direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4×Dic5, C24.57D10, C10⋊6(C4×D4), (D4×C10)⋊23C4, C20⋊7(C22×C4), C23⋊4(C2×Dic5), C4⋊1(C22×Dic5), (C2×D4).250D10, C10.65(C23×C4), C4⋊Dic5⋊75C22, (C23×Dic5)⋊7C2, (C22×D4).13D5, C22.145(D4×D5), C2.6(C23×Dic5), (C2×C10).290C24, (C2×C20).539C23, (C4×Dic5)⋊66C22, C10.128(C22×D4), (C22×C4).377D10, C23.D5⋊56C22, C22⋊1(C22×Dic5), C22.44(C23×D5), (D4×C10).268C22, (C23×C10).72C22, C23.203(C22×D5), C22.76(D4⋊2D5), (C22×C10).226C23, (C22×C20).272C22, (C2×Dic5).290C23, (C22×Dic5)⋊47C22, C5⋊7(C2×C4×D4), C2.6(C2×D4×D5), (D4×C2×C10).7C2, (C2×C20)⋊27(C2×C4), (C5×D4)⋊29(C2×C4), (C2×C4×Dic5)⋊10C2, (C2×C4)⋊7(C2×Dic5), (C2×C4⋊Dic5)⋊44C2, (C2×C10)⋊7(C22×C4), C2.6(C2×D4⋊2D5), (C22×C10)⋊18(C2×C4), C10.102(C2×C4○D4), (C2×C10).404(C2×D4), (C2×C23.D5)⋊23C2, (C2×C4).622(C22×D5), (C2×C10).174(C4○D4), SmallGroup(320,1467)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1102 in 426 conjugacy classes, 215 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×10], C22, C22 [×14], C22 [×24], C5, C2×C4 [×6], C2×C4 [×34], D4 [×16], C23, C23 [×12], C23 [×8], C10 [×3], C10 [×4], C10 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×20], C2×D4 [×12], C24 [×2], Dic5 [×4], Dic5 [×6], C20 [×4], C2×C10, C2×C10 [×14], C2×C10 [×24], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C2×Dic5 [×12], C2×Dic5 [×22], C2×C20 [×6], C5×D4 [×16], C22×C10, C22×C10 [×12], C22×C10 [×8], C2×C4×D4, C4×Dic5 [×4], C4⋊Dic5 [×4], C23.D5 [×8], C22×Dic5 [×2], C22×Dic5 [×10], C22×Dic5 [×8], C22×C20, D4×C10 [×12], C23×C10 [×2], C2×C4×Dic5, C2×C4⋊Dic5, D4×Dic5 [×8], C2×C23.D5 [×2], C23×Dic5 [×2], D4×C2×C10, C2×D4×Dic5
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D5, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, Dic5 [×8], D10 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×Dic5 [×28], C22×D5 [×7], C2×C4×D4, D4×D5 [×2], D4⋊2D5 [×2], C22×Dic5 [×14], C23×D5, D4×Dic5 [×4], C2×D4×D5, C2×D4⋊2D5, C23×Dic5, C2×D4×Dic5
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►On 160 points
Generators in S160
(1 68)(2 69)(3 70)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 122)(12 123)(13 124)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 121)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 88)(42 89)(43 90)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(91 142)(92 143)(93 144)(94 145)(95 146)(96 147)(97 148)(98 149)(99 150)(100 141)(101 140)(102 131)(103 132)(104 133)(105 134)(106 135)(107 136)(108 137)(109 138)(110 139)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 85 28 78)(2 86 29 79)(3 87 30 80)(4 88 21 71)(5 89 22 72)(6 90 23 73)(7 81 24 74)(8 82 25 75)(9 83 26 76)(10 84 27 77)(11 102 151 91)(12 103 152 92)(13 104 153 93)(14 105 154 94)(15 106 155 95)(16 107 156 96)(17 108 157 97)(18 109 158 98)(19 110 159 99)(20 101 160 100)(31 61 41 51)(32 62 42 52)(33 63 43 53)(34 64 44 54)(35 65 45 55)(36 66 46 56)(37 67 47 57)(38 68 48 58)(39 69 49 59)(40 70 50 60)(111 142 122 131)(112 143 123 132)(113 144 124 133)(114 145 125 134)(115 146 126 135)(116 147 127 136)(117 148 128 137)(118 149 129 138)(119 150 130 139)(120 141 121 140)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 51)(10 52)(11 127)(12 128)(13 129)(14 130)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 81)(50 82)(91 136)(92 137)(93 138)(94 139)(95 140)(96 131)(97 132)(98 133)(99 134)(100 135)(101 146)(102 147)(103 148)(104 149)(105 150)(106 141)(107 142)(108 143)(109 144)(110 145)(111 156)(112 157)(113 158)(114 159)(115 160)(116 151)(117 152)(118 153)(119 154)(120 155)
(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)
(1 131 6 136)(2 140 7 135)(3 139 8 134)(4 138 9 133)(5 137 10 132)(11 33 16 38)(12 32 17 37)(13 31 18 36)(14 40 19 35)(15 39 20 34)(21 149 26 144)(22 148 27 143)(23 147 28 142)(24 146 29 141)(25 145 30 150)(41 158 46 153)(42 157 47 152)(43 156 48 151)(44 155 49 160)(45 154 50 159)(51 98 56 93)(52 97 57 92)(53 96 58 91)(54 95 59 100)(55 94 60 99)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 129 76 124)(72 128 77 123)(73 127 78 122)(74 126 79 121)(75 125 80 130)(81 115 86 120)(82 114 87 119)(83 113 88 118)(84 112 89 117)(85 111 90 116)
