direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D4.D5, C20⋊11SD16, C42.208D10, C5⋊6(C4×SD16), (C4×D4).7D5, D4.5(C4×D5), (D4×C20).8C2, C4⋊C4.245D10, (C2×C20).255D4, C10.101(C4×D4), Dic10⋊19(C2×C4), (C4×Dic10)⋊19C2, (C2×D4).192D10, C4.38(C4○D20), C10.90(C4○D8), C20.52(C4○D4), C10.Q16⋊44C2, C20.58(C22×C4), (C4×C20).88C22, C20.Q8⋊44C2, C10.52(C2×SD16), (C2×C20).339C23, D4⋊Dic5.17C2, C2.4(D4.8D10), (D4×C10).234C22, C4⋊Dic5.328C22, (C2×Dic10).272C22, (C4×C5⋊2C8)⋊9C2, C4.23(C2×C4×D5), C5⋊2C8⋊21(C2×C4), C2.17(C4×C5⋊D4), C2.3(C2×D4.D5), (C5×D4).26(C2×C4), (C2×C10).470(C2×D4), (C2×D4.D5).10C2, C22.77(C2×C5⋊D4), (C2×C4).102(C5⋊D4), (C5×C4⋊C4).276C22, (C2×C4).439(C22×D5), (C2×C5⋊2C8).253C22, SmallGroup(320,644)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D4.D5
G = < a,b,c,d,e | a4=b4=c2=d5=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 358 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C4×SD16, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, D4.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C4×C5⋊2C8, C20.Q8, C10.Q16, D4⋊Dic5, C4×Dic10, C2×D4.D5, D4×C20, C4×D4.D5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, SD16, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×SD16, C4○D8, C4×D5, C5⋊D4, C22×D5, C4×SD16, D4.D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C2×D4.D5, D4.8D10, C4×D4.D5
►On 160 points
Generators in S160
(1 61 21 41)(2 62 22 42)(3 63 23 43)(4 64 24 44)(5 65 25 45)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 141 101 121)(82 142 102 122)(83 143 103 123)(84 144 104 124)(85 145 105 125)(86 146 106 126)(87 147 107 127)(88 148 108 128)(89 149 109 129)(90 150 110 130)(91 151 111 131)(92 152 112 132)(93 153 113 133)(94 154 114 134)(95 155 115 135)(96 156 116 136)(97 157 117 137)(98 158 118 138)(99 159 119 139)(100 160 120 140)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(81 106)(82 107)(83 108)(84 109)(85 110)(86 101)(87 102)(88 103)(89 104)(90 105)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 146)(122 147)(123 148)(124 149)(125 150)(126 141)(127 142)(128 143)(129 144)(130 145)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(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 88 6 83)(2 87 7 82)(3 86 8 81)(4 90 9 85)(5 89 10 84)(11 98 16 93)(12 97 17 92)(13 96 18 91)(14 100 19 95)(15 99 20 94)(21 108 26 103)(22 107 27 102)(23 106 28 101)(24 110 29 105)(25 109 30 104)(31 118 36 113)(32 117 37 112)(33 116 38 111)(34 120 39 115)(35 119 40 114)(41 128 46 123)(42 127 47 122)(43 126 48 121)(44 130 49 125)(45 129 50 124)(51 138 56 133)(52 137 57 132)(53 136 58 131)(54 140 59 135)(55 139 60 134)(61 148 66 143)(62 147 67 142)(63 146 68 141)(64 150 69 145)(65 149 70 144)(71 158 76 153)(72 157 77 152)(73 156 78 151)(74 160 79 155)(75 159 80 154)
G:=sub<Sym(160)| (1,61,21,41)(2,62,22,42)(3,63,23,43)(4,64,24,44)(5,65,25,45)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,141,101,121)(82,142,102,122)(83,143,103,123)(84,144,104,124)(85,145,105,125)(86,146,106,126)(87,147,107,127)(88,148,108,128)(89,149,109,129)(90,150,110,130)(91,151,111,131)(92,152,112,132)(93,153,113,133)(94,154,114,134)(95,155,115,135)(96,156,116,136)(97,157,117,137)(98,158,118,138)(99,159,119,139)(100,160,120,140), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120)(121,131,126,136)(122,132,127,137)(123,133,128,138)(124,134,129,139)(125,135,130,140)(141,151,146,156)(142,152,147,157)(143,153,148,158)(144,154,149,159)(145,155,150,160), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(81,106)(82,107)(83,108)(84,109)(85,110)(86,101)(87,102)(88,103)(89,104)(90,105)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,146)(122,147)(123,148)(124,149)(125,150)(126,141)(127,142)(128,143)(129,144)(130,145)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (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), (1,88,6,83)(2,87,7,82)(3,86,8,81)(4,90,9,85)(5,89,10,84)(11,98,16,93)(12,97,17,92)(13,96,18,91)(14,100,19,95)(15,99,20,94)(21,108,26,103)(22,107,27,102)(23,106,28,101)(24,110,29,105)(25,109,30,104)(31,118,36,113)(32,117,37,112)(33,116,38,111)(34,120,39,115)(35,119,40,114)(41,128,46,123)(42,127,47,122)(43,126,48,121)(44,130,49,125)(45,129,50,124)(51,138,56,133)(52,137,57,132)(53,136,58,131)(54,140,59,135)(55,139,60,134)(61,148,66,143)(62,147,67,142)(63,146,68,141)(64,150,69,145)(65,149,70,144)(71,158,76,153)(72,157,77,152)(73,156,78,151)(74,160,79,155)(75,159,80,154)>;
G:=Group( (1,61,21,41)(2,62,22,42)(3,63,23,43)(4,64,24,44)(5,65,25,45)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,141,101,121)(82,142,102,122)(83,143,103,123)(84,144,104,124)(85,145,105,125)(86,146,106,126)(87,147,107,127)(88,148,108,128)(89,149,109,129)(90,150,110,130)(91,151,111,131)(92,152,112,132)(93,153,113,133)(94,154,114,134)(95,155,115,135)(96,156,116,136)(97,157,117,137)(98,158,118,138)(99,159,119,139)(100,160,120,140), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120)(121,131,126,136)(122,132,127,137)(123,133,128,138)(124,134,129,139)(125,135,130,140)(141,151,146,156)(142,152,147,157)(143,153,148,158)(144,154,149,159)(145,155,150,160), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(81,106)(82,107)(83,108)(84,109)(85,110)(86,101)(87,102)(88,103)(89,104)(90,105)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,146)(122,147)(123,148)(124,149)(125,150)(126,141)(127,142)(128,143)(129,144)(130,145)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (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), (1,88,6,83)(2,87,7,82)(3,86,8,81)(4,90,9,85)(5,89,10,84)(11,98,16,93)(12,97,17,92)(13,96,18,91)(14,100,19,95)(15,99,20,94)(21,108,26,103)(22,107,27,102)(23,106,28,101)(24,110,29,105)(25,109,30,104)(31,118,36,113)(32,117,37,112)(33,116,38,111)(34,120,39,115)(35,119,40,114)(41,128,46,123)(42,127,47,122)(43,126,48,121)(44,130,49,125)(45,129,50,124)(51,138,56,133)(52,137,57,132)(53,136,58,131)(54,140,59,135)(55,139,60,134)(61,148,66,143)(62,147,67,142)(63,146,68,141)(64,150,69,145)(65,149,70,144)(71,158,76,153)(72,157,77,152)(73,156,78,151)(74,160,79,155)(75,159,80,154) );
G=PermutationGroup([[(1,61,21,41),(2,62,22,42),(3,63,23,43),(4,64,24,44),(5,65,25,45),(6,66,26,46),(7,67,27,47),(8,68,28,48),(9,69,29,49),(10,70,30,50),(11,71,31,51),(12,72,32,52),(13,73,33,53),(14,74,34,54),(15,75,35,55),(16,76,36,56),(17,77,37,57),(18,78,38,58),(19,79,39,59),(20,80,40,60),(81,141,101,121),(82,142,102,122),(83,143,103,123),(84,144,104,124),(85,145,105,125),(86,146,106,126),(87,147,107,127),(88,148,108,128),(89,149,109,129),(90,150,110,130),(91,151,111,131),(92,152,112,132),(93,153,113,133),(94,154,114,134),(95,155,115,135),(96,156,116,136),(97,157,117,137),(98,158,118,138),(99,159,119,139),(100,160,120,140)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75),(81,91,86,96),(82,92,87,97),(83,93,88,98),(84,94,89,99),(85,95,90,100),(101,111,106,116),(102,112,107,117),(103,113,108,118),(104,114,109,119),(105,115,110,120),(121,131,126,136),(122,132,127,137),(123,133,128,138),(124,134,129,139),(125,135,130,140),(141,151,146,156),(142,152,147,157),(143,153,148,158),(144,154,149,159),(145,155,150,160)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70),(81,106),(82,107),(83,108),(84,109),(85,110),(86,101),(87,102),(88,103),(89,104),(90,105),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(121,146),(122,147),(123,148),(124,149),(125,150),(126,141),(127,142),(128,143),(129,144),(130,145),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(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)], [(1,88,6,83),(2,87,7,82),(3,86,8,81),(4,90,9,85),(5,89,10,84),(11,98,16,93),(12,97,17,92),(13,96,18,91),(14,100,19,95),(15,99,20,94),(21,108,26,103),(22,107,27,102),(23,106,28,101),(24,110,29,105),(25,109,30,104),(31,118,36,113),(32,117,37,112),(33,116,38,111),(34,120,39,115),(35,119,40,114),(41,128,46,123),(42,127,47,122),(43,126,48,121),(44,130,49,125),(45,129,50,124),(51,138,56,133),(52,137,57,132),(53,136,58,131),(54,140,59,135),(55,139,60,134),(61,148,66,143),(62,147,67,142),(63,146,68,141),(64,150,69,145),(65,149,70,144),(71,158,76,153),(72,157,77,152),(73,156,78,151),(74,160,79,155),(75,159,80,154)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C4×D5 | C4○D20 | D4.D5 | D4.8D10 |
| kernel | C4×D4.D5 | C4×C5⋊2C8 | C20.Q8 | C10.Q16 | D4⋊Dic5 | C4×Dic10 | C2×D4.D5 | D4×C20 | D4.D5 | C2×C20 | C4×D4 | C20 | C20 | C42 | C4⋊C4 | C2×D4 | C10 | C2×C4 | D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C4×D4.D5 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 3 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 |
| 38 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 3 | 0 | 0 |
| 38 | 25 | 0 | 0 |
| 0 | 0 | 15 | 26 |
| 0 | 0 | 26 | 26 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[1,3,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[16,38,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[16,38,0,0,3,25,0,0,0,0,15,26,0,0,26,26] >;
C4×D4.D5 in GAP, Magma, Sage, TeX
C_4\times D_4.D_5
% in TeX
G:=Group("C4xD4.D5"); // GroupNames label
G:=SmallGroup(320,644);
// by ID
G=gap.SmallGroup(320,644);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,58,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^5=1,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^-1>;
// generators/relations