metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.49D20, C4○D20⋊12C4, D20⋊28(C2×C4), C20.419(C2×D4), (C2×C8).190D10, (C2×C20).175D4, (C2×C4).154D20, D20⋊5C4⋊40C2, Dic10⋊26(C2×C4), C2.5(C8⋊D10), (C2×M4(2))⋊13D5, C22.58(C2×D20), C20.44D4⋊40C2, C20.75(C22⋊C4), C10.21(C8⋊C22), (C10×M4(2))⋊21C2, (C2×C40).320C22, (C2×C20).774C23, C20.175(C22×C4), C2.5(C8.D10), (C22×C4).141D10, (C22×C10).102D4, C5⋊5(C23.36D4), C4.39(D10⋊C4), (C2×D20).207C22, C10.21(C8.C22), C4⋊Dic5.285C22, C22.3(D10⋊C4), (C22×C20).190C22, (C2×Dic10).227C22, C4.74(C2×C4×D5), (C2×C4).54(C4×D5), (C2×C4⋊Dic5)⋊33C2, C4.112(C2×C5⋊D4), (C2×C20).283(C2×C4), (C2×C4○D20).13C2, (C2×C10).164(C2×D4), (C2×C4).78(C5⋊D4), C2.32(C2×D10⋊C4), C10.101(C2×C22⋊C4), (C2×C4).723(C22×D5), (C2×C10).86(C22⋊C4), SmallGroup(320,760)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.49D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, dad-1=ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >
Subgroups: 670 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.36D4, C4⋊Dic5, C4⋊Dic5, C2×C40, C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, C20.44D4, D20⋊5C4, C2×C4⋊Dic5, C10×M4(2), C2×C4○D20, C23.49D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8⋊C22, C8.C22, C4×D5, D20, C5⋊D4, C22×D5, C23.36D4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C8⋊D10, C8.D10, C2×D10⋊C4, C23.49D20
►On 160 points
Generators in S160
(1 123)(2 144)(3 125)(4 146)(5 127)(6 148)(7 129)(8 150)(9 131)(10 152)(11 133)(12 154)(13 135)(14 156)(15 137)(16 158)(17 139)(18 160)(19 141)(20 122)(21 143)(22 124)(23 145)(24 126)(25 147)(26 128)(27 149)(28 130)(29 151)(30 132)(31 153)(32 134)(33 155)(34 136)(35 157)(36 138)(37 159)(38 140)(39 121)(40 142)(41 89)(42 110)(43 91)(44 112)(45 93)(46 114)(47 95)(48 116)(49 97)(50 118)(51 99)(52 120)(53 101)(54 82)(55 103)(56 84)(57 105)(58 86)(59 107)(60 88)(61 109)(62 90)(63 111)(64 92)(65 113)(66 94)(67 115)(68 96)(69 117)(70 98)(71 119)(72 100)(73 81)(74 102)(75 83)(76 104)(77 85)(78 106)(79 87)(80 108)
(1 108)(2 109)(3 110)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 118)(12 119)(13 120)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 131)(49 132)(50 133)(51 134)(52 135)(53 136)(54 137)(55 138)(56 139)(57 140)(58 141)(59 142)(60 143)(61 144)(62 145)(63 146)(64 147)(65 148)(66 149)(67 150)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)(76 159)(77 160)(78 121)(79 122)(80 123)
(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 87 88 40)(2 39 89 86)(3 85 90 38)(4 37 91 84)(5 83 92 36)(6 35 93 82)(7 81 94 34)(8 33 95 120)(9 119 96 32)(10 31 97 118)(11 117 98 30)(12 29 99 116)(13 115 100 28)(14 27 101 114)(15 113 102 26)(16 25 103 112)(17 111 104 24)(18 23 105 110)(19 109 106 22)(20 21 107 108)(41 58 144 121)(42 160 145 57)(43 56 146 159)(44 158 147 55)(45 54 148 157)(46 156 149 53)(47 52 150 155)(48 154 151 51)(49 50 152 153)(59 80 122 143)(60 142 123 79)(61 78 124 141)(62 140 125 77)(63 76 126 139)(64 138 127 75)(65 74 128 137)(66 136 129 73)(67 72 130 135)(68 134 131 71)(69 70 132 133)