G:=sub<Sym(160)| (1,68)(2,69)(3,70)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,121)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,88)(42,89)(43,90)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,141)(101,140)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,85,28,78)(2,86,29,79)(3,87,30,80)(4,88,21,71)(5,89,22,72)(6,90,23,73)(7,81,24,74)(8,82,25,75)(9,83,26,76)(10,84,27,77)(11,102,151,91)(12,103,152,92)(13,104,153,93)(14,105,154,94)(15,106,155,95)(16,107,156,96)(17,108,157,97)(18,109,158,98)(19,110,159,99)(20,101,160,100)(31,61,41,51)(32,62,42,52)(33,63,43,53)(34,64,44,54)(35,65,45,55)(36,66,46,56)(37,67,47,57)(38,68,48,58)(39,69,49,59)(40,70,50,60)(111,142,122,131)(112,143,123,132)(113,144,124,133)(114,145,125,134)(115,146,126,135)(116,147,127,136)(117,148,128,137)(118,149,129,138)(119,150,130,139)(120,141,121,140), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,127)(12,128)(13,129)(14,130)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,81)(50,82)(91,136)(92,137)(93,138)(94,139)(95,140)(96,131)(97,132)(98,133)(99,134)(100,135)(101,146)(102,147)(103,148)(104,149)(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)(112,157)(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)(120,155), (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), (1,131,6,136)(2,140,7,135)(3,139,8,134)(4,138,9,133)(5,137,10,132)(11,33,16,38)(12,32,17,37)(13,31,18,36)(14,40,19,35)(15,39,20,34)(21,149,26,144)(22,148,27,143)(23,147,28,142)(24,146,29,141)(25,145,30,150)(41,158,46,153)(42,157,47,152)(43,156,48,151)(44,155,49,160)(45,154,50,159)(51,98,56,93)(52,97,57,92)(53,96,58,91)(54,95,59,100)(55,94,60,99)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,129,76,124)(72,128,77,123)(73,127,78,122)(74,126,79,121)(75,125,80,130)(81,115,86,120)(82,114,87,119)(83,113,88,118)(84,112,89,117)(85,111,90,116)>;
G:=Group( (1,68)(2,69)(3,70)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,121)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,88)(42,89)(43,90)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,141)(101,140)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,85,28,78)(2,86,29,79)(3,87,30,80)(4,88,21,71)(5,89,22,72)(6,90,23,73)(7,81,24,74)(8,82,25,75)(9,83,26,76)(10,84,27,77)(11,102,151,91)(12,103,152,92)(13,104,153,93)(14,105,154,94)(15,106,155,95)(16,107,156,96)(17,108,157,97)(18,109,158,98)(19,110,159,99)(20,101,160,100)(31,61,41,51)(32,62,42,52)(33,63,43,53)(34,64,44,54)(35,65,45,55)(36,66,46,56)(37,67,47,57)(38,68,48,58)(39,69,49,59)(40,70,50,60)(111,142,122,131)(112,143,123,132)(113,144,124,133)(114,145,125,134)(115,146,126,135)(116,147,127,136)(117,148,128,137)(118,149,129,138)(119,150,130,139)(120,141,121,140), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,127)(12,128)(13,129)(14,130)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,81)(50,82)(91,136)(92,137)(93,138)(94,139)(95,140)(96,131)(97,132)(98,133)(99,134)(100,135)(101,146)(102,147)(103,148)(104,149)(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)(112,157)(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)(120,155), (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), (1,131,6,136)(2,140,7,135)(3,139,8,134)(4,138,9,133)(5,137,10,132)(11,33,16,38)(12,32,17,37)(13,31,18,36)(14,40,19,35)(15,39,20,34)(21,149,26,144)(22,148,27,143)(23,147,28,142)(24,146,29,141)(25,145,30,150)(41,158,46,153)(42,157,47,152)(43,156,48,151)(44,155,49,160)(45,154,50,159)(51,98,56,93)(52,97,57,92)(53,96,58,91)(54,95,59,100)(55,94,60,99)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,129,76,124)(72,128,77,123)(73,127,78,122)(74,126,79,121)(75,125,80,130)(81,115,86,120)(82,114,87,119)(83,113,88,118)(84,112,89,117)(85,111,90,116) );
