metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊2D20, C24.15D10, (C2×C20)⋊6D4, (C2×Dic5)⋊5D4, (C22×D5)⋊4D4, (C22×C10)⋊7D4, (C22×D20)⋊3C2, C5⋊2(C23⋊2D4), C10.33C22≀C2, C2.6(C20⋊D4), C2.7(C20⋊7D4), (C22×C4).35D10, C22.242(D4×D5), C2.8(C23⋊D10), C10.59(C4⋊D4), C10.13(C4⋊1D4), C22.126(C2×D20), C2.34(C22⋊D20), C2.34(D10⋊D4), (C23×C10).43C22, (C22×C20).61C22, (C23×D5).16C22, C23.372(C22×D5), C10.10C42⋊32C2, C22.100(C4○D20), (C22×C10).334C23, (C22×Dic5).46C22, (C2×C4)⋊3(C5⋊D4), (C2×C22⋊C4)⋊8D5, (C22×C5⋊D4)⋊1C2, (C2×D10⋊C4)⋊8C2, (C10×C22⋊C4)⋊11C2, (C2×C10).325(C2×D4), (C2×C10).80(C4○D4), C22.128(C2×C5⋊D4), SmallGroup(320,587)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊2D20
G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1526 in 322 conjugacy classes, 67 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×7], C22 [×3], C22 [×4], C22 [×30], C5, C2×C4 [×2], C2×C4 [×13], D4 [×24], C23, C23 [×2], C23 [×22], D5 [×4], C10 [×3], C10 [×4], C10 [×2], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×2], C2×D4 [×18], C24, C24 [×2], Dic5 [×4], C20 [×3], D10 [×20], C2×C10 [×3], C2×C10 [×4], C2×C10 [×10], C2.C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C22×D4 [×3], D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×2], C2×C20 [×5], C22×D5 [×4], C22×D5 [×12], C22×C10, C22×C10 [×2], C22×C10 [×6], C23⋊2D4, D10⋊C4 [×4], C5×C22⋊C4 [×2], C2×D20 [×6], C22×Dic5 [×2], C2×C5⋊D4 [×12], C22×C20 [×2], C23×D5 [×2], C23×C10, C10.10C42, C2×D10⋊C4 [×2], C10×C22⋊C4, C22×D20, C22×C5⋊D4 [×2], C23⋊2D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, D5, C2×D4 [×6], C4○D4, D10 [×3], C22≀C2 [×3], C4⋊D4 [×3], C4⋊1D4, D20 [×2], C5⋊D4 [×2], C22×D5, C23⋊2D4, C2×D20, C4○D20, D4×D5 [×4], C2×C5⋊D4, C22⋊D20 [×2], D10⋊D4 [×2], C20⋊7D4, C23⋊D10, C20⋊D4, C23⋊2D20
►On 160 points
Generators in S160
(1 58)(2 77)(3 60)(4 79)(5 42)(6 61)(7 44)(8 63)(9 46)(10 65)(11 48)(12 67)(13 50)(14 69)(15 52)(16 71)(17 54)(18 73)(19 56)(20 75)(21 152)(22 139)(23 154)(24 121)(25 156)(26 123)(27 158)(28 125)(29 160)(30 127)(31 142)(32 129)(33 144)(34 131)(35 146)(36 133)(37 148)(38 135)(39 150)(40 137)(41 108)(43 110)(45 112)(47 114)(49 116)(51 118)(53 120)(55 102)(57 104)(59 106)(62 111)(64 113)(66 115)(68 117)(70 119)(72 101)(74 103)(76 105)(78 107)(80 109)(81 153)(82 140)(83 155)(84 122)(85 157)(86 124)(87 159)(88 126)(89 141)(90 128)(91 143)(92 130)(93 145)(94 132)(95 147)(96 134)(97 149)(98 136)(99 151)(100 138)
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 121)(120 122)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 101)(18 102)(19 103)(20 104)(21 100)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 99)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(121 155)(122 156)(123 157)(124 158)(125 159)(126 160)(127 141)(128 142)(129 143)(130 144)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 151)(138 152)(139 153)(140 154)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 64)(22 63)(23 62)(24 61)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 85)(42 84)(43 83)(44 82)(45 81)(46 100)(47 99)(48 98)(49 97)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(133 140)(134 139)(135 138)(136 137)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)
G:=sub<Sym(160)| (1,58)(2,77)(3,60)(4,79)(5,42)(6,61)(7,44)(8,63)(9,46)(10,65)(11,48)(12,67)(13,50)(14,69)(15,52)(16,71)(17,54)(18,73)(19,56)(20,75)(21,152)(22,139)(23,154)(24,121)(25,156)(26,123)(27,158)(28,125)(29,160)(30,127)(31,142)(32,129)(33,144)(34,131)(35,146)(36,133)(37,148)(38,135)(39,150)(40,137)(41,108)(43,110)(45,112)(47,114)(49,116)(51,118)(53,120)(55,102)(57,104)(59,106)(62,111)(64,113)(66,115)(68,117)(70,119)(72,101)(74,103)(76,105)(78,107)(80,109)(81,153)(82,140)(83,155)(84,122)(85,157)(86,124)(87,159)(88,126)(89,141)(90,128)(91,143)(92,130)(93,145)(94,132)(95,147)(96,134)(97,149)(98,136)(99,151)(100,138), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,121)(120,122), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,101)(18,102)(19,103)(20,104)(21,100)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,64)(22,63)(23,62)(24,61)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,85)(42,84)(43,83)(44,82)(45,81)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(101,108)(102,107)(103,106)(104,105)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,140)(134,139)(135,138)(136,137)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)>;
G:=Group( (1,58)(2,77)(3,60)(4,79)(5,42)(6,61)(7,44)(8,63)(9,46)(10,65)(11,48)(12,67)(13,50)(14,69)(15,52)(16,71)(17,54)(18,73)(19,56)(20,75)(21,152)(22,139)(23,154)(24,121)(25,156)(26,123)(27,158)(28,125)(29,160)(30,127)(31,142)(32,129)(33,144)(34,131)(35,146)(36,133)(37,148)(38,135)(39,150)(40,137)(41,108)(43,110)(45,112)(47,114)(49,116)(51,118)(53,120)(55,102)(57,104)(59,106)(62,111)(64,113)(66,115)(68,117)(70,119)(72,101)(74,103)(76,105)(78,107)(80,109)(81,153)(82,140)(83,155)(84,122)(85,157)(86,124)(87,159)(88,126)(89,141)(90,128)(91,143)(92,130)(93,145)(94,132)(95,147)(96,134)(97,149)(98,136)(99,151)(100,138), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,121)(120,122), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,101)(18,102)(19,103)(20,104)(21,100)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,64)(22,63)(23,62)(24,61)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,85)(42,84)(43,83)(44,82)(45,81)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(101,108)(102,107)(103,106)(104,105)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,140)(134,139)(135,138)(136,137)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151) );
G=PermutationGroup([(1,58),(2,77),(3,60),(4,79),(5,42),(6,61),(7,44),(8,63),(9,46),(10,65),(11,48),(12,67),(13,50),(14,69),(15,52),(16,71),(17,54),(18,73),(19,56),(20,75),(21,152),(22,139),(23,154),(24,121),(25,156),(26,123),(27,158),(28,125),(29,160),(30,127),(31,142),(32,129),(33,144),(34,131),(35,146),(36,133),(37,148),(38,135),(39,150),(40,137),(41,108),(43,110),(45,112),(47,114),(49,116),(51,118),(53,120),(55,102),(57,104),(59,106),(62,111),(64,113),(66,115),(68,117),(70,119),(72,101),(74,103),(76,105),(78,107),(80,109),(81,153),(82,140),(83,155),(84,122),(85,157),(86,124),(87,159),(88,126),(89,141),(90,128),(91,143),(92,130),(93,145),(94,132),(95,147),(96,134),(97,149),(98,136),(99,151),(100,138)], [(1,141),(2,142),(3,143),(4,144),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,92),(42,93),(43,94),(44,95),(45,96),(46,97),(47,98),(48,99),(49,100),(50,81),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(57,88),(58,89),(59,90),(60,91),(101,123),(102,124),(103,125),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134),(113,135),(114,136),(115,137),(116,138),(117,139),(118,140),(119,121),(120,122)], [(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,101),(18,102),(19,103),(20,104),(21,100),(22,81),(23,82),(24,83),(25,84),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(37,96),(38,97),(39,98),(40,99),(41,79),(42,80),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(121,155),(122,156),(123,157),(124,158),(125,159),(126,160),(127,141),(128,142),(129,143),(130,144),(131,145),(132,146),(133,147),(134,148),(135,149),(136,150),(137,151),(138,152),(139,153),(140,154)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,64),(22,63),(23,62),(24,61),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65),(41,85),(42,84),(43,83),(44,82),(45,81),(46,100),(47,99),(48,98),(49,97),(50,96),(51,95),(52,94),(53,93),(54,92),(55,91),(56,90),(57,89),(58,88),(59,87),(60,86),(101,108),(102,107),(103,106),(104,105),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115),(121,132),(122,131),(123,130),(124,129),(125,128),(126,127),(133,140),(134,139),(135,138),(136,137),(141,160),(142,159),(143,158),(144,157),(145,156),(146,155),(147,154),(148,153),(149,152),(150,151)])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | C4○D20 | D4×D5 |
| kernel | C23⋊2D20 | C10.10C42 | C2×D10⋊C4 | C10×C22⋊C4 | C22×D20 | C22×C5⋊D4 | C2×Dic5 | C2×C20 | C22×D5 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 8 | 8 |
Matrix representation of C23⋊2D20 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 0 | 0 | 0 | 0 | 40 | 17 |
| 40 | 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 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 | 2 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 14 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 32 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,24,40,0,0,0,0,1,17],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,2,40,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,40,0,0,0,0,0,0,9,30,0,0,0,0,11,32] >;
C23⋊2D20 in GAP, Magma, Sage, TeX
C_2^3\rtimes_2D_{20} % in TeX
G:=Group("C2^3:2D20"); // GroupNames label
G:=SmallGroup(320,587);
// by ID
G=gap.SmallGroup(320,587);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations