direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4×C5⋊D4, C24.69D10, C10⋊5(C4×D4), C20⋊16(C2×D4), (C2×C20)⋊39D4, (C23×C4)⋊3D5, C23⋊5(C4×D5), (C23×C20)⋊12C2, D10⋊6(C22×C4), (C22×C4)⋊42D10, C10.60(C23×C4), Dic5⋊4(C22×C4), (C2×C10).286C24, (C2×C20).886C23, (C22×C20)⋊63C22, (C4×Dic5)⋊82C22, C10.132(C22×D4), C23.D5⋊68C22, D10⋊C4⋊76C22, C22.43(C23×D5), C22.81(C4○D20), C10.D4⋊78C22, C23.336(C22×D5), (C23×C10).108C22, (C22×C10).415C23, (C2×Dic5).289C23, (C23×D5).124C22, (C22×D5).247C23, (C22×Dic5).253C22, C5⋊6(C2×C4×D4), C22⋊3(C2×C4×D5), (C2×C4×Dic5)⋊38C2, C2.6(C2×C4○D20), (C2×C4×D5)⋊71C22, (D5×C22×C4)⋊25C2, C2.39(D5×C22×C4), (C2×C10)⋊9(C22×C4), C10.61(C2×C4○D4), C2.3(C22×C5⋊D4), (C22×C10)⋊21(C2×C4), (C2×Dic5)⋊27(C2×C4), (C2×C10).573(C2×D4), (C22×D5)⋊18(C2×C4), (C2×C23.D5)⋊33C2, (C2×D10⋊C4)⋊46C2, (C2×C10.D4)⋊52C2, (C2×C4).830(C22×D5), (C22×C5⋊D4).16C2, C22.102(C2×C5⋊D4), (C2×C10).112(C4○D4), (C2×C5⋊D4).160C22, SmallGroup(320,1460)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4×C5⋊D4
G = < a,b,c,d,e | a2=b4=c5=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 1262 in 426 conjugacy classes, 183 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C23×C4, C22×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C4×D4, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, C22×C20, C22×C20, C23×D5, C23×C10, C2×C4×Dic5, C2×C10.D4, C2×D10⋊C4, C4×C5⋊D4, C2×C23.D5, D5×C22×C4, C22×C5⋊D4, C23×C20, C2×C4×C5⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, C24, D10, C4×D4, C23×C4, C22×D4, C2×C4○D4, C4×D5, C5⋊D4, C22×D5, C2×C4×D4, C2×C4×D5, C4○D20, C2×C5⋊D4, C23×D5, C4×C5⋊D4, D5×C22×C4, C2×C4○D20, C22×C5⋊D4, C2×C4×C5⋊D4
►On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(1 36 11 26)(2 37 12 27)(3 38 13 28)(4 39 14 29)(5 40 15 30)(6 31 16 21)(7 32 17 22)(8 33 18 23)(9 34 19 24)(10 35 20 25)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)(81 116 91 106)(82 117 92 107)(83 118 93 108)(84 119 94 109)(85 120 95 110)(86 111 96 101)(87 112 97 102)(88 113 98 103)(89 114 99 104)(90 115 100 105)(121 156 131 146)(122 157 132 147)(123 158 133 148)(124 159 134 149)(125 160 135 150)(126 151 136 141)(127 152 137 142)(128 153 138 143)(129 154 139 144)(130 155 140 145)
(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)
(1 121 6 126)(2 125 7 130)(3 124 8 129)(4 123 9 128)(5 122 10 127)(11 131 16 136)(12 135 17 140)(13 134 18 139)(14 133 19 138)(15 132 20 137)(21 141 26 146)(22 145 27 150)(23 144 28 149)(24 143 29 148)(25 142 30 147)(31 151 36 156)(32 155 37 160)(33 154 38 159)(34 153 39 158)(35 152 40 157)(41 86 46 81)(42 90 47 85)(43 89 48 84)(44 88 49 83)(45 87 50 82)(51 96 56 91)(52 100 57 95)(53 99 58 94)(54 98 59 93)(55 97 60 92)(61 106 66 101)(62 110 67 105)(63 109 68 104)(64 108 69 103)(65 107 70 102)(71 116 76 111)(72 120 77 115)(73 119 78 114)(74 118 79 113)(75 117 80 112)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)(82 85)(83 84)(87 90)(88 89)(92 95)(93 94)(97 100)(98 99)(102 105)(103 104)(107 110)(108 109)(112 115)(113 114)(117 120)(118 119)(121 126)(122 130)(123 129)(124 128)(125 127)(131 136)(132 140)(133 139)(134 138)(135 137)(141 146)(142 150)(143 149)(144 148)(145 147)(151 156)(152 160)(153 159)(154 158)(155 157)