G=PermutationGroup([(1,68),(2,69),(3,70),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,122),(12,123),(13,124),(14,125),(15,126),(16,127),(17,128),(18,129),(19,130),(20,121),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,88),(42,89),(43,90),(44,81),(45,82),(46,83),(47,84),(48,85),(49,86),(50,87),(91,142),(92,143),(93,144),(94,145),(95,146),(96,147),(97,148),(98,149),(99,150),(100,141),(101,140),(102,131),(103,132),(104,133),(105,134),(106,135),(107,136),(108,137),(109,138),(110,139),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160)], [(1,85,28,78),(2,86,29,79),(3,87,30,80),(4,88,21,71),(5,89,22,72),(6,90,23,73),(7,81,24,74),(8,82,25,75),(9,83,26,76),(10,84,27,77),(11,102,151,91),(12,103,152,92),(13,104,153,93),(14,105,154,94),(15,106,155,95),(16,107,156,96),(17,108,157,97),(18,109,158,98),(19,110,159,99),(20,101,160,100),(31,61,41,51),(32,62,42,52),(33,63,43,53),(34,64,44,54),(35,65,45,55),(36,66,46,56),(37,67,47,57),(38,68,48,58),(39,69,49,59),(40,70,50,60),(111,142,122,131),(112,143,123,132),(113,144,124,133),(114,145,125,134),(115,146,126,135),(116,147,127,136),(117,148,128,137),(118,149,129,138),(119,150,130,139),(120,141,121,140)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,51),(10,52),(11,127),(12,128),(13,129),(14,130),(15,121),(16,122),(17,123),(18,124),(19,125),(20,126),(21,66),(22,67),(23,68),(24,69),(25,70),(26,61),(27,62),(28,63),(29,64),(30,65),(31,76),(32,77),(33,78),(34,79),(35,80),(36,71),(37,72),(38,73),(39,74),(40,75),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,81),(50,82),(91,136),(92,137),(93,138),(94,139),(95,140),(96,131),(97,132),(98,133),(99,134),(100,135),(101,146),(102,147),(103,148),(104,149),(105,150),(106,141),(107,142),(108,143),(109,144),(110,145),(111,156),(112,157),(113,158),(114,159),(115,160),(116,151),(117,152),(118,153),(119,154),(120,155)], [(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)], [(1,131,6,136),(2,140,7,135),(3,139,8,134),(4,138,9,133),(5,137,10,132),(11,33,16,38),(12,32,17,37),(13,31,18,36),(14,40,19,35),(15,39,20,34),(21,149,26,144),(22,148,27,143),(23,147,28,142),(24,146,29,141),(25,145,30,150),(41,158,46,153),(42,157,47,152),(43,156,48,151),(44,155,49,160),(45,154,50,159),(51,98,56,93),(52,97,57,92),(53,96,58,91),(54,95,59,100),(55,94,60,99),(61,109,66,104),(62,108,67,103),(63,107,68,102),(64,106,69,101),(65,105,70,110),(71,129,76,124),(72,128,77,123),(73,127,78,122),(74,126,79,121),(75,125,80,130),(81,115,86,120),(82,114,87,119),(83,113,88,118),(84,112,89,117),(85,111,90,116)])
Matrix representation ►G ⊆ 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 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 37 |
| 0 | 0 | 0 | 0 | 21 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 20 | 40 |
| 23 | 0 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 31 | 0 | 0 |
| 0 | 0 | 14 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,21,0,0,0,0,37,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,20,0,0,0,0,0,40],[23,0,0,0,0,0,0,25,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,4,14,0,0,0,0,31,37,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4X | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | Dic5 | D10 | D10 | D4×D5 | D4⋊2D5 |
| kernel | C2×D4×Dic5 | C2×C4×Dic5 | C2×C4⋊Dic5 | D4×Dic5 | C2×C23.D5 | C23×Dic5 | D4×C2×C10 | D4×C10 | C2×Dic5 | C22×D4 | C2×C10 | C22×C4 | C2×D4 | C2×D4 | C24 | C22 | C22 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 16 | 4 | 2 | 4 | 2 | 16 | 8 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_4\times Dic_5
% in TeX
G:=Group("C2xD4xDic5"); // GroupNames label
G:=SmallGroup(320,1467);
// by ID
G=gap.SmallGroup(320,1467);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^10=1,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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ℤ
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