metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.47D20, M4(2)⋊1Dic5, C40⋊13(C2×C4), C40⋊6C4⋊3C2, C8⋊1(C2×Dic5), C40⋊5C4⋊17C2, (C2×C8).75D10, C20.52(C4⋊C4), C20.76(C2×Q8), (C2×C20).26Q8, (C2×C20).167D4, (C2×C4).149D20, (C5×M4(2))⋊7C4, C4.6(C4⋊Dic5), C2.3(C8⋊D10), (C2×C40).61C22, C4.42(C2×Dic10), (C2×C4).15Dic10, C22.56(C2×D20), (C2×M4(2)).1D5, C10.19(C8⋊C22), C5⋊6(M4(2)⋊C4), (C2×C20).772C23, C20.232(C22×C4), C2.4(C8.D10), (C22×C10).100D4, (C22×C4).133D10, (C10×M4(2)).1C2, C22.6(C4⋊Dic5), C4.27(C22×Dic5), C10.20(C8.C22), C4⋊Dic5.284C22, (C22×C20).180C22, C23.21D10.17C2, C10.73(C2×C4⋊C4), C2.14(C2×C4⋊Dic5), (C2×C10).44(C4⋊C4), (C2×C20).273(C2×C4), (C2×C10).162(C2×D4), (C2×C4⋊Dic5).40C2, (C2×C4).21(C2×Dic5), (C2×C4).720(C22×D5), SmallGroup(320,748)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.47D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d19 >
Subgroups: 382 in 118 conjugacy classes, 71 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C40, C2×Dic5, C2×C20, C2×C20, C22×C10, M4(2)⋊C4, C4×Dic5, C4⋊Dic5, C4⋊Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C22×Dic5, C22×C20, C40⋊6C4, C40⋊5C4, C2×C4⋊Dic5, C23.21D10, C10×M4(2), C23.47D20
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, D10, C2×C4⋊C4, C8⋊C22, C8.C22, Dic10, D20, C2×Dic5, C22×D5, M4(2)⋊C4, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C8⋊D10, C8.D10, C2×C4⋊Dic5, C23.47D20
►On 160 points
Generators in S160
(1 109)(2 90)(3 111)(4 92)(5 113)(6 94)(7 115)(8 96)(9 117)(10 98)(11 119)(12 100)(13 81)(14 102)(15 83)(16 104)(17 85)(18 106)(19 87)(20 108)(21 89)(22 110)(23 91)(24 112)(25 93)(26 114)(27 95)(28 116)(29 97)(30 118)(31 99)(32 120)(33 101)(34 82)(35 103)(36 84)(37 105)(38 86)(39 107)(40 88)(41 149)(42 130)(43 151)(44 132)(45 153)(46 134)(47 155)(48 136)(49 157)(50 138)(51 159)(52 140)(53 121)(54 142)(55 123)(56 144)(57 125)(58 146)(59 127)(60 148)(61 129)(62 150)(63 131)(64 152)(65 133)(66 154)(67 135)(68 156)(69 137)(70 158)(71 139)(72 160)(73 141)(74 122)(75 143)(76 124)(77 145)(78 126)(79 147)(80 128)
(1 109)(2 110)(3 111)(4 112)(5 113)(6 114)(7 115)(8 116)(9 117)(10 118)(11 119)(12 120)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(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)
(1 129 89 61)(2 148 90 80)(3 127 91 59)(4 146 92 78)(5 125 93 57)(6 144 94 76)(7 123 95 55)(8 142 96 74)(9 121 97 53)(10 140 98 72)(11 159 99 51)(12 138 100 70)(13 157 101 49)(14 136 102 68)(15 155 103 47)(16 134 104 66)(17 153 105 45)(18 132 106 64)(19 151 107 43)(20 130 108 62)(21 149 109 41)(22 128 110 60)(23 147 111 79)(24 126 112 58)(25 145 113 77)(26 124 114 56)(27 143 115 75)(28 122 116 54)(29 141 117 73)(30 160 118 52)(31 139 119 71)(32 158 120 50)(33 137 81 69)(34 156 82 48)(35 135 83 67)(36 154 84 46)(37 133 85 65)(38 152 86 44)(39 131 87 63)(40 150 88 42)
