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, C2, C4, C22, C22, C22, C5, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C22×C10, C22×C10, C23.34D4, C23.D5, C5×C22⋊C4, C22×Dic5, C22×Dic5, C22×C20, C23×C10, C10.10C42, C2×C23.D5, C10×C22⋊C4, C23×Dic5, C23.42D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C22⋊C4, C42⋊C2, C22.D4, C4×D5, D20, C5⋊D4, C22×D5, C23.34D4, D10⋊C4, C2×C4×D5, C2×D20, D4⋊2D5, C2×C5⋊D4, C23.11D10, C22.D20, C2×D10⋊C4, C23.18D10, C23.42D20
►On 160 points
Generators in S160
(1 34)(2 144)(3 36)(4 146)(5 38)(6 148)(7 40)(8 150)(9 22)(10 152)(11 24)(12 154)(13 26)(14 156)(15 28)(16 158)(17 30)(18 160)(19 32)(20 142)(21 111)(23 113)(25 115)(27 117)(29 119)(31 101)(33 103)(35 105)(37 107)(39 109)(41 69)(42 81)(43 71)(44 83)(45 73)(46 85)(47 75)(48 87)(49 77)(50 89)(51 79)(52 91)(53 61)(54 93)(55 63)(56 95)(57 65)(58 97)(59 67)(60 99)(62 122)(64 124)(66 126)(68 128)(70 130)(72 132)(74 134)(76 136)(78 138)(80 140)(82 131)(84 133)(86 135)(88 137)(90 139)(92 121)(94 123)(96 125)(98 127)(100 129)(102 141)(104 143)(106 145)(108 147)(110 149)(112 151)(114 153)(116 155)(118 157)(120 159)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(81 130)(82 131)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 121)(93 122)(94 123)(95 124)(96 125)(97 126)(98 127)(99 128)(100 129)(101 160)(102 141)(103 142)(104 143)(105 144)(106 145)(107 146)(108 147)(109 148)(110 149)(111 150)(112 151)(113 152)(114 153)(115 154)(116 155)(117 156)(118 157)(119 158)(120 159)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 101)(19 102)(20 103)(21 150)(22 151)(23 152)(24 153)(25 154)(26 155)(27 156)(28 157)(29 158)(30 159)(31 160)(32 141)(33 142)(34 143)(35 144)(36 145)(37 146)(38 147)(39 148)(40 149)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 91)
(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 99 143 60)(2 127 144 67)(3 97 145 58)(4 125 146 65)(5 95 147 56)(6 123 148 63)(7 93 149 54)(8 121 150 61)(9 91 151 52)(10 139 152 79)(11 89 153 50)(12 137 154 77)(13 87 155 48)(14 135 156 75)(15 85 157 46)(16 133 158 73)(17 83 159 44)(18 131 160 71)(19 81 141 42)(20 129 142 69)(21 92 111 53)(22 140 112 80)(23 90 113 51)(24 138 114 78)(25 88 115 49)(26 136 116 76)(27 86 117 47)(28 134 118 74)(29 84 119 45)(30 132 120 72)(31 82 101 43)(32 130 102 70)(33 100 103 41)(34 128 104 68)(35 98 105 59)(36 126 106 66)(37 96 107 57)(38 124 108 64)(39 94 109 55)(40 122 110 62)
G:=sub<Sym(160)| (1,34)(2,144)(3,36)(4,146)(5,38)(6,148)(7,40)(8,150)(9,22)(10,152)(11,24)(12,154)(13,26)(14,156)(15,28)(16,158)(17,30)(18,160)(19,32)(20,142)(21,111)(23,113)(25,115)(27,117)(29,119)(31,101)(33,103)(35,105)(37,107)(39,109)(41,69)(42,81)(43,71)(44,83)(45,73)(46,85)(47,75)(48,87)(49,77)(50,89)(51,79)(52,91)(53,61)(54,93)(55,63)(56,95)(57,65)(58,97)(59,67)(60,99)(62,122)(64,124)(66,126)(68,128)(70,130)(72,132)(74,134)(76,136)(78,138)(80,140)(82,131)(84,133)(86,135)(88,137)(90,139)(92,121)(94,123)(96,125)(98,127)(100,129)(102,141)(104,143)(106,145)(108,147)(110,149)(112,151)(114,153)(116,155)(118,157)(120,159), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,160)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,101)(19,102)(20,103)(21,150)(22,151)(23,152)(24,153)(25,154)(26,155)(27,156)(28,157)(29,158)(30,159)(31,160)(32,141)(33,142)(34,143)(35,144)(36,145)(37,146)(38,147)(39,148)(40,149)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91), (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), (1,99,143,60)(2,127,144,67)(3,97,145,58)(4,125,146,65)(5,95,147,56)(6,123,148,63)(7,93,149,54)(8,121,150,61)(9,91,151,52)(10,139,152,79)(11,89,153,50)(12,137,154,77)(13,87,155,48)(14,135,156,75)(15,85,157,46)(16,133,158,73)(17,83,159,44)(18,131,160,71)(19,81,141,42)(20,129,142,69)(21,92,111,53)(22,140,112,80)(23,90,113,51)(24,138,114,78)(25,88,115,49)(26,136,116,76)(27,86,117,47)(28,134,118,74)(29,84,119,45)(30,132,120,72)(31,82,101,43)(32,130,102,70)(33,100,103,41)(34,128,104,68)(35,98,105,59)(36,126,106,66)(37,96,107,57)(38,124,108,64)(39,94,109,55)(40,122,110,62)>;
G:=Group( (1,34)(2,144)(3,36)(4,146)(5,38)(6,148)(7,40)(8,150)(9,22)(10,152)(11,24)(12,154)(13,26)(14,156)(15,28)(16,158)(17,30)(18,160)(19,32)(20,142)(21,111)(23,113)(25,115)(27,117)(29,119)(31,101)(33,103)(35,105)(37,107)(39,109)(41,69)(42,81)(43,71)(44,83)(45,73)(46,85)(47,75)(48,87)(49,77)(50,89)(51,79)(52,91)(53,61)(54,93)(55,63)(56,95)(57,65)(58,97)(59,67)(60,99)(62,122)(64,124)(66,126)(68,128)(70,130)(72,132)(74,134)(76,136)(78,138)(80,140)(82,131)(84,133)(86,135)(88,137)(90,139)(92,121)(94,123)(96,125)(98,127)(100,129)(102,141)(104,143)(106,145)(108,147)(110,149)(112,151)(114,153)(116,155)(118,157)(120,159), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,160)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,101)(19,102)(20,103)(21,150)(22,151)(23,152)(24,153)(25,154)(26,155)(27,156)(28,157)(29,158)(30,159)(31,160)(32,141)(33,142)(34,143)(35,144)(36,145)(37,146)(38,147)(39,148)(40,149)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91), (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), (1,99,143,60)(2,127,144,67)(3,97,145,58)(4,125,146,65)(5,95,147,56)(6,123,148,63)(7,93,149,54)(8,121,150,61)(9,91,151,52)(10,139,152,79)(11,89,153,50)(12,137,154,77)(13,87,155,48)(14,135,156,75)(15,85,157,46)(16,133,158,73)(17,83,159,44)(18,131,160,71)(19,81,141,42)(20,129,142,69)(21,92,111,53)(22,140,112,80)(23,90,113,51)(24,138,114,78)(25,88,115,49)(26,136,116,76)(27,86,117,47)(28,134,118,74)(29,84,119,45)(30,132,120,72)(31,82,101,43)(32,130,102,70)(33,100,103,41)(34,128,104,68)(35,98,105,59)(36,126,106,66)(37,96,107,57)(38,124,108,64)(39,94,109,55)(40,122,110,62) );
G=PermutationGroup([[(1,34),(2,144),(3,36),(4,146),(5,38),(6,148),(7,40),(8,150),(9,22),(10,152),(11,24),(12,154),(13,26),(14,156),(15,28),(16,158),(17,30),(18,160),(19,32),(20,142),(21,111),(23,113),(25,115),(27,117),(29,119),(31,101),(33,103),(35,105),(37,107),(39,109),(41,69),(42,81),(43,71),(44,83),(45,73),(46,85),(47,75),(48,87),(49,77),(50,89),(51,79),(52,91),(53,61),(54,93),(55,63),(56,95),(57,65),(58,97),(59,67),(60,99),(62,122),(64,124),(66,126),(68,128),(70,130),(72,132),(74,134),(76,136),(78,138),(80,140),(82,131),(84,133),(86,135),(88,137),(90,139),(92,121),(94,123),(96,125),(98,127),(100,129),(102,141),(104,143),(106,145),(108,147),(110,149),(112,151),(114,153),(116,155),(118,157),(120,159)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(41,69),(42,70),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,80),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(81,130),(82,131),(83,132),(84,133),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,121),(93,122),(94,123),(95,124),(96,125),(97,126),(98,127),(99,128),(100,129),(101,160),(102,141),(103,142),(104,143),(105,144),(106,145),(107,146),(108,147),(109,148),(110,149),(111,150),(112,151),(113,152),(114,153),(115,154),(116,155),(117,156),(118,157),(119,158),(120,159)], [(1,104),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,112),(10,113),(11,114),(12,115),(13,116),(14,117),(15,118),(16,119),(17,120),(18,101),(19,102),(20,103),(21,150),(22,151),(23,152),(24,153),(25,154),(26,155),(27,156),(28,157),(29,158),(30,159),(31,160),(32,141),(33,142),(34,143),(35,144),(36,145),(37,146),(38,147),(39,148),(40,149),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,121),(54,122),(55,123),(56,124),(57,125),(58,126),(59,127),(60,128),(61,92),(62,93),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,91)], [(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)], [(1,99,143,60),(2,127,144,67),(3,97,145,58),(4,125,146,65),(5,95,147,56),(6,123,148,63),(7,93,149,54),(8,121,150,61),(9,91,151,52),(10,139,152,79),(11,89,153,50),(12,137,154,77),(13,87,155,48),(14,135,156,75),(15,85,157,46),(16,133,158,73),(17,83,159,44),(18,131,160,71),(19,81,141,42),(20,129,142,69),(21,92,111,53),(22,140,112,80),(23,90,113,51),(24,138,114,78),(25,88,115,49),(26,136,116,76),(27,86,117,47),(28,134,118,74),(29,84,119,45),(30,132,120,72),(31,82,101,43),(32,130,102,70),(33,100,103,41),(34,128,104,68),(35,98,105,59),(36,126,106,66),(37,96,107,57),(38,124,108,64),(39,94,109,55),(40,122,110,62)]])
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