G:=sub<Sym(160)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,96)(12,97)(13,98)(14,99)(15,100)(16,91)(17,92)(18,93)(19,94)(20,95)(21,106)(22,107)(23,108)(24,109)(25,110)(26,101)(27,102)(28,103)(29,104)(30,105)(31,116)(32,117)(33,118)(34,119)(35,120)(36,111)(37,112)(38,113)(39,114)(40,115)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,136)(52,137)(53,138)(54,139)(55,140)(56,131)(57,132)(58,133)(59,134)(60,135)(61,146)(62,147)(63,148)(64,149)(65,150)(66,141)(67,142)(68,143)(69,144)(70,145)(71,156)(72,157)(73,158)(74,159)(75,160)(76,151)(77,152)(78,153)(79,154)(80,155), (1,36,11,26)(2,37,12,27)(3,38,13,28)(4,39,14,29)(5,40,15,30)(6,31,16,21)(7,32,17,22)(8,33,18,23)(9,34,19,24)(10,35,20,25)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65)(81,116,91,106)(82,117,92,107)(83,118,93,108)(84,119,94,109)(85,120,95,110)(86,111,96,101)(87,112,97,102)(88,113,98,103)(89,114,99,104)(90,115,100,105)(121,156,131,146)(122,157,132,147)(123,158,133,148)(124,159,134,149)(125,160,135,150)(126,151,136,141)(127,152,137,142)(128,153,138,143)(129,154,139,144)(130,155,140,145), (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), (1,121,6,126)(2,125,7,130)(3,124,8,129)(4,123,9,128)(5,122,10,127)(11,131,16,136)(12,135,17,140)(13,134,18,139)(14,133,19,138)(15,132,20,137)(21,141,26,146)(22,145,27,150)(23,144,28,149)(24,143,29,148)(25,142,30,147)(31,151,36,156)(32,155,37,160)(33,154,38,159)(34,153,39,158)(35,152,40,157)(41,86,46,81)(42,90,47,85)(43,89,48,84)(44,88,49,83)(45,87,50,82)(51,96,56,91)(52,100,57,95)(53,99,58,94)(54,98,59,93)(55,97,60,92)(61,106,66,101)(62,110,67,105)(63,109,68,104)(64,108,69,103)(65,107,70,102)(71,116,76,111)(72,120,77,115)(73,119,78,114)(74,118,79,113)(75,117,80,112), (2,5)(3,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)(121,126)(122,130)(123,129)(124,128)(125,127)(131,136)(132,140)(133,139)(134,138)(135,137)(141,146)(142,150)(143,149)(144,148)(145,147)(151,156)(152,160)(153,159)(154,158)(155,157)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,96)(12,97)(13,98)(14,99)(15,100)(16,91)(17,92)(18,93)(19,94)(20,95)(21,106)(22,107)(23,108)(24,109)(25,110)(26,101)(27,102)(28,103)(29,104)(30,105)(31,116)(32,117)(33,118)(34,119)(35,120)(36,111)(37,112)(38,113)(39,114)(40,115)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,136)(52,137)(53,138)(54,139)(55,140)(56,131)(57,132)(58,133)(59,134)(60,135)(61,146)(62,147)(63,148)(64,149)(65,150)(66,141)(67,142)(68,143)(69,144)(70,145)(71,156)(72,157)(73,158)(74,159)(75,160)(76,151)(77,152)(78,153)(79,154)(80,155), (1,36,11,26)(2,37,12,27)(3,38,13,28)(4,39,14,29)(5,40,15,30)(6,31,16,21)(7,32,17,22)(8,33,18,23)(9,34,19,24)(10,35,20,25)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65)(81,116,91,106)(82,117,92,107)(83,118,93,108)(84,119,94,109)(85,120,95,110)(86,111,96,101)(87,112,97,102)(88,113,98,103)(89,114,99,104)(90,115,100,105)(121,156,131,146)(122,157,132,147)(123,158,133,148)(124,159,134,149)(125,160,135,150)(126,151,136,141)(127,152,137,142)(128,153,138,143)(129,154,139,144)(130,155,140,145), (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), (1,121,6,126)(2,125,7,130)(3,124,8,129)(4,123,9,128)(5,122,10,127)(11,131,16,136)(12,135,17,140)(13,134,18,139)(14,133,19,138)(15,132,20,137)(21,141,26,146)(22,145,27,150)(23,144,28,149)(24,143,29,148)(25,142,30,147)(31,151,36,156)(32,155,37,160)(33,154,38,159)(34,153,39,158)(35,152,40,157)(41,86,46,81)(42,90,47,85)(43,89,48,84)(44,88,49,83)(45,87,50,82)(51,96,56,91)(52,100,57,95)(53,99,58,94)(54,98,59,93)(55,97,60,92)(61,106,66,101)(62,110,67,105)(63,109,68,104)(64,108,69,103)(65,107,70,102)(71,116,76,111)(72,120,77,115)(73,119,78,114)(74,118,79,113)(75,117,80,112), (2,5)(3,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)(121,126)(122,130)(123,129)(124,128)(125,127)(131,136)(132,140)(133,139)(134,138)(135,137)(141,146)(142,150)(143,149)(144,148)(145,147)(151,156)(152,160)(153,159)(154,158)(155,157) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,81),(7,82),(8,