G:=sub<Sym(160)| (1,123)(2,144)(3,125)(4,146)(5,127)(6,148)(7,129)(8,150)(9,131)(10,152)(11,133)(12,154)(13,135)(14,156)(15,137)(16,158)(17,139)(18,160)(19,141)(20,122)(21,143)(22,124)(23,145)(24,126)(25,147)(26,128)(27,149)(28,130)(29,151)(30,132)(31,153)(32,134)(33,155)(34,136)(35,157)(36,138)(37,159)(38,140)(39,121)(40,142)(41,89)(42,110)(43,91)(44,112)(45,93)(46,114)(47,95)(48,116)(49,97)(50,118)(51,99)(52,120)(53,101)(54,82)(55,103)(56,84)(57,105)(58,86)(59,107)(60,88)(61,109)(62,90)(63,111)(64,92)(65,113)(66,94)(67,115)(68,96)(69,117)(70,98)(71,119)(72,100)(73,81)(74,102)(75,83)(76,104)(77,85)(78,106)(79,87)(80,108), (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,132)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,140)(58,141)(59,142)(60,143)(61,144)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,121)(79,122)(80,123), (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), (1,87,88,40)(2,39,89,86)(3,85,90,38)(4,37,91,84)(5,83,92,36)(6,35,93,82)(7,81,94,34)(8,33,95,120)(9,119,96,32)(10,31,97,118)(11,117,98,30)(12,29,99,116)(13,115,100,28)(14,27,101,114)(15,113,102,26)(16,25,103,112)(17,111,104,24)(18,23,105,110)(19,109,106,22)(20,21,107,108)(41,58,144,121)(42,160,145,57)(43,56,146,159)(44,158,147,55)(45,54,148,157)(46,156,149,53)(47,52,150,155)(48,154,151,51)(49,50,152,153)(59,80,122,143)(60,142,123,79)(61,78,124,141)(62,140,125,77)(63,76,126,139)(64,138,127,75)(65,74,128,137)(66,136,129,73)(67,72,130,135)(68,134,131,71)(69,70,132,133)>;
G:=Group( (1,123)(2,144)(3,125)(4,146)(5,127)(6,148)(7,129)(8,150)(9,131)(10,152)(11,133)(12,154)(13,135)(14,156)(15,137)(16,158)(17,139)(18,160)(19,141)(20,122)(21,143)(22,124)(23,145)(24,126)(25,147)(26,128)(27,149)(28,130)(29,151)(30,132)(31,153)(32,134)(33,155)(34,136)(35,157)(36,138)(37,159)(38,140)(39,121)(40,142)(41,89)(42,110)(43,91)(44,112)(45,93)(46,114)(47,95)(48,116)(49,97)(50,118)(51,99)(52,120)(53,101)(54,82)(55,103)(56,84)(57,105)(58,86)(59,107)(60,88)(61,109)(62,90)(63,111)(64,92)(65,113)(66,94)(67,115)(68,96)(69,117)(70,98)(71,119)(72,100)(73,81)(74,102)(75,83)(76,104)(77,85)(78,106)(79,87)(80,108), (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,132)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,140)(58,141)(59,142)(60,143)(61,144)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,121)(79,122)(80,123), (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), (1,87,88,40)(2,39,89,86)(3,85,90,38)(4,37,91,84)(5,83,92,36)(6,35,93,82)(7,81,94,34)(8,33,95,120)(9,119,96,32)(10,31,97,118)(11,117,98,30)(12,29,99,116)(13,115,100,28)(14,27,101,114)(15,113,102,26)(16,25,103,112)(17,111,104,24)(18,23,105,110)(19,109,106,22)(20,21,107,108)(41,58,144,121)(42,160,145,57)(43,56,146,159)(44,158,147,55)(45,54,148,157)(46,156,149,53)(47,52,150,155)(48,154,151,51)(49,50,152,153)(59,80,122,143)(60,142,123,79)(61,78,124,141)(62,140,125,77)(63,76,126,139)(64,138,127,75)(65,74,128,137)(66,136,129,73)(67,72,130,135)(68,134,131,71)(69,70,132,133) );
G=PermutationGroup([[(1,123),(2,144),(3,125),(4,146),(5,127),(6,148),(7,129),(8,150),(9,131),(10,152),(11,133),(12,154),(13,135),(14,156),(15,137),(16,158),(17,139),(18,160),(19,141),(20,122),(21,143),(22,124),(23,145),(24,126),(25,147),(26,128),(27,149),(28,130),(29,151),(30,132),(31,153),(32,134),(33,155),(34,136),(35,157),(36,138),(37,159),(38,140),(39,121),(40,142),(41,89),(42,110),(43,91),(44,112),(45,93),(46,114),(47,95),(48,116),(49,97),(50,118),(51,99),(52,120),(53,101),(54,82),(55,103),(56,84),(57,105),(58,86),(59,107),(60,88),(61,109),(62,90),(63,111),(64,92),(65,113),(66,94),(67,115),(68,96),(69,117),(70,98),(71,119),(72,100),(73,81),(74,102),(75,83),(76,104),(77,85),(78,106),(79,87),(80,108)], [(1,108),(2,109),(3,110),(4,111),(5,112),(6,113),(7,114),(8,115),(9,116),(10,117),(11,118),(12,119),(13,120),(14,81),(15,82),(16,83),(17,84),(18,85),(19,86),(20,87),(21,88),(22,89),(23,90),(24,91),(25,92),(26,93),(27,94),(28,95),(29,96),(30,97),(31,98),(32,99),(33,100),(34,101),(35,102),(36,103),(37,104),(38,105),(39,106),(40,107),(41,124),(42,125),(43,126),(44,127),(45,128),(46,129),(47,130),(48,131),(49,132),(50,133),(51,134),(52,135),(53,136),(54,137),(55,138),(56,139),(57,140),(58,141),(59,142),(60,143),(61,144),(62,145),(63,146),(64,147),(65,148),(66,149),(67,150),(68,151),(69,152),(70,153),(71,154),(72,155),(73,156),(74,157),(75,158),(76,159),(77,160),(78,121),(79,122),(80,123)], [(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)], [(1,87,88,40),(2,39,89,86),(3,85,90,38),(4,37,91,84),(5,83,92,36),(6,35,93,82),(7,81,94,34),(8,33,95,120),(9,119,96,32),(10,31,97,118),(11,117,98,30),(12,29,99,116),(13,115,100,28),(14,27,101,114),(15,113,102,26),(16,25,103,112),(17,111,104,24),(18,23,105,110),(19,109,106,22),(20,21,107,108),(41,58,144,121),(42,160,145,57),(43,56,146,159),(44,158,147,55),(45,54,148,157),(46,156,149,53),(47,52,150,155),(48,154,151,51),(49,50,152,153),(59,80,122,143),(60,142,123,79),(61,78,124,141),(62,140,125,77),(63,76,126,139),(64,138,127,75),(65,74,128,137),(66,136,129,73),(67,72,130,135),(68,134,131,71),(69,70,132,133)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 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 | 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 | 20 | 20 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D20 | C8⋊C22 | C8.C22 | C8⋊D10 | C8.D10 |
| kernel | C23.49D20 | C20.44D4 | D20⋊5C4 | C2×C4⋊Dic5 | C10×M4(2) | C2×C4○D20 | C4○D20 | C2×C20 | C22×C10 | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 2 | 8 | 4 | 8 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of C23.49D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 28 | 23 | 34 |
| 0 | 0 | 13 | 9 | 7 | 15 |
| 0 | 0 | 34 | 1 | 23 | 13 |
| 0 | 0 | 40 | 0 | 28 | 32 |
| 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 33 | 14 | 39 |
| 0 | 0 | 8 | 1 | 2 | 0 |
| 0 | 0 | 7 | 26 | 25 | 8 |
| 0 | 0 | 15 | 25 | 33 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 22 | 39 | 14 |
| 0 | 0 | 40 | 15 | 0 | 2 |
| 0 | 0 | 19 | 31 | 8 | 25 |
| 0 | 0 | 0 | 22 | 40 | 33 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,13,34,40,0,0,28,9,1,0,0,0,23,7,23,28,0,0,34,15,13,32],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,8,7,15,0,0,33,1,26,25,0,0,14,2,25,33,0,0,39,0,8,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,26,40,19,0,0,0,22,15,31,22,0,0,39,0,8,40,0,0,14,2,25,33] >;
C23.49D20 in GAP, Magma, Sage, TeX
C_2^3._{49}D_{20} % in TeX
G:=Group("C2^3.49D20"); // GroupNames label
G:=SmallGroup(320,760);
// by ID
G=gap.SmallGroup(320,760);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,387,142,1123,136,12550]);
// 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=a*c=c*a,a*e=e*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^19>;
// generators/relations