1,98,11,88)(2,99,12,89)(3,100,13,90)(4,81,14,91)(5,82,15,92)(6,83,16,93)(7,84,17,94)(8,85,18,95)(9,86,19,96)(10,87,20,97)(21,61,31,71)(22,62,32,72)(23,63,33,73)(24,64,34,74)(25,65,35,75)(26,66,36,76)(27,67,37,77)(28,68,38,78)(29,69,39,79)(30,70,40,80)(41,135,51,125)(42,136,52,126)(43,137,53,127)(44,138,54,128)(45,139,55,129)(46,140,56,130)(47,121,57,131)(48,122,58,132)(49,123,59,133)(50,124,60,134)(101,158,111,148)(102,159,112,149)(103,160,113,150)(104,141,114,151)(105,142,115,152)(106,143,116,153)(107,144,117,154)(108,145,118,155)(109,146,119,156)(110,147,120,157), (1,38)(2,39)(3,40)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(41,116)(42,117)(43,118)(44,119)(45,120)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,64,31,74)(22,73,32,63)(23,62,33,72)(24,71,34,61)(25,80,35,70)(26,69,36,79)(27,78,37,68)(28,67,38,77)(29,76,39,66)(30,65,40,75)(41,122,51,132)(42,131,52,121)(43,140,53,130)(44,129,54,139)(45,138,55,128)(46,127,56,137)(47,136,57,126)(48,125,58,135)(49,134,59,124)(50,123,60,133)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(101,118,111,108)(102,107,112,117)(103,116,113,106)(104,105,114,115)(109,110,119,120)(141,152,151,142)(143,150,153,160)(144,159,154,149)(145,148,155,158)(146,157,156,147) );
G=PermutationGroup([[(1,110),(2,111),(3,112),(4,113),(5,114),(6,115),(7,116),(8,117),(9,118),(10,119),(11,120),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,109),(21,58),(22,59),(23,60),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(61,132),(62,133),(63,134),(64,135),(65,136),(66,137),(67,138),(68,139),(69,140),(70,121),(71,122),(72,123),(73,124),(74,125),(75,126),(76,127),(77,128),(78,129),(79,130),(80,131),(81,150),(82,151),(83,152),(84,153),(85,154),(86,155),(87,156),(88,157),(89,158),(90,159),(91,160),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149)], [(1,98,11,88),(2,99,12,89),(3,100,13,90),(4,81,14,91),(5,82,15,92),(6,83,16,93),(7,84,17,94),(8,85,18,95),(9,86,19,96),(10,87,20,97),(21,61,31,71),(22,62,32,72),(23,63,33,73),(24,64,34,74),(25,65,35,75),(26,66,36,76),(27,67,37,77),(28,68,38,78),(29,69,39,79),(30,70,40,80),(41,135,51,125),(42,136,52,126),(43,137,53,127),(44,138,54,128),(45,139,55,129),(46,140,56,130),(47,121,57,131),(48,122,58,132),(49,123,59,133),(50,124,60,134),(101,158,111,148),(102,159,112,149),(103,160,113,150),(104,141,114,151),(105,142,115,152),(106,143,116,153),(107,144,117,154),(108,145,118,155),(109,146,119,156),(110,147,120,157)], [(1,38),(2,39),(3,40),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(41,116),(42,117),(43,118),(44,119),(45,120),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112),(58,113),(59,114),(60,115),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,81),(72,82),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,141),(134,142),(135,143),(136,144),(137,145),(138,146),(139,147),(140,148)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,64,31,74),(22,73,32,63),(23,62,33,72),(24,71,34,61),(25,80,35,70),(26,69,36,79),(27,78,37,68),(28,67,38,77),(29,76,39,66),(30,65,40,75),(41,122,51,132),(42,131,52,121),(43,140,53,130),(44,129,54,139),(45,138,55,128),(46,127,56,137),(47,136,57,126),(48,125,58,135),(49,134,59,124),(50,123,60,133),(81,84,91,94),(82,93,92,83),(85,100,95,90),(86,89,96,99),(87,98,97,88),(101,118,111,108),(102,107,112,117),(103,116,113,106),(104,105,114,115),(109,110,119,120),(141,152,151,142),(143,150,153,160),(144,159,154,149),(145,148,155,158),(146,157,156,147)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | D4.8D10 |
| kernel | C2×D4.8D10 | C22×C5⋊2C8 | C2×D4⋊D5 | C2×D4.D5 | C2×Q8⋊D5 | C2×C5⋊Q16 | D4.8D10 | C2×C4○D20 | C10×C4○D4 | C2×C20 | C22×C10 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 3 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 12 | 4 | 8 |
Matrix representation of C2×D4.8D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 17 |
| 0 | 0 | 0 | 0 | 12 | 24 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 21 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 32 | 32 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,4,1,0,0,0,0,0,0,17,12,0,0,0,0,17,24],[1,6,0,0,0,0,35,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,35,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40,0,0,0,0,0,0,9,32,0,0,0,0,0,32] >;
C2×D4.8D10 in GAP, Magma, Sage, TeX
C_2\times D_4._8D_{10} % in TeX
G:=Group("C2xD4.8D10"); // GroupNames label
G:=SmallGroup(320,1493);
// by ID
G=gap.SmallGroup(320,1493);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^9>;
// generators/relations