83),(9,84),(10,85),(11,96),(12,97),(13,98),(14,99),(15,100),(16,91),(17,92),(18,93),(19,94),(20,95),(21,106),(22,107),(23,108),(24,109),(25,110),(26,101),(27,102),(28,103),(29,104),(30,105),(31,116),(32,117),(33,118),(34,119),(35,120),(36,111),(37,112),(38,113),(39,114),(40,115),(41,126),(42,127),(43,128),(44,129),(45,130),(46,121),(47,122),(48,123),(49,124),(50,125),(51,136),(52,137),(53,138),(54,139),(55,140),(56,131),(57,132),(58,133),(59,134),(60,135),(61,146),(62,147),(63,148),(64,149),(65,150),(66,141),(67,142),(68,143),(69,144),(70,145),(71,156),(72,157),(73,158),(74,159),(75,160),(76,151),(77,152),(78,153),(79,154),(80,155)], [(1,36,11,26),(2,37,12,27),(3,38,13,28),(4,39,14,29),(5,40,15,30),(6,31,16,21),(7,32,17,22),(8,33,18,23),(9,34,19,24),(10,35,20,25),(41,76,51,66),(42,77,52,67),(43,78,53,68),(44,79,54,69),(45,80,55,70),(46,71,56,61),(47,72,57,62),(48,73,58,63),(49,74,59,64),(50,75,60,65),(81,116,91,106),(82,117,92,107),(83,118,93,108),(84,119,94,109),(85,120,95,110),(86,111,96,101),(87,112,97,102),(88,113,98,103),(89,114,99,104),(90,115,100,105),(121,156,131,146),(122,157,132,147),(123,158,133,148),(124,159,134,149),(125,160,135,150),(126,151,136,141),(127,152,137,142),(128,153,138,143),(129,154,139,144),(130,155,140,145)], [(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)], [(1,121,6,126),(2,125,7,130),(3,124,8,129),(4,123,9,128),(5,122,10,127),(11,131,16,136),(12,135,17,140),(13,134,18,139),(14,133,19,138),(15,132,20,137),(21,141,26,146),(22,145,27,150),(23,144,28,149),(24,143,29,148),(25,142,30,147),(31,151,36,156),(32,155,37,160),(33,154,38,159),(34,153,39,158),(35,152,40,157),(41,86,46,81),(42,90,47,85),(43,89,48,84),(44,88,49,83),(45,87,50,82),(51,96,56,91),(52,100,57,95),(53,99,58,94),(54,98,59,93),(55,97,60,92),(61,106,66,101),(62,110,67,105),(63,109,68,104),(64,108,69,103),(65,107,70,102),(71,116,76,111),(72,120,77,115),(73,119,78,114),(74,118,79,113),(75,117,80,112)], [(2,5),(3,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57),(61,66),(62,70),(63,69),(64,68),(65,67),(71,76),(72,80),(73,79),(74,78),(75,77),(82,85),(83,84),(87,90),(88,89),(92,95),(93,94),(97,100),(98,99),(102,105),(103,104),(107,110),(108,109),(112,115),(113,114),(117,120),(118,119),(121,126),(122,130),(123,129),(124,128),(125,127),(131,136),(132,140),(133,139),(134,138),(135,137),(141,146),(142,150),(143,149),(144,148),(145,147),(151,156),(152,160),(153,159),(154,158),(155,157)]])
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4X | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C4×D5 | C4○D20 |
| kernel | C2×C4×C5⋊D4 | C2×C4×Dic5 | C2×C10.D4 | C2×D10⋊C4 | C4×C5⋊D4 | C2×C23.D5 | D5×C22×C4 | C22×C5⋊D4 | C23×C20 | C2×C5⋊D4 | C2×C20 | C23×C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 16 | 4 | 2 | 4 | 12 | 2 | 16 | 16 | 16 |
Matrix representation of C2×C4×C5⋊D4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 40 | 40 |
| 0 | 0 | 0 | 8 | 7 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 21 |
| 0 | 0 | 0 | 21 | 38 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 34 | 35 |
| 0 | 0 | 0 | 8 | 7 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,40,34,0,0,0,0,0,40,8,0,0,0,40,7],[1,0,0,0,0,0,40,0,0,0,0,7,1,0,0,0,0,0,3,21,0,0,0,21,38],[40,0,0,0,0,0,1,0,0,0,0,34,40,0,0,0,0,0,34,8,0,0,0,35,7] >;
C2×C4×C5⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_5\rtimes D_4
% in TeX
G:=Group("C2xC4xC5:D4"); // GroupNames label
G:=SmallGroup(320,1460);
// by ID
G=gap.SmallGroup(320,1460);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^5=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations