metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.13D20, C22⋊C8⋊7D5, C40⋊6C4⋊9C2, C40⋊5C4⋊5C2, (C2×C8).110D10, C20⋊7D4.8C2, (C2×C20).242D4, (C2×C4).120D20, D20⋊5C4⋊11C2, C10.10(C4○D8), (C22×C4).87D10, (C22×C10).57D4, C20.284(C4○D4), C2.15(C8⋊D10), C10.12(C8⋊C22), (C2×C40).121C22, (C2×C20).747C23, (C2×D20).14C22, C22.110(C2×D20), C5⋊1(C23.19D4), C4⋊Dic5.14C22, C4.108(D4⋊2D5), C2.12(D40⋊7C2), C23.21D10⋊1C2, (C22×C20).98C22, C10.19(C22.D4), C2.15(C22.D20), (C5×C22⋊C8)⋊9C2, (C2×C10).130(C2×D4), (C2×C4).692(C22×D5), SmallGroup(320,364)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.13D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, eae-1=ac=ca, dad-1=abc, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >
Subgroups: 494 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×4], C23, C23, D5, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4 [×2], Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C40 [×2], D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C23.19D4, C4×Dic5, C4⋊Dic5 [×3], D10⋊C4, C23.D5, C2×C40 [×2], C2×D20, C2×C5⋊D4, C22×C20, C40⋊6C4, C40⋊5C4, D20⋊5C4 [×2], C5×C22⋊C8, C23.21D10, C20⋊7D4, C23.13D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8⋊C22, D20 [×2], C22×D5, C23.19D4, C2×D20, D4⋊2D5 [×2], C22.D20, D40⋊7C2, C8⋊D10, C23.13D20
►On 160 points
Generators in S160
(1 77)(2 160)(3 79)(4 122)(5 41)(6 124)(7 43)(8 126)(9 45)(10 128)(11 47)(12 130)(13 49)(14 132)(15 51)(16 134)(17 53)(18 136)(19 55)(20 138)(21 57)(22 140)(23 59)(24 142)(25 61)(26 144)(27 63)(28 146)(29 65)(30 148)(31 67)(32 150)(33 69)(34 152)(35 71)(36 154)(37 73)(38 156)(39 75)(40 158)(42 103)(44 105)(46 107)(48 109)(50 111)(52 113)(54 115)(56 117)(58 119)(60 81)(62 83)(64 85)(66 87)(68 89)(70 91)(72 93)(74 95)(76 97)(78 99)(80 101)(82 143)(84 145)(86 147)(88 149)(90 151)(92 153)(94 155)(96 157)(98 159)(100 121)(102 123)(104 125)(106 127)(108 129)(110 131)(112 133)(114 135)(116 137)(118 139)(120 141)
(1 118)(2 119)(3 120)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 143)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 159)(58 160)(59 121)(60 122)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 141)(80 142)
(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 97 98 40)(2 39 99 96)(3 95 100 38)(4 37 101 94)(5 93 102 36)(6 35 103 92)(7 91 104 34)(8 33 105 90)(9 89 106 32)(10 31 107 88)(11 87 108 30)(12 29 109 86)(13 85 110 28)(14 27 111 84)(15 83 112 26)(16 25 113 82)(17 81 114 24)(18 23 115 120)(19 119 116 22)(20 21 117 118)(41 52 123 134)(42 133 124 51)(43 50 125 132)(44 131 126 49)(45 48 127 130)(46 129 128 47)(53 80 135 122)(54 121 136 79)(55 78 137 160)(56 159 138 77)(57 76 139 158)(58 157 140 75)(59 74 141 156)(60 155 142 73)(61 72 143 154)(62 153 144 71)(63 70 145 152)(64 151 146 69)(65 68 147 150)(66 149 148 67)
