metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.42D20, C24.45D10, C23.50(C4×D5), (C22×C10).61D4, (C22×C4).23D10, C22.41(C2×D20), (C22×Dic5)⋊12C4, C23.50(C5⋊D4), C5⋊3(C23.34D4), (C23×Dic5).3C2, (C22×C20).22C22, (C23×C10).26C22, C23.276(C22×D5), C10.10C42⋊10C2, C10.44(C42⋊C2), C22.42(D4⋊2D5), (C22×C10).318C23, C2.3(C22.D20), C22.22(D10⋊C4), C10.70(C22.D4), C2.1(C23.18D10), C2.12(C23.11D10), (C22×Dic5).206C22, (C2×C22⋊C4).5D5, C22.122(C2×C4×D5), C2.7(C2×D10⋊C4), (C2×C10).149(C2×D4), (C10×C22⋊C4).6C2, C10.75(C2×C22⋊C4), C22.46(C2×C5⋊D4), (C2×C23.D5).5C2, (C2×C10).139(C4○D4), (C2×C10).77(C22⋊C4), (C22×C10).114(C2×C4), (C2×C10).205(C22×C4), (C2×Dic5).148(C2×C4), SmallGroup(320,570)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.42D20
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 686 in 218 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×28], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22×C4 [×2], C22×C4 [×12], C24, Dic5 [×6], C20 [×2], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×4], C2×C22⋊C4, C2×C22⋊C4, C23×C4, C2×Dic5 [×4], C2×Dic5 [×18], C2×C20 [×6], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.34D4, C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×8], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42 [×4], C2×C23.D5, C10×C22⋊C4, C23×Dic5, C23.42D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], D10 [×3], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.34D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, D4⋊2D5 [×4], C2×C5⋊D4, C23.11D10 [×2], C22.D20 [×2], C2×D10⋊C4, C23.18D10 [×2], C23.42D20
►On 160 points
Generators in S160
(1 27)(2 78)(3 29)(4 80)(5 31)(6 62)(7 33)(8 64)(9 35)(10 66)(11 37)(12 68)(13 39)(14 70)(15 21)(16 72)(17 23)(18 74)(19 25)(20 76)(22 97)(24 99)(26 81)(28 83)(30 85)(32 87)(34 89)(36 91)(38 93)(40 95)(41 135)(42 151)(43 137)(44 153)(45 139)(46 155)(47 121)(48 157)(49 123)(50 159)(51 125)(52 141)(53 127)(54 143)(55 129)(56 145)(57 131)(58 147)(59 133)(60 149)(61 86)(63 88)(65 90)(67 92)(69 94)(71 96)(73 98)(75 100)(77 82)(79 84)(101 142)(102 128)(103 144)(104 130)(105 146)(106 132)(107 148)(108 134)(109 150)(110 136)(111 152)(112 138)(113 154)(114 140)(115 156)(116 122)(117 158)(118 124)(119 160)(120 126)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 21)(16 22)(17 23)(18 24)(19 25)(20 26)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 81)(77 82)(78 83)(79 84)(80 85)(101 127)(102 128)(103 129)(104 130)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 81)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)
(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)
(1 142 77 101)(2 52 78 126)(3 160 79 119)(4 50 80 124)(5 158 61 117)(6 48 62 122)(7 156 63 115)(8 46 64 140)(9 154 65 113)(10 44 66 138)(11 152 67 111)(12 42 68 136)(13 150 69 109)(14 60 70 134)(15 148 71 107)(16 58 72 132)(17 146 73 105)(18 56 74 130)(19 144 75 103)(20 54 76 128)(21 59 96 133)(22 147 97 106)(23 57 98 131)(24 145 99 104)(25 55 100 129)(26 143 81 102)(27 53 82 127)(28 141 83 120)(29 51 84 125)(30 159 85 118)(31 49 86 123)(32 157 87 116)(33 47 88 121)(34 155 89 114)(35 45 90 139)(36 153 91 112)(37 43 92 137)(38 151 93 110)(39 41 94 135)(40 149 95 108)
G:=sub<Sym(160)| (1,27)(2,78)(3,29)(4,80)(5,31)(6,62)(7,33)(8,64)(9,35)(10,66)(11,37)(12,68)(13,39)(14,70)(15,21)(16,72)(17,23)(18,74)(19,25)(20,76)(22,97)(24,99)(26,81)(28,83)(30,85)(32,87)(34,89)(36,91)(38,93)(40,95)(41,135)(42,151)(43,137)(44,153)(45,139)(46,155)(47,121)(48,157)(49,123)(50,159)(51,125)(52,141)(53,127)(54,143)(55,129)(56,145)(57,131)(58,147)(59,133)(60,149)(61,86)(63,88)(65,90)(67,92)(69,94)(71,96)(73,98)(75,100)(77,82)(79,84)(101,142)(102,128)(103,144)(104,130)(105,146)(106,132)(107,148)(108,134)(109,150)(110,136)(111,152)(112,138)(113,154)(114,140)(115,156)(116,122)(117,158)(118,124)(119,160)(120,126), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,81)(77,82)(78,83)(79,84)(80,85)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,81)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155), (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), (1,142,77,101)(2,52,78,126)(3,160,79,119)(4,50,80,124)(5,158,61,117)(6,48,62,122)(7,156,63,115)(8,46,64,140)(9,154,65,113)(10,44,66,138)(11,152,67,111)(12,42,68,136)(13,150,69,109)(14,60,70,134)(15,148,71,107)(16,58,72,132)(17,146,73,105)(18,56,74,130)(19,144,75,103)(20,54,76,128)(21,59,96,133)(22,147,97,106)(23,57,98,131)(24,145,99,104)(25,55,100,129)(26,143,81,102)(27,53,82,127)(28,141,83,120)(29,51,84,125)(30,159,85,118)(31,49,86,123)(32,157,87,116)(33,47,88,121)(34,155,89,114)(35,45,90,139)(36,153,91,112)(37,43,92,137)(38,151,93,110)(39,41,94,135)(40,149,95,108)>;
G:=Group( (1,27)(2,78)(3,29)(4,80)(5,31)(6,62)(7,33)(8,64)(9,35)(10,66)(11,37)(12,68)(13,39)(14,70)(15,21)(16,72)(17,23)(18,74)(19,25)(20,76)(22,97)(24,99)(26,81)(28,83)(30,85)(32,87)(34,89)(36,91)(38,93)(40,95)(41,135)(42,151)(43,137)(44,153)(45,139)(46,155)(47,121)(48,157)(49,123)(50,159)(51,125)(52,141)(53,127)(54,143)(55,129)(56,145)(57,131)(58,147)(59,133)(60,149)(61,86)(63,88)(65,90)(67,92)(69,94)(71,96)(73,98)(75,100)(77,82)(79,84)(101,142)(102,128)(103,144)(104,130)(105,146)(106,132)(107,148)(108,134)(109,150)(110,136)(111,152)(112,138)(113,154)(114,140)(115,156)(116,122)(117,158)(118,124)(119,160)(120,126), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,81)(77,82)(78,83)(79,84)(80,85)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,81)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155), (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), (1,142,77,101)(2,52,78,126)(3,160,79,119)(4,50,80,124)(5,158,61,117)(6,48,62,122)(7,156,63,115)(8,46,64,140)(9,154,65,113)(10,44,66,138)(11,152,67,111)(12,42,68,136)(13,150,69,109)(14,60,70,134)(15,148,71,107)(16,58,72,132)(17,146,73,105)(18,56,74,130)(19,144,75,103)(20,54,76,128)(21,59,96,133)(22,147,97,106)(23,57,98,131)(24,145,99,104)(25,55,100,129)(26,143,81,102)(27,53,82,127)(28,141,83,120)(29,51,84,125)(30,159,85,118)(31,49,86,123)(32,157,87,116)(33,47,88,121)(34,155,89,114)(35,45,90,139)(36,153,91,112)(37,43,92,137)(38,151,93,110)(39,41,94,135)(40,149,95,108) );
G=PermutationGroup([(1,27),(2,78),(3,29),(4,80),(5,31),(6,62),(7,33),(8,64),(9,35),(10,66),(11,37),(12,68),(13,39),(14,70),(15,21),(16,72),(17,23),(18,74),(19,25),(20,76),(22,97),(24,99),(26,81),(28,83),(30,85),(32,87),(34,89),(36,91),(38,93),(40,95),(41,135),(42,151),(43,137),(44,153),(45,139),(46,155),(47,121),(48,157),(49,123),(50,159),(51,125),(52,141),(53,127),(54,143),(55,129),(56,145),(57,131),(58,147),(59,133),(60,149),(61,86),(63,88),(65,90),(67,92),(69,94),(71,96),(73,98),(75,100),(77,82),(79,84),(101,142),(102,128),(103,144),(104,130),(105,146),(106,132),(107,148),(108,134),(109,150),(110,136),(111,152),(112,138),(113,154),(114,140),(115,156),(116,122),(117,158),(118,124),(119,160),(120,126)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,21),(16,22),(17,23),(18,24),(19,25),(20,26),(41,150),(42,151),(43,152),(44,153),(45,154),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,86),(62,87),(63,88),(64,89),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,97),(73,98),(74,99),(75,100),(76,81),(77,82),(78,83),(79,84),(80,85),(101,127),(102,128),(103,129),(104,130),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138),(113,139),(114,140),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,81),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,117),(50,118),(51,119),(52,120),(53,101),(54,102),(55,103),(56,104),(57,105),(58,106),(59,107),(60,108),(121,156),(122,157),(123,158),(124,159),(125,160),(126,141),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150),(136,151),(137,152),(138,153),(139,154),(140,155)], [(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)], [(1,142,77,101),(2,52,78,126),(3,160,79,119),(4,50,80,124),(5,158,61,117),(6,48,62,122),(7,156,63,115),(8,46,64,140),(9,154,65,113),(10,44,66,138),(11,152,67,111),(12,42,68,136),(13,150,69,109),(14,60,70,134),(15,148,71,107),(16,58,72,132),(17,146,73,105),(18,56,74,130),(19,144,75,103),(20,54,76,128),(21,59,96,133),(22,147,97,106),(23,57,98,131),(24,145,99,104),(25,55,100,129),(26,143,81,102),(27,53,82,127),(28,141,83,120),(29,51,84,125),(30,159,85,118),(31,49,86,123),(32,157,87,116),(33,47,88,121),(34,155,89,114),(35,45,90,139),(36,153,91,112),(37,43,92,137),(38,151,93,110),(39,41,94,135),(40,149,95,108)])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D4⋊2D5 |
| kernel | C23.42D20 | C10.10C42 | C2×C23.D5 | C10×C22⋊C4 | C23×Dic5 | C22×Dic5 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C23 | C23 | C23 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 2 | 8 | 4 | 2 | 8 | 8 | 8 | 8 |
Matrix representation of C23.42D20 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 13 | 2 | 0 | 0 |
| 0 | 39 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 12 | 18 | 0 | 0 |
| 0 | 8 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 32 | 0 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,13,39,0,0,0,2,25,0,0,0,0,0,0,1,0,0,0,1,0],[32,0,0,0,0,0,12,8,0,0,0,18,29,0,0,0,0,0,0,32,0,0,0,32,0] >;
C23.42D20 in GAP, Magma, Sage, TeX
C_2^3._{42}D_{20} % in TeX
G:=Group("C2^3.42D20"); // GroupNames label
G:=SmallGroup(320,570);
// by ID
G=gap.SmallGroup(320,570);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,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=b*d^-1>;
// generators/relations