metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.38D20, C22⋊C8⋊9D5, C40⋊6C4⋊8C2, (C2×C20).44D4, (C2×C4).33D20, (C2×C8).109D10, C20⋊7D4.2C2, D20⋊5C4⋊10C2, C10.8(C2×SD16), (C2×C10).14SD16, (C22×C4).85D10, (C22×C10).55D4, C20.282(C4○D4), C2.14(C8⋊D10), C10.11(C8⋊C22), (C2×C40).120C22, (C2×C20).745C23, (C2×D20).12C22, C22.108(C2×D20), C22.3(C40⋊C2), C5⋊1(C23.46D4), C4.106(D4⋊2D5), C4⋊Dic5.270C22, (C22×C20).52C22, C10.17(C22.D4), C2.13(C22.D20), (C2×C4⋊Dic5)⋊5C2, (C5×C22⋊C8)⋊11C2, C2.11(C2×C40⋊C2), (C2×C10).128(C2×D4), (C2×C4).690(C22×D5), SmallGroup(320,362)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.38D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, dad-1=ab=ba, ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >
Subgroups: 542 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.46D4, C4⋊Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C2×C40, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C40⋊6C4, D20⋊5C4, C5×C22⋊C8, C2×C4⋊Dic5, C20⋊7D4, C23.38D20
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C22.D4, C2×SD16, C8⋊C22, D20, C22×D5, C23.46D4, C40⋊C2, C2×D20, D4⋊2D5, C22.D20, C2×C40⋊C2, C8⋊D10, C23.38D20
►On 160 points
Generators in S160
(1 127)(2 82)(3 129)(4 84)(5 131)(6 86)(7 133)(8 88)(9 135)(10 90)(11 137)(12 92)(13 139)(14 94)(15 141)(16 96)(17 143)(18 98)(19 145)(20 100)(21 147)(22 102)(23 149)(24 104)(25 151)(26 106)(27 153)(28 108)(29 155)(30 110)(31 157)(32 112)(33 159)(34 114)(35 121)(36 116)(37 123)(38 118)(39 125)(40 120)(41 122)(42 117)(43 124)(44 119)(45 126)(46 81)(47 128)(48 83)(49 130)(50 85)(51 132)(52 87)(53 134)(54 89)(55 136)(56 91)(57 138)(58 93)(59 140)(60 95)(61 142)(62 97)(63 144)(64 99)(65 146)(66 101)(67 148)(68 103)(69 150)(70 105)(71 152)(72 107)(73 154)(74 109)(75 156)(76 111)(77 158)(78 113)(79 160)(80 115)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 41)(37 42)(38 43)(39 44)(40 45)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)
(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 65 66 40)(2 39 67 64)(3 63 68 38)(4 37 69 62)(5 61 70 36)(6 35 71 60)(7 59 72 34)(8 33 73 58)(9 57 74 32)(10 31 75 56)(11 55 76 30)(12 29 77 54)(13 53 78 28)(14 27 79 52)(15 51 80 26)(16 25 41 50)(17 49 42 24)(18 23 43 48)(19 47 44 22)(20 21 45 46)(81 100 147 126)(82 125 148 99)(83 98 149 124)(84 123 150 97)(85 96 151 122)(86 121 152 95)(87 94 153 160)(88 159 154 93)(89 92 155 158)(90 157 156 91)(101 120 127 146)(102 145 128 119)(103 118 129 144)(104 143 130 117)(105 116 131 142)(106 141 132 115)(107 114 133 140)(108 139 134 113)(109 112 135 138)(110 137 136 111)
G:=sub<Sym(160)| (1,127)(2,82)(3,129)(4,84)(5,131)(6,86)(7,133)(8,88)(9,135)(10,90)(11,137)(12,92)(13,139)(14,94)(15,141)(16,96)(17,143)(18,98)(19,145)(20,100)(21,147)(22,102)(23,149)(24,104)(25,151)(26,106)(27,153)(28,108)(29,155)(30,110)(31,157)(32,112)(33,159)(34,114)(35,121)(36,116)(37,123)(38,118)(39,125)(40,120)(41,122)(42,117)(43,124)(44,119)(45,126)(46,81)(47,128)(48,83)(49,130)(50,85)(51,132)(52,87)(53,134)(54,89)(55,136)(56,91)(57,138)(58,93)(59,140)(60,95)(61,142)(62,97)(63,144)(64,99)(65,146)(66,101)(67,148)(68,103)(69,150)(70,105)(71,152)(72,107)(73,154)(74,109)(75,156)(76,111)(77,158)(78,113)(79,160)(80,115), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,41)(37,42)(38,43)(39,44)(40,45)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (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), (1,65,66,40)(2,39,67,64)(3,63,68,38)(4,37,69,62)(5,61,70,36)(6,35,71,60)(7,59,72,34)(8,33,73,58)(9,57,74,32)(10,31,75,56)(11,55,76,30)(12,29,77,54)(13,53,78,28)(14,27,79,52)(15,51,80,26)(16,25,41,50)(17,49,42,24)(18,23,43,48)(19,47,44,22)(20,21,45,46)(81,100,147,126)(82,125,148,99)(83,98,149,124)(84,123,150,97)(85,96,151,122)(86,121,152,95)(87,94,153,160)(88,159,154,93)(89,92,155,158)(90,157,156,91)(101,120,127,146)(102,145,128,119)(103,118,129,144)(104,143,130,117)(105,116,131,142)(106,141,132,115)(107,114,133,140)(108,139,134,113)(109,112,135,138)(110,137,136,111)>;
G:=Group( (1,127)(2,82)(3,129)(4,84)(5,131)(6,86)(7,133)(8,88)(9,135)(10,90)(11,137)(12,92)(13,139)(14,94)(15,141)(16,96)(17,143)(18,98)(19,145)(20,100)(21,147)(22,102)(23,149)(24,104)(25,151)(26,106)(27,153)(28,108)(29,155)(30,110)(31,157)(32,112)(33,159)(34,114)(35,121)(36,116)(37,123)(38,118)(39,125)(40,120)(41,122)(42,117)(43,124)(44,119)(45,126)(46,81)(47,128)(48,83)(49,130)(50,85)(51,132)(52,87)(53,134)(54,89)(55,136)(56,91)(57,138)(58,93)(59,140)(60,95)(61,142)(62,97)(63,144)(64,99)(65,146)(66,101)(67,148)(68,103)(69,150)(70,105)(71,152)(72,107)(73,154)(74,109)(75,156)(76,111)(77,158)(78,113)(79,160)(80,115), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,41)(37,42)(38,43)(39,44)(40,45)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (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), (1,65,66,40)(2,39,67,64)(3,63,68,38)(4,37,69,62)(5,61,70,36)(6,35,71,60)(7,59,72,34)(8,33,73,58)(9,57,74,32)(10,31,75,56)(11,55,76,30)(12,29,77,54)(13,53,78,28)(14,27,79,52)(15,51,80,26)(16,25,41,50)(17,49,42,24)(18,23,43,48)(19,47,44,22)(20,21,45,46)(81,100,147,126)(82,125,148,99)(83,98,149,124)(84,123,150,97)(85,96,151,122)(86,121,152,95)(87,94,153,160)(88,159,154,93)(89,92,155,158)(90,157,156,91)(101,120,127,146)(102,145,128,119)(103,118,129,144)(104,143,130,117)(105,116,131,142)(106,141,132,115)(107,114,133,140)(108,139,134,113)(109,112,135,138)(110,137,136,111) );
G=PermutationGroup([[(1,127),(2,82),(3,129),(4,84),(5,131),(6,86),(7,133),(8,88),(9,135),(10,90),(11,137),(12,92),(13,139),(14,94),(15,141),(16,96),(17,143),(18,98),(19,145),(20,100),(21,147),(22,102),(23,149),(24,104),(25,151),(26,106),(27,153),(28,108),(29,155),(30,110),(31,157),(32,112),(33,159),(34,114),(35,121),(36,116),(37,123),(38,118),(39,125),(40,120),(41,122),(42,117),(43,124),(44,119),(45,126),(46,81),(47,128),(48,83),(49,130),(50,85),(51,132),(52,87),(53,134),(54,89),(55,136),(56,91),(57,138),(58,93),(59,140),(60,95),(61,142),(62,97),(63,144),(64,99),(65,146),(66,101),(67,148),(68,103),(69,150),(70,105),(71,152),(72,107),(73,154),(74,109),(75,156),(76,111),(77,158),(78,113),(79,160),(80,115)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,41),(37,42),(38,43),(39,44),(40,45),(81,127),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,159),(114,160),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126)], [(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)], [(1,65,66,40),(2,39,67,64),(3,63,68,38),(4,37,69,62),(5,61,70,36),(6,35,71,60),(7,59,72,34),(8,33,73,58),(9,57,74,32),(10,31,75,56),(11,55,76,30),(12,29,77,54),(13,53,78,28),(14,27,79,52),(15,51,80,26),(16,25,41,50),(17,49,42,24),(18,23,43,48),(19,47,44,22),(20,21,45,46),(81,100,147,126),(82,125,148,99),(83,98,149,124),(84,123,150,97),(85,96,151,122),(86,121,152,95),(87,94,153,160),(88,159,154,93),(89,92,155,158),(90,157,156,91),(101,120,127,146),(102,145,128,119),(103,118,129,144),(104,143,130,117),(105,116,131,142),(106,141,132,115),(107,114,133,140),(108,139,134,113),(109,112,135,138),(110,137,136,111)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 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 | 2 | 2 | 40 | 2 | 2 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | SD16 | D10 | D10 | D20 | D20 | C40⋊C2 | C8⋊C22 | D4⋊2D5 | C8⋊D10 |
| kernel | C23.38D20 | C40⋊6C4 | D20⋊5C4 | C5×C22⋊C8 | C2×C4⋊Dic5 | C20⋊7D4 | C2×C20 | C22×C10 | C22⋊C8 | C20 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C10 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C23.38D20 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 5 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 13 | 14 | 0 | 0 |
| 27 | 29 | 0 | 0 |
| 0 | 0 | 32 | 21 |
| 0 | 0 | 0 | 9 |
| 29 | 16 | 0 | 0 |
| 14 | 12 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,5,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[13,27,0,0,14,29,0,0,0,0,32,0,0,0,21,9],[29,14,0,0,16,12,0,0,0,0,32,0,0,0,0,32] >;
C23.38D20 in GAP, Magma, Sage, TeX
C_2^3._{38}D_{20} % in TeX
G:=Group("C2^3.38D20"); // GroupNames label
G:=SmallGroup(320,362);
// by ID
G=gap.SmallGroup(320,362);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*b=b*c,d*a*d^-1=a*b=b*a,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