direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.D10, Q16⋊11D10, D40⋊17C22, C40.34C23, C20.12C24, D20.7C23, C4.48(D4×D5), (C2×D40)⋊20C2, C10⋊4(C4○D8), (C2×Q16)⋊13D5, (C10×Q16)⋊8C2, (C4×D5).70D4, C20.87(C2×D4), D10.23(C2×D4), (C2×C8).247D10, (C8×D5)⋊15C22, Q8⋊D5⋊10C22, (C5×Q16)⋊9C22, C4.12(C23×D5), C8.40(C22×D5), (C5×Q8).6C23, Q8.6(C22×D5), (C2×C40).99C22, C5⋊2C8.23C23, (C2×Q8).154D10, Q8⋊2D5⋊7C22, (C22×D5).94D4, (C4×D5).64C23, C22.144(D4×D5), (C2×C20).529C23, (C2×Dic5).285D4, Dic5.125(C2×D4), C10.113(C22×D4), (C2×D20).186C22, (Q8×C10).151C22, (D5×C2×C8)⋊6C2, C5⋊4(C2×C4○D8), C2.86(C2×D4×D5), (C2×Q8⋊D5)⋊28C2, (C2×Q8⋊2D5)⋊16C2, (C2×C10).402(C2×D4), (C2×C4×D5).330C22, (C2×C4).617(C22×D5), (C2×C5⋊2C8).294C22, SmallGroup(320,1437)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8.D10
G = < a,b,c,d,e | a2=b4=e2=1, c2=d10=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=b2d9 >
Subgroups: 1054 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C5⋊2C8, C40, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×D5, C2×C4○D8, C8×D5, D40, C2×C5⋊2C8, Q8⋊D5, C2×C40, C5×Q16, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5, Q8⋊2D5, Q8×C10, D5×C2×C8, C2×D40, Q8.D10, C2×Q8⋊D5, C10×Q16, C2×Q8⋊2D5, C2×Q8.D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C4○D8, C22×D4, C22×D5, C2×C4○D8, D4×D5, C23×D5, Q8.D10, C2×D4×D5, C2×Q8.D10
►On 160 points
Generators in S160
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 41)(18 42)(19 43)(20 44)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 121)(38 122)(39 123)(40 124)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 156)(82 157)(83 158)(84 159)(85 160)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)
(1 30 11 40)(2 21 12 31)(3 32 13 22)(4 23 14 33)(5 34 15 24)(6 25 16 35)(7 36 17 26)(8 27 18 37)(9 38 19 28)(10 29 20 39)(41 130 51 140)(42 121 52 131)(43 132 53 122)(44 123 54 133)(45 134 55 124)(46 125 56 135)(47 136 57 126)(48 127 58 137)(49 138 59 128)(50 129 60 139)(61 145 71 155)(62 156 72 146)(63 147 73 157)(64 158 74 148)(65 149 75 159)(66 160 76 150)(67 151 77 141)(68 142 78 152)(69 153 79 143)(70 144 80 154)(81 102 91 112)(82 113 92 103)(83 104 93 114)(84 115 94 105)(85 106 95 116)(86 117 96 107)(87 108 97 118)(88 119 98 109)(89 110 99 120)(90 101 100 111)
(1 142 11 152)(2 79 12 69)(3 144 13 154)(4 61 14 71)(5 146 15 156)(6 63 16 73)(7 148 17 158)(8 65 18 75)(9 150 19 160)(10 67 20 77)(21 153 31 143)(22 80 32 70)(23 155 33 145)(24 62 34 72)(25 157 35 147)(26 64 36 74)(27 159 37 149)(28 66 38 76)(29 141 39 151)(30 68 40 78)(41 83 51 93)(42 105 52 115)(43 85 53 95)(44 107 54 117)(45 87 55 97)(46 109 56 119)(47 89 57 99)(48 111 58 101)(49 91 59 81)(50 113 60 103)(82 139 92 129)(84 121 94 131)(86 123 96 133)(88 125 98 135)(90 127 100 137)(102 128 112 138)(104 130 114 140)(106 132 116 122)(108 134 118 124)(110 136 120 126)
(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)
(1 107)(2 106)(3 105)(4 104)(5 103)(6 102)(7 101)(8 120)(9 119)(10 118)(11 117)(12 116)(13 115)(14 114)(15 113)(16 112)(17 111)(18 110)(19 109)(20 108)(21 85)(22 84)(23 83)(24 82)(25 81)(26 100)(27 99)(28 98)(29 97)(30 96)(31 95)(32 94)(33 93)(34 92)(35 91)(36 90)(37 89)(38 88)(39 87)(40 86)(41 61)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(121 144)(122 143)(123 142)(124 141)(125 160)(126 159)(127 158)(128 157)(129 156)(130 155)(131 154)(132 153)(133 152)(134 151)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)
G:=sub<Sym(160)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,121)(38,122)(39,123)(40,124)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,156)(82,157)(83,158)(84,159)(85,160)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155), (1,30,11,40)(2,21,12,31)(3,32,13,22)(4,23,14,33)(5,34,15,24)(6,25,16,35)(7,36,17,26)(8,27,18,37)(9,38,19,28)(10,29,20,39)(41,130,51,140)(42,121,52,131)(43,132,53,122)(44,123,54,133)(45,134,55,124)(46,125,56,135)(47,136,57,126)(48,127,58,137)(49,138,59,128)(50,129,60,139)(61,145,71,155)(62,156,72,146)(63,147,73,157)(64,158,74,148)(65,149,75,159)(66,160,76,150)(67,151,77,141)(68,142,78,152)(69,153,79,143)(70,144,80,154)(81,102,91,112)(82,113,92,103)(83,104,93,114)(84,115,94,105)(85,106,95,116)(86,117,96,107)(87,108,97,118)(88,119,98,109)(89,110,99,120)(90,101,100,111), (1,142,11,152)(2,79,12,69)(3,144,13,154)(4,61,14,71)(5,146,15,156)(6,63,16,73)(7,148,17,158)(8,65,18,75)(9,150,19,160)(10,67,20,77)(21,153,31,143)(22,80,32,70)(23,155,33,145)(24,62,34,72)(25,157,35,147)(26,64,36,74)(27,159,37,149)(28,66,38,76)(29,141,39,151)(30,68,40,78)(41,83,51,93)(42,105,52,115)(43,85,53,95)(44,107,54,117)(45,87,55,97)(46,109,56,119)(47,89,57,99)(48,111,58,101)(49,91,59,81)(50,113,60,103)(82,139,92,129)(84,121,94,131)(86,123,96,133)(88,125,98,135)(90,127,100,137)(102,128,112,138)(104,130,114,140)(106,132,116,122)(108,134,118,124)(110,136,120,126), (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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,85)(22,84)(23,83)(24,82)(25,81)(26,100)(27,99)(28,98)(29,97)(30,96)(31,95)(32,94)(33,93)(34,92)(35,91)(36,90)(37,89)(38,88)(39,87)(40,86)(41,61)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(121,144)(122,143)(123,142)(124,141)(125,160)(126,159)(127,158)(128,157)(129,156)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,121)(38,122)(39,123)(40,124)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,156)(82,157)(83,158)(84,159)(85,160)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155), (1,30,11,40)(2,21,12,31)(3,32,13,22)(4,23,14,33)(5,34,15,24)(6,25,16,35)(7,36,17,26)(8,27,18,37)(9,38,19,28)(10,29,20,39)(41,130,51,140)(42,121,52,131)(43,132,53,122)(44,123,54,133)(45,134,55,124)(46,125,56,135)(47,136,57,126)(48,127,58,137)(49,138,59,128)(50,129,60,139)(61,145,71,155)(62,156,72,146)(63,147,73,157)(64,158,74,148)(65,149,75,159)(66,160,76,150)(67,151,77,141)(68,142,78,152)(69,153,79,143)(70,144,80,154)(81,102,91,112)(82,113,92,103)(83,104,93,114)(84,115,94,105)(85,106,95,116)(86,117,96,107)(87,108,97,118)(88,119,98,109)(89,110,99,120)(90,101,100,111), (1,142,11,152)(2,79,12,69)(3,144,13,154)(4,61,14,71)(5,146,15,156)(6,63,16,73)(7,148,17,158)(8,65,18,75)(9,150,19,160)(10,67,20,77)(21,153,31,143)(22,80,32,70)(23,155,33,145)(24,62,34,72)(25,157,35,147)(26,64,36,74)(27,159,37,149)(28,66,38,76)(29,141,39,151)(30,68,40,78)(41,83,51,93)(42,105,52,115)(43,85,53,95)(44,107,54,117)(45,87,55,97)(46,109,56,119)(47,89,57,99)(48,111,58,101)(49,91,59,81)(50,113,60,103)(82,139,92,129)(84,121,94,131)(86,123,96,133)(88,125,98,135)(90,127,100,137)(102,128,112,138)(104,130,114,140)(106,132,116,122)(108,134,118,124)(110,136,120,126), (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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,85)(22,84)(23,83)(24,82)(25,81)(26,100)(27,99)(28,98)(29,97)(30,96)(31,95)(32,94)(33,93)(34,92)(35,91)(36,90)(37,89)(38,88)(39,87)(40,86)(41,61)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(121,144)(122,143)(123,142)(124,141)(125,160)(126,159)(127,158)(128,157)(129,156)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145) );
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,41),(18,42),(19,43),(20,44),(21,125),(22,126),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,121),(38,122),(39,123),(40,124),(61,111),(62,112),(63,113),(64,114),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,156),(82,157),(83,158),(84,159),(85,160),(86,141),(87,142),(88,143),(89,144),(90,145),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,154),(100,155)], [(1,30,11,40),(2,21,12,31),(3,32,13,22),(4,23,14,33),(5,34,15,24),(6,25,16,35),(7,36,17,26),(8,27,18,37),(9,38,19,28),(10,29,20,39),(41,130,51,140),(42,121,52,131),(43,132,53,122),(44,123,54,133),(45,134,55,124),(46,125,56,135),(47,136,57,126),(48,127,58,137),(49,138,59,128),(50,129,60,139),(61,145,71,155),(62,156,72,146),(63,147,73,157),(64,158,74,148),(65,149,75,159),(66,160,76,150),(67,151,77,141),(68,142,78,152),(69,153,79,143),(70,144,80,154),(81,102,91,112),(82,113,92,103),(83,104,93,114),(84,115,94,105),(85,106,95,116),(86,117,96,107),(87,108,97,118),(88,119,98,109),(89,110,99,120),(90,101,100,111)], [(1,142,11,152),(2,79,12,69),(3,144,13,154),(4,61,14,71),(5,146,15,156),(6,63,16,73),(7,148,17,158),(8,65,18,75),(9,150,19,160),(10,67,20,77),(21,153,31,143),(22,80,32,70),(23,155,33,145),(24,62,34,72),(25,157,35,147),(26,64,36,74),(27,159,37,149),(28,66,38,76),(29,141,39,151),(30,68,40,78),(41,83,51,93),(42,105,52,115),(43,85,53,95),(44,107,54,117),(45,87,55,97),(46,109,56,119),(47,89,57,99),(48,111,58,101),(49,91,59,81),(50,113,60,103),(82,139,92,129),(84,121,94,131),(86,123,96,133),(88,125,98,135),(90,127,100,137),(102,128,112,138),(104,130,114,140),(106,132,116,122),(108,134,118,124),(110,136,120,126)], [(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)], [(1,107),(2,106),(3,105),(4,104),(5,103),(6,102),(7,101),(8,120),(9,119),(10,118),(11,117),(12,116),(13,115),(14,114),(15,113),(16,112),(17,111),(18,110),(19,109),(20,108),(21,85),(22,84),(23,83),(24,82),(25,81),(26,100),(27,99),(28,98),(29,97),(30,96),(31,95),(32,94),(33,93),(34,92),(35,91),(36,90),(37,89),(38,88),(39,87),(40,86),(41,61),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,72),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(121,144),(122,143),(123,142),(124,141),(125,160),(126,159),(127,158),(128,157),(129,156),(130,155),(131,154),(132,153),(133,152),(134,151),(135,150),(136,149),(137,148),(138,147),(139,146),(140,145)]])
56 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 | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| 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 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | D4×D5 | D4×D5 | Q8.D10 |
| kernel | C2×Q8.D10 | D5×C2×C8 | C2×D40 | Q8.D10 | C2×Q8⋊D5 | C10×Q16 | C2×Q8⋊2D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 4 | 8 | 2 | 2 | 8 |
Matrix representation of C2×Q8.D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 6 | 35 | 0 | 0 |
| 6 | 1 | 0 | 0 |
| 0 | 0 | 15 | 15 |
| 0 | 0 | 15 | 26 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 29 | 29 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,9],[6,6,0,0,35,1,0,0,0,0,15,15,0,0,15,26],[0,40,0,0,40,0,0,0,0,0,12,29,0,0,29,29] >;
C2×Q8.D10 in GAP, Magma, Sage, TeX
C_2\times Q_8.D_{10} % in TeX
G:=Group("C2xQ8.D10"); // GroupNames label
G:=SmallGroup(320,1437);
// by ID
G=gap.SmallGroup(320,1437);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,1123,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=b^2*d^9>;
// generators/relations