G:=sub<Sym(160)| (1,77)(2,160)(3,79)(4,122)(5,41)(6,124)(7,43)(8,126)(9,45)(10,128)(11,47)(12,130)(13,49)(14,132)(15,51)(16,134)(17,53)(18,136)(19,55)(20,138)(21,57)(22,140)(23,59)(24,142)(25,61)(26,144)(27,63)(28,146)(29,65)(30,148)(31,67)(32,150)(33,69)(34,152)(35,71)(36,154)(37,73)(38,156)(39,75)(40,158)(42,103)(44,105)(46,107)(48,109)(50,111)(52,113)(54,115)(56,117)(58,119)(60,81)(62,83)(64,85)(66,87)(68,89)(70,91)(72,93)(74,95)(76,97)(78,99)(80,101)(82,143)(84,145)(86,147)(88,149)(90,151)(92,153)(94,155)(96,157)(98,159)(100,121)(102,123)(104,125)(106,127)(108,129)(110,131)(112,133)(114,135)(116,137)(118,139)(120,141), (1,118)(2,119)(3,120)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142), (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), (1,97,98,40)(2,39,99,96)(3,95,100,38)(4,37,101,94)(5,93,102,36)(6,35,103,92)(7,91,104,34)(8,33,105,90)(9,89,106,32)(10,31,107,88)(11,87,108,30)(12,29,109,86)(13,85,110,28)(14,27,111,84)(15,83,112,26)(16,25,113,82)(17,81,114,24)(18,23,115,120)(19,119,116,22)(20,21,117,118)(41,52,123,134)(42,133,124,51)(43,50,125,132)(44,131,126,49)(45,48,127,130)(46,129,128,47)(53,80,135,122)(54,121,136,79)(55,78,137,160)(56,159,138,77)(57,76,139,158)(58,157,140,75)(59,74,141,156)(60,155,142,73)(61,72,143,154)(62,153,144,71)(63,70,145,152)(64,151,146,69)(65,68,147,150)(66,149,148,67)>;
G:=Group( (1,77)(2,160)(3,79)(4,122)(5,41)(6,124)(7,43)(8,126)(9,45)(10,128)(11,47)(12,130)(13,49)(14,132)(15,51)(16,134)(17,53)(18,136)(19,55)(20,138)(21,57)(22,140)(23,59)(24,142)(25,61)(26,144)(27,63)(28,146)(29,65)(30,148)(31,67)(32,150)(33,69)(34,152)(35,71)(36,154)(37,73)(38,156)(39,75)(40,158)(42,103)(44,105)(46,107)(48,109)(50,111)(52,113)(54,115)(56,117)(58,119)(60,81)(62,83)(64,85)(66,87)(68,89)(70,91)(72,93)(74,95)(76,97)(78,99)(80,101)(82,143)(84,145)(86,147)(88,149)(90,151)(92,153)(94,155)(96,157)(98,159)(100,121)(102,123)(104,125)(106,127)(108,129)(110,131)(112,133)(114,135)(116,137)(118,139)(120,141), (1,118)(2,119)(3,120)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142), (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), (1,97,98,40)(2,39,99,96)(3,95,100,38)(4,37,101,94)(5,93,102,36)(6,35,103,92)(7,91,104,34)(8,33,105,90)(9,89,106,32)(10,31,107,88)(11,87,108,30)(12,29,109,86)(13,85,110,28)(14,27,111,84)(15,83,112,26)(16,25,113,82)(17,81,114,24)(18,23,115,120)(19,119,116,22)(20,21,117,118)(41,52,123,134)(42,133,124,51)(43,50,125,132)(44,131,126,49)(45,48,127,130)(46,129,128,47)(53,80,135,122)(54,121,136,79)(55,78,137,160)(56,159,138,77)(57,76,139,158)(58,157,140,75)(59,74,141,156)(60,155,142,73)(61,72,143,154)(62,153,144,71)(63,70,145,152)(64,151,146,69)(65,68,147,150)(66,149,148,67) );
G=PermutationGroup([(1,77),(2,160),(3,79),(4,122),(5,41),(6,124),(7,43),(8,126),(9,45),(10,128),(11,47),(12,130),(13,49),(14,132),(15,51),(16,134),(17,53),(18,136),(19,55),(20,138),(21,57),(22,140),(23,59),(24,142),(25,61),(26,144),(27,63),(28,146),(29,65),(30,148),(31,67),(32,150),(33,69),(34,152),(35,71),(36,154),(37,73),(38,156),(39,75),(40,158),(42,103),(44,105),(46,107),(48,109),(50,111),(52,113),(54,115),(56,117),(58,119),(60,81),(62,83),(64,85),(66,87),(68,89),(70,91),(72,93),(74,95),(76,97),(78,99),(80,101),(82,143),(84,145),(86,147),(88,149),(90,151),(92,153),(94,155),(96,157),(98,159),(100,121),(102,123),(104,125),(106,127),(108,129),(110,131),(112,133),(114,135),(116,137),(118,139),(120,141)], [(1,118),(2,119),(3,120),(4,81),(5,82),(6,83),(7,84),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,97),(21,98),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,106),(30,107),(31,108),(32,109),(33,110),(34,111),(35,112),(36,113),(37,114),(38,115),(39,116),(40,117),(41,143),(42,144),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(57,159),(58,160),(59,121),(60,122),(61,123),(62,124),(63,125),(64,126),(65,127),(66,128),(67,129),(68,130),(69,131),(70,132),(71,133),(72,134),(73,135),(74,136),(75,137),(76,138),(77,139),(78,140),(79,141),(80,142)], [(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)], [(1,97,98,40),(2,39,99,96),(3,95,100,38),(4,37,101,94),(5,93,102,36),(6,35,103,92),(7,91,104,34),(8,33,105,90),(9,89,106,32),(10,31,107,88),(11,87,108,30),(12,29,109,86),(13,85,110,28),(14,27,111,84),(15,83,112,26),(16,25,113,82),(17,81,114,24),(18,23,115,120),(19,119,116,22),(20,21,117,118),(41,52,123,134),(42,133,124,51),(43,50,125,132),(44,131,126,49),(45,48,127,130),(46,129,128,47),(53,80,135,122),(54,121,136,79),(55,78,137,160),(56,159,138,77),(57,76,139,158),(58,157,140,75),(59,74,141,156),(60,155,142,73),(61,72,143,154),(62,153,144,71),(63,70,145,152),(64,151,146,69),(65,68,147,150),(66,149,148,67)])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 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 | 4 | 40 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | D20 | D20 | D40⋊7C2 | C8⋊C22 | D4⋊2D5 | C8⋊D10 |
| kernel | C23.13D20 | C40⋊6C4 | C40⋊5C4 | D20⋊5C4 | C5×C22⋊C8 | C23.21D10 | C20⋊7D4 | C2×C20 | C22×C10 | C22⋊C8 | C20 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C2 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C23.13D20 ►in GL4(𝔽41) generated by
| 17 | 7 | 0 | 0 |
| 35 | 24 | 0 | 0 |
| 0 | 0 | 16 | 31 |
| 0 | 0 | 5 | 25 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 27 | 0 | 0 |
| 12 | 6 | 0 | 0 |
| 0 | 0 | 32 | 33 |
| 0 | 0 | 0 | 9 |
| 38 | 23 | 0 | 0 |
| 5 | 3 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [17,35,0,0,7,24,0,0,0,0,16,5,0,0,31,25],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[20,12,0,0,27,6,0,0,0,0,32,0,0,0,33,9],[38,5,0,0,23,3,0,0,0,0,32,0,0,0,0,32] >;
C23.13D20 in GAP, Magma, Sage, TeX
C_2^3._{13}D_{20} % in TeX
G:=Group("C2^3.13D20"); // GroupNames label
G:=SmallGroup(320,364);
// by ID
G=gap.SmallGroup(320,364);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,555,142,1123,136,12550]);
// 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,e*a*e^-1=a*c=c*a,d*a*d^-1=a*b*c,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