G:=sub<Sym(160)| (1,109)(2,90)(3,111)(4,92)(5,113)(6,94)(7,115)(8,96)(9,117)(10,98)(11,119)(12,100)(13,81)(14,102)(15,83)(16,104)(17,85)(18,106)(19,87)(20,108)(21,89)(22,110)(23,91)(24,112)(25,93)(26,114)(27,95)(28,116)(29,97)(30,118)(31,99)(32,120)(33,101)(34,82)(35,103)(36,84)(37,105)(38,86)(39,107)(40,88)(41,149)(42,130)(43,151)(44,132)(45,153)(46,134)(47,155)(48,136)(49,157)(50,138)(51,159)(52,140)(53,121)(54,142)(55,123)(56,144)(57,125)(58,146)(59,127)(60,148)(61,129)(62,150)(63,131)(64,152)(65,133)(66,154)(67,135)(68,156)(69,137)(70,158)(71,139)(72,160)(73,141)(74,122)(75,143)(76,124)(77,145)(78,126)(79,147)(80,128), (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (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), (1,129,89,61)(2,148,90,80)(3,127,91,59)(4,146,92,78)(5,125,93,57)(6,144,94,76)(7,123,95,55)(8,142,96,74)(9,121,97,53)(10,140,98,72)(11,159,99,51)(12,138,100,70)(13,157,101,49)(14,136,102,68)(15,155,103,47)(16,134,104,66)(17,153,105,45)(18,132,106,64)(19,151,107,43)(20,130,108,62)(21,149,109,41)(22,128,110,60)(23,147,111,79)(24,126,112,58)(25,145,113,77)(26,124,114,56)(27,143,115,75)(28,122,116,54)(29,141,117,73)(30,160,118,52)(31,139,119,71)(32,158,120,50)(33,137,81,69)(34,156,82,48)(35,135,83,67)(36,154,84,46)(37,133,85,65)(38,152,86,44)(39,131,87,63)(40,150,88,42)>;
G:=Group( (1,109)(2,90)(3,111)(4,92)(5,113)(6,94)(7,115)(8,96)(9,117)(10,98)(11,119)(12,100)(13,81)(14,102)(15,83)(16,104)(17,85)(18,106)(19,87)(20,108)(21,89)(22,110)(23,91)(24,112)(25,93)(26,114)(27,95)(28,116)(29,97)(30,118)(31,99)(32,120)(33,101)(34,82)(35,103)(36,84)(37,105)(38,86)(39,107)(40,88)(41,149)(42,130)(43,151)(44,132)(45,153)(46,134)(47,155)(48,136)(49,157)(50,138)(51,159)(52,140)(53,121)(54,142)(55,123)(56,144)(57,125)(58,146)(59,127)(60,148)(61,129)(62,150)(63,131)(64,152)(65,133)(66,154)(67,135)(68,156)(69,137)(70,158)(71,139)(72,160)(73,141)(74,122)(75,143)(76,124)(77,145)(78,126)(79,147)(80,128), (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (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), (1,129,89,61)(2,148,90,80)(3,127,91,59)(4,146,92,78)(5,125,93,57)(6,144,94,76)(7,123,95,55)(8,142,96,74)(9,121,97,53)(10,140,98,72)(11,159,99,51)(12,138,100,70)(13,157,101,49)(14,136,102,68)(15,155,103,47)(16,134,104,66)(17,153,105,45)(18,132,106,64)(19,151,107,43)(20,130,108,62)(21,149,109,41)(22,128,110,60)(23,147,111,79)(24,126,112,58)(25,145,113,77)(26,124,114,56)(27,143,115,75)(28,122,116,54)(29,141,117,73)(30,160,118,52)(31,139,119,71)(32,158,120,50)(33,137,81,69)(34,156,82,48)(35,135,83,67)(36,154,84,46)(37,133,85,65)(38,152,86,44)(39,131,87,63)(40,150,88,42) );
G=PermutationGroup([[(1,109),(2,90),(3,111),(4,92),(5,113),(6,94),(7,115),(8,96),(9,117),(10,98),(11,119),(12,100),(13,81),(14,102),(15,83),(16,104),(17,85),(18,106),(19,87),(20,108),(21,89),(22,110),(23,91),(24,112),(25,93),(26,114),(27,95),(28,116),(29,97),(30,118),(31,99),(32,120),(33,101),(34,82),(35,103),(36,84),(37,105),(38,86),(39,107),(40,88),(41,149),(42,130),(43,151),(44,132),(45,153),(46,134),(47,155),(48,136),(49,157),(50,138),(51,159),(52,140),(53,121),(54,142),(55,123),(56,144),(57,125),(58,146),(59,127),(60,148),(61,129),(62,150),(63,131),(64,152),(65,133),(66,154),(67,135),(68,156),(69,137),(70,158),(71,139),(72,160),(73,141),(74,122),(75,143),(76,124),(77,145),(78,126),(79,147),(80,128)], [(1,109),(2,110),(3,111),(4,112),(5,113),(6,114),(7,115),(8,116),(9,117),(10,118),(11,119),(12,120),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(64,152),(65,153),(66,154),(67,155),(68,156),(69,157),(70,158),(71,159),(72,160),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(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)], [(1,129,89,61),(2,148,90,80),(3,127,91,59),(4,146,92,78),(5,125,93,57),(6,144,94,76),(7,123,95,55),(8,142,96,74),(9,121,97,53),(10,140,98,72),(11,159,99,51),(12,138,100,70),(13,157,101,49),(14,136,102,68),(15,155,103,47),(16,134,104,66),(17,153,105,45),(18,132,106,64),(19,151,107,43),(20,130,108,62),(21,149,109,41),(22,128,110,60),(23,147,111,79),(24,126,112,58),(25,145,113,77),(26,124,114,56),(27,143,115,75),(28,122,116,54),(29,141,117,73),(30,160,118,52),(31,139,119,71),(32,158,120,50),(33,137,81,69),(34,156,82,48),(35,135,83,67),(36,154,84,46),(37,133,85,65),(38,152,86,44),(39,131,87,63),(40,150,88,42)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D5 | D10 | Dic5 | D10 | Dic10 | D20 | D20 | C8⋊C22 | C8.C22 | C8⋊D10 | C8.D10 |
| kernel | C23.47D20 | C40⋊6C4 | C40⋊5C4 | C2×C4⋊Dic5 | C23.21D10 | C10×M4(2) | C5×M4(2) | C2×C20 | C2×C20 | C22×C10 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C23 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 8 | 2 | 8 | 4 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of C23.47D20 ►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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 36 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 32 | 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 6 |
| 0 | 0 | 0 | 0 | 21 | 18 |
| 0 | 0 | 18 | 35 | 0 | 0 |
| 0 | 0 | 20 | 23 | 0 | 0 |
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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,36,0,0,0,0,40,6,0,0,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,1,0,0,0,0,0,0,1,0,0],[32,0,0,0,0,0,9,9,0,0,0,0,0,0,0,0,18,20,0,0,0,0,35,23,0,0,23,21,0,0,0,0,6,18,0,0] >;
C23.47D20 in GAP, Magma, Sage, TeX
C_2^3._{47}D_{20} % in TeX
G:=Group("C2^3.47D20"); // GroupNames label
G:=SmallGroup(320,748);
// by ID
G=gap.SmallGroup(320,748);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^19>;
// generators/relations