direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C22.D20, C23.51D20, C24.55D10, C22⋊C4⋊42D10, C2.9(C22×D20), C10.7(C22×D4), (C2×C10).36C24, C4⋊Dic5⋊52C22, (C23×Dic5)⋊4C2, C22.66(C2×D20), (C2×C20).129C23, (C22×C10).117D4, (C22×C4).171D10, (C2×Dic5).9C23, D10⋊C4⋊48C22, (C22×D5).8C23, C22.75(C23×D5), C10⋊2(C22.D4), (C22×C20).72C22, (C23×C10).62C22, (C23×D5).33C22, C23.147(C22×D5), C22.69(D4⋊2D5), (C22×C10).126C23, (C22×Dic5)⋊42C22, (C2×C4⋊Dic5)⋊20C2, (C2×C22⋊C4)⋊15D5, C10.68(C2×C4○D4), (C2×C10).48(C2×D4), (C10×C22⋊C4)⋊14C2, C5⋊2(C2×C22.D4), C2.11(C2×D4⋊2D5), (C2×D10⋊C4)⋊19C2, (C5×C22⋊C4)⋊47C22, (C2×C4).135(C22×D5), (C2×C5⋊D4).99C22, (C22×C5⋊D4).12C2, (C2×C10).168(C4○D4), SmallGroup(320,1164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C22.D20
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 1214 in 342 conjugacy classes, 127 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C22.D4, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C22×Dic5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, C22.D20, C2×C4⋊Dic5, C2×D10⋊C4, C10×C22⋊C4, C23×Dic5, C22×C5⋊D4, C2×C22.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22.D4, C22×D4, C2×C4○D4, D20, C22×D5, C2×C22.D4, C2×D20, D4⋊2D5, C23×D5, C22.D20, C22×D20, C2×D4⋊2D5, C2×C22.D20
►On 160 points
Generators in S160
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 141)(19 142)(20 143)(21 60)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 121)(79 122)(80 123)(81 116)(82 117)(83 118)(84 119)(85 120)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)
(1 29)(2 81)(3 31)(4 83)(5 33)(6 85)(7 35)(8 87)(9 37)(10 89)(11 39)(12 91)(13 21)(14 93)(15 23)(16 95)(17 25)(18 97)(19 27)(20 99)(22 136)(24 138)(26 140)(28 122)(30 124)(32 126)(34 128)(36 130)(38 132)(40 134)(41 73)(42 158)(43 75)(44 160)(45 77)(46 142)(47 79)(48 144)(49 61)(50 146)(51 63)(52 148)(53 65)(54 150)(55 67)(56 152)(57 69)(58 154)(59 71)(60 156)(62 117)(64 119)(66 101)(68 103)(70 105)(72 107)(74 109)(76 111)(78 113)(80 115)(82 125)(84 127)(86 129)(88 131)(90 133)(92 135)(94 137)(96 139)(98 121)(100 123)(102 151)(104 153)(106 155)(108 157)(110 159)(112 141)(114 143)(116 145)(118 147)(120 149)
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 121)(20 122)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 108)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 101)(55 102)(56 103)(57 104)(58 105)(59 106)(60 107)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 151)(68 152)(69 153)(70 154)(71 155)(72 156)(73 157)(74 158)(75 159)(76 160)(77 141)(78 142)(79 143)(80 144)
(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)
(1 122 123 20)(2 19 124 121)(3 140 125 18)(4 17 126 139)(5 138 127 16)(6 15 128 137)(7 136 129 14)(8 13 130 135)(9 134 131 12)(10 11 132 133)(21 36 92 87)(22 86 93 35)(23 34 94 85)(24 84 95 33)(25 32 96 83)(26 82 97 31)(27 30 98 81)(28 100 99 29)(37 40 88 91)(38 90 89 39)(41 101 108 54)(42 53 109 120)(43 119 110 52)(44 51 111 118)(45 117 112 50)(46 49 113 116)(47 115 114 48)(55 107 102 60)(56 59 103 106)(57 105 104 58)(61 78 145 142)(62 141 146 77)(63 76 147 160)(64 159 148 75)(65 74 149 158)(66 157 150 73)(67 72 151 156)(68 155 152 71)(69 70 153 154)(79 80 143 144)
G:=sub<Sym(160)| (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,141)(19,142)(20,143)(21,60)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,121)(79,122)(80,123)(81,116)(82,117)(83,118)(84,119)(85,120)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115), (1,29)(2,81)(3,31)(4,83)(5,33)(6,85)(7,35)(8,87)(9,37)(10,89)(11,39)(12,91)(13,21)(14,93)(15,23)(16,95)(17,25)(18,97)(19,27)(20,99)(22,136)(24,138)(26,140)(28,122)(30,124)(32,126)(34,128)(36,130)(38,132)(40,134)(41,73)(42,158)(43,75)(44,160)(45,77)(46,142)(47,79)(48,144)(49,61)(50,146)(51,63)(52,148)(53,65)(54,150)(55,67)(56,152)(57,69)(58,154)(59,71)(60,156)(62,117)(64,119)(66,101)(68,103)(70,105)(72,107)(74,109)(76,111)(78,113)(80,115)(82,125)(84,127)(86,129)(88,131)(90,133)(92,135)(94,137)(96,139)(98,121)(100,123)(102,151)(104,153)(106,155)(108,157)(110,159)(112,141)(114,143)(116,145)(118,147)(120,149), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,121)(20,122)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,141)(78,142)(79,143)(80,144), (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), (1,122,123,20)(2,19,124,121)(3,140,125,18)(4,17,126,139)(5,138,127,16)(6,15,128,137)(7,136,129,14)(8,13,130,135)(9,134,131,12)(10,11,132,133)(21,36,92,87)(22,86,93,35)(23,34,94,85)(24,84,95,33)(25,32,96,83)(26,82,97,31)(27,30,98,81)(28,100,99,29)(37,40,88,91)(38,90,89,39)(41,101,108,54)(42,53,109,120)(43,119,110,52)(44,51,111,118)(45,117,112,50)(46,49,113,116)(47,115,114,48)(55,107,102,60)(56,59,103,106)(57,105,104,58)(61,78,145,142)(62,141,146,77)(63,76,147,160)(64,159,148,75)(65,74,149,158)(66,157,150,73)(67,72,151,156)(68,155,152,71)(69,70,153,154)(79,80,143,144)>;
G:=Group( (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,141)(19,142)(20,143)(21,60)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,121)(79,122)(80,123)(81,116)(82,117)(83,118)(84,119)(85,120)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115), (1,29)(2,81)(3,31)(4,83)(5,33)(6,85)(7,35)(8,87)(9,37)(10,89)(11,39)(12,91)(13,21)(14,93)(15,23)(16,95)(17,25)(18,97)(19,27)(20,99)(22,136)(24,138)(26,140)(28,122)(30,124)(32,126)(34,128)(36,130)(38,132)(40,134)(41,73)(42,158)(43,75)(44,160)(45,77)(46,142)(47,79)(48,144)(49,61)(50,146)(51,63)(52,148)(53,65)(54,150)(55,67)(56,152)(57,69)(58,154)(59,71)(60,156)(62,117)(64,119)(66,101)(68,103)(70,105)(72,107)(74,109)(76,111)(78,113)(80,115)(82,125)(84,127)(86,129)(88,131)(90,133)(92,135)(94,137)(96,139)(98,121)(100,123)(102,151)(104,153)(106,155)(108,157)(110,159)(112,141)(114,143)(116,145)(118,147)(120,149), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,121)(20,122)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,141)(78,142)(79,143)(80,144), (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), (1,122,123,20)(2,19,124,121)(3,140,125,18)(4,17,126,139)(5,138,127,16)(6,15,128,137)(7,136,129,14)(8,13,130,135)(9,134,131,12)(10,11,132,133)(21,36,92,87)(22,86,93,35)(23,34,94,85)(24,84,95,33)(25,32,96,83)(26,82,97,31)(27,30,98,81)(28,100,99,29)(37,40,88,91)(38,90,89,39)(41,101,108,54)(42,53,109,120)(43,119,110,52)(44,51,111,118)(45,117,112,50)(46,49,113,116)(47,115,114,48)(55,107,102,60)(56,59,103,106)(57,105,104,58)(61,78,145,142)(62,141,146,77)(63,76,147,160)(64,159,148,75)(65,74,149,158)(66,157,150,73)(67,72,151,156)(68,155,152,71)(69,70,153,154)(79,80,143,144) );
G=PermutationGroup([[(1,144),(2,145),(3,146),(4,147),(5,148),(6,149),(7,150),(8,151),(9,152),(10,153),(11,154),(12,155),(13,156),(14,157),(15,158),(16,159),(17,160),(18,141),(19,142),(20,143),(21,60),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135),(73,136),(74,137),(75,138),(76,139),(77,140),(78,121),(79,122),(80,123),(81,116),(82,117),(83,118),(84,119),(85,120),(86,101),(87,102),(88,103),(89,104),(90,105),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115)], [(1,29),(2,81),(3,31),(4,83),(5,33),(6,85),(7,35),(8,87),(9,37),(10,89),(11,39),(12,91),(13,21),(14,93),(15,23),(16,95),(17,25),(18,97),(19,27),(20,99),(22,136),(24,138),(26,140),(28,122),(30,124),(32,126),(34,128),(36,130),(38,132),(40,134),(41,73),(42,158),(43,75),(44,160),(45,77),(46,142),(47,79),(48,144),(49,61),(50,146),(51,63),(52,148),(53,65),(54,150),(55,67),(56,152),(57,69),(58,154),(59,71),(60,156),(62,117),(64,119),(66,101),(68,103),(70,105),(72,107),(74,109),(76,111),(78,113),(80,115),(82,125),(84,127),(86,129),(88,131),(90,133),(92,135),(94,137),(96,139),(98,121),(100,123),(102,151),(104,153),(106,155),(108,157),(110,159),(112,141),(114,143),(116,145),(118,147),(120,149)], [(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,129),(8,130),(9,131),(10,132),(11,133),(12,134),(13,135),(14,136),(15,137),(16,138),(17,139),(18,140),(19,121),(20,122),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,108),(42,109),(43,110),(44,111),(45,112),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,101),(55,102),(56,103),(57,104),(58,105),(59,106),(60,107),(61,145),(62,146),(63,147),(64,148),(65,149),(66,150),(67,151),(68,152),(69,153),(70,154),(71,155),(72,156),(73,157),(74,158),(75,159),(76,160),(77,141),(78,142),(79,143),(80,144)], [(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)], [(1,122,123,20),(2,19,124,121),(3,140,125,18),(4,17,126,139),(5,138,127,16),(6,15,128,137),(7,136,129,14),(8,13,130,135),(9,134,131,12),(10,11,132,133),(21,36,92,87),(22,86,93,35),(23,34,94,85),(24,84,95,33),(25,32,96,83),(26,82,97,31),(27,30,98,81),(28,100,99,29),(37,40,88,91),(38,90,89,39),(41,101,108,54),(42,53,109,120),(43,119,110,52),(44,51,111,118),(45,117,112,50),(46,49,113,116),(47,115,114,48),(55,107,102,60),(56,59,103,106),(57,105,104,58),(61,78,145,142),(62,141,146,77),(63,76,147,160),(64,159,148,75),(65,74,149,158),(66,157,150,73),(67,72,151,156),(68,155,152,71),(69,70,153,154),(79,80,143,144)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 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 | 20 | 20 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D20 | D4⋊2D5 |
| kernel | C2×C22.D20 | C22.D20 | C2×C4⋊Dic5 | C2×D10⋊C4 | C10×C22⋊C4 | C23×Dic5 | C22×C5⋊D4 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22⋊C4 | C22×C4 | C24 | C23 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 8 | 4 | 2 | 16 | 8 |
Matrix representation of C2×C22.D20 ►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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 39 |
| 0 | 0 | 0 | 0 | 16 | 27 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 16 |
| 0 | 0 | 0 | 0 | 36 | 9 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,11,16,0,0,0,0,39,27],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,32,36,0,0,0,0,16,9] >;
C2×C22.D20 in GAP, Magma, Sage, TeX
C_2\times C_2^2.D_{20} % in TeX
G:=Group("C2xC2^2.D20"); // GroupNames label
G:=SmallGroup(320,1164);
// by ID
G=gap.SmallGroup(320,1164);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations