metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.27D20, C24.64D10, C23.16Dic10, (C23×C4).8D5, (C22×C20)⋊25C4, (C2×C20).475D4, C4⋊2(C23.D5), (C22×C4)⋊8Dic5, C20⋊10(C22⋊C4), C2.4(C20⋊7D4), (C23×C20).11C2, C22⋊2(C4⋊Dic5), C22.60(C2×D20), (C22×C10).26Q8, C10.79(C4⋊D4), C5⋊5(C23.7Q8), (C22×C4).433D10, (C22×C10).143D4, C10.68(C22⋊Q8), C23.31(C2×Dic5), C2.5(C20.48D4), C22.63(C4○D20), (C23×C10).99C22, C22.32(C2×Dic10), C23.303(C22×D5), C10.10C42⋊24C2, C10.68(C42⋊C2), (C22×C10).363C23, (C22×C20).484C22, C22.50(C22×Dic5), (C22×Dic5).66C22, C2.12(C23.21D10), C10.79(C2×C4⋊C4), (C2×C10)⋊10(C4⋊C4), (C2×C4⋊Dic5)⋊16C2, C2.16(C2×C4⋊Dic5), (C2×C10).44(C2×Q8), (C2×C20).455(C2×C4), C2.6(C2×C23.D5), (C2×C10).549(C2×D4), (C2×C4).85(C2×Dic5), C22.87(C2×C5⋊D4), (C2×C10).91(C4○D4), (C2×C4).260(C5⋊D4), C10.111(C2×C22⋊C4), (C2×C23.D5).18C2, (C2×C10).294(C22×C4), (C22×C10).204(C2×C4), SmallGroup(320,839)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.27D20
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 638 in 234 conjugacy classes, 103 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×8], C2×C4 [×22], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C24, Dic5 [×4], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C2×Dic5 [×12], C2×C20 [×8], C2×C20 [×10], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.7Q8, C4⋊Dic5 [×4], C23.D5 [×4], C22×Dic5 [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C10.10C42 [×2], C2×C4⋊Dic5 [×2], C2×C23.D5 [×2], C23×C20, C23.27D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, D5, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.7Q8, C4⋊Dic5 [×4], C23.D5 [×4], C2×Dic10, C2×D20, C4○D20 [×2], C22×Dic5, C2×C5⋊D4 [×2], C20.48D4 [×2], C2×C4⋊Dic5, C23.21D10, C20⋊7D4 [×2], C2×C23.D5, C23.27D20
►On 160 points
Generators in S160
(1 37)(2 38)(3 39)(4 40)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 157)(48 158)(49 159)(50 160)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 81)(70 82)(71 83)(72 84)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 121)(119 122)(120 123)
(1 37)(2 38)(3 39)(4 40)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 141)(97 142)(98 143)(99 144)(100 145)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 121)(119 122)(120 123)
(1 140)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 81)(57 82)(58 83)(59 84)(60 85)(61 158)(62 159)(63 160)(64 141)(65 142)(66 143)(67 144)(68 145)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 155)(79 156)(80 157)
(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)
(1 150 37 85)(2 149 38 84)(3 148 39 83)(4 147 40 82)(5 146 21 81)(6 145 22 100)(7 144 23 99)(8 143 24 98)(9 142 25 97)(10 141 26 96)(11 160 27 95)(12 159 28 94)(13 158 29 93)(14 157 30 92)(15 156 31 91)(16 155 32 90)(17 154 33 89)(18 153 34 88)(19 152 35 87)(20 151 36 86)(41 139 74 116)(42 138 75 115)(43 137 76 114)(44 136 77 113)(45 135 78 112)(46 134 79 111)(47 133 80 110)(48 132 61 109)(49 131 62 108)(50 130 63 107)(51 129 64 106)(52 128 65 105)(53 127 66 104)(54 126 67 103)(55 125 68 102)(56 124 69 101)(57 123 70 120)(58 122 71 119)(59 121 72 118)(60 140 73 117)
G:=sub<Sym(160)| (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,159)(50,160)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,121)(119,122)(120,123), (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,141)(97,142)(98,143)(99,144)(100,145)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,121)(119,122)(120,123), (1,140)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,81)(57,82)(58,83)(59,84)(60,85)(61,158)(62,159)(63,160)(64,141)(65,142)(66,143)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157), (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), (1,150,37,85)(2,149,38,84)(3,148,39,83)(4,147,40,82)(5,146,21,81)(6,145,22,100)(7,144,23,99)(8,143,24,98)(9,142,25,97)(10,141,26,96)(11,160,27,95)(12,159,28,94)(13,158,29,93)(14,157,30,92)(15,156,31,91)(16,155,32,90)(17,154,33,89)(18,153,34,88)(19,152,35,87)(20,151,36,86)(41,139,74,116)(42,138,75,115)(43,137,76,114)(44,136,77,113)(45,135,78,112)(46,134,79,111)(47,133,80,110)(48,132,61,109)(49,131,62,108)(50,130,63,107)(51,129,64,106)(52,128,65,105)(53,127,66,104)(54,126,67,103)(55,125,68,102)(56,124,69,101)(57,123,70,120)(58,122,71,119)(59,121,72,118)(60,140,73,117)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,159)(50,160)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,121)(119,122)(120,123), (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,141)(97,142)(98,143)(99,144)(100,145)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,121)(119,122)(120,123), (1,140)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,81)(57,82)(58,83)(59,84)(60,85)(61,158)(62,159)(63,160)(64,141)(65,142)(66,143)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157), (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), (1,150,37,85)(2,149,38,84)(3,148,39,83)(4,147,40,82)(5,146,21,81)(6,145,22,100)(7,144,23,99)(8,143,24,98)(9,142,25,97)(10,141,26,96)(11,160,27,95)(12,159,28,94)(13,158,29,93)(14,157,30,92)(15,156,31,91)(16,155,32,90)(17,154,33,89)(18,153,34,88)(19,152,35,87)(20,151,36,86)(41,139,74,116)(42,138,75,115)(43,137,76,114)(44,136,77,113)(45,135,78,112)(46,134,79,111)(47,133,80,110)(48,132,61,109)(49,131,62,108)(50,130,63,107)(51,129,64,106)(52,128,65,105)(53,127,66,104)(54,126,67,103)(55,125,68,102)(56,124,69,101)(57,123,70,120)(58,122,71,119)(59,121,72,118)(60,140,73,117) );
G=PermutationGroup([(1,37),(2,38),(3,39),(4,40),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(41,151),(42,152),(43,153),(44,154),(45,155),(46,156),(47,157),(48,158),(49,159),(50,160),(51,141),(52,142),(53,143),(54,144),(55,145),(56,146),(57,147),(58,148),(59,149),(60,150),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,81),(70,82),(71,83),(72,84),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131),(109,132),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(116,139),(117,140),(118,121),(119,122),(120,123)], [(1,37),(2,38),(3,39),(4,40),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(81,146),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,153),(89,154),(90,155),(91,156),(92,157),(93,158),(94,159),(95,160),(96,141),(97,142),(98,143),(99,144),(100,145),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131),(109,132),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(116,139),(117,140),(118,121),(119,122),(120,123)], [(1,140),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(49,94),(50,95),(51,96),(52,97),(53,98),(54,99),(55,100),(56,81),(57,82),(58,83),(59,84),(60,85),(61,158),(62,159),(63,160),(64,141),(65,142),(66,143),(67,144),(68,145),(69,146),(70,147),(71,148),(72,149),(73,150),(74,151),(75,152),(76,153),(77,154),(78,155),(79,156),(80,157)], [(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)], [(1,150,37,85),(2,149,38,84),(3,148,39,83),(4,147,40,82),(5,146,21,81),(6,145,22,100),(7,144,23,99),(8,143,24,98),(9,142,25,97),(10,141,26,96),(11,160,27,95),(12,159,28,94),(13,158,29,93),(14,157,30,92),(15,156,31,91),(16,155,32,90),(17,154,33,89),(18,153,34,88),(19,152,35,87),(20,151,36,86),(41,139,74,116),(42,138,75,115),(43,137,76,114),(44,136,77,113),(45,135,78,112),(46,134,79,111),(47,133,80,110),(48,132,61,109),(49,131,62,108),(50,130,63,107),(51,129,64,106),(52,128,65,105),(53,127,66,104),(54,126,67,103),(55,125,68,102),(56,124,69,101),(57,123,70,120),(58,122,71,119),(59,121,72,118),(60,140,73,117)])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | - | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D5 | C4○D4 | Dic5 | D10 | D10 | C5⋊D4 | Dic10 | D20 | C4○D20 |
| kernel | C23.27D20 | C10.10C42 | C2×C4⋊Dic5 | C2×C23.D5 | C23×C20 | C22×C20 | C2×C20 | C22×C10 | C22×C10 | C23×C4 | C2×C10 | C22×C4 | C22×C4 | C24 | C2×C4 | C23 | C23 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 2 | 2 | 2 | 4 | 8 | 4 | 2 | 16 | 8 | 8 | 16 |
Matrix representation of C23.27D20 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 18 | 0 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,18,0,0,0,0,0,16],[9,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,0,18,0,0,0,16,0] >;
C23.27D20 in GAP, Magma, Sage, TeX
C_2^3._{27}D_{20} % in TeX
G:=Group("C2^3.27D20"); // GroupNames label
G:=SmallGroup(320,839);
// by ID
G=gap.SmallGroup(320,839);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations