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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.19D4, C4×Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C2×C40, C2×D20, C2×C5⋊D4, C22×C20, C40⋊6C4, C40⋊5C4, D20⋊5C4, C5×C22⋊C8, C23.21D10, C20⋊7D4, C23.13D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8⋊C22, D20, C22×D5, C23.19D4, C2×D20, D4⋊2D5, C22.D20, D40⋊7C2, C8⋊D10, C23.13D20
►On 160 points
Generators in S160
(1 107)(2 126)(3 109)(4 128)(5 111)(6 130)(7 113)(8 132)(9 115)(10 134)(11 117)(12 136)(13 119)(14 138)(15 81)(16 140)(17 83)(18 142)(19 85)(20 144)(21 87)(22 146)(23 89)(24 148)(25 91)(26 150)(27 93)(28 152)(29 95)(30 154)(31 97)(32 156)(33 99)(34 158)(35 101)(36 160)(37 103)(38 122)(39 105)(40 124)(41 92)(42 151)(43 94)(44 153)(45 96)(46 155)(47 98)(48 157)(49 100)(50 159)(51 102)(52 121)(53 104)(54 123)(55 106)(56 125)(57 108)(58 127)(59 110)(60 129)(61 112)(62 131)(63 114)(64 133)(65 116)(66 135)(67 118)(68 137)(69 120)(70 139)(71 82)(72 141)(73 84)(74 143)(75 86)(76 145)(77 88)(78 147)(79 90)(80 149)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(81 159)(82 160)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 145)(108 146)(109 147)(110 148)(111 149)(112 150)(113 151)(114 152)(115 153)(116 154)(117 155)(118 156)(119 157)(120 158)
(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 55 56 40)(2 39 57 54)(3 53 58 38)(4 37 59 52)(5 51 60 36)(6 35 61 50)(7 49 62 34)(8 33 63 48)(9 47 64 32)(10 31 65 46)(11 45 66 30)(12 29 67 44)(13 43 68 28)(14 27 69 42)(15 41 70 26)(16 25 71 80)(17 79 72 24)(18 23 73 78)(19 77 74 22)(20 21 75 76)(81 112 139 130)(82 129 140 111)(83 110 141 128)(84 127 142 109)(85 108 143 126)(86 125 144 107)(87 106 145 124)(88 123 146 105)(89 104 147 122)(90 121 148 103)(91 102 149 160)(92 159 150 101)(93 100 151 158)(94 157 152 99)(95 98 153 156)(96 155 154 97)(113 120 131 138)(114 137 132 119)(115 118 133 136)(116 135 134 117)
G:=sub<Sym(160)| (1,107)(2,126)(3,109)(4,128)(5,111)(6,130)(7,113)(8,132)(9,115)(10,134)(11,117)(12,136)(13,119)(14,138)(15,81)(16,140)(17,83)(18,142)(19,85)(20,144)(21,87)(22,146)(23,89)(24,148)(25,91)(26,150)(27,93)(28,152)(29,95)(30,154)(31,97)(32,156)(33,99)(34,158)(35,101)(36,160)(37,103)(38,122)(39,105)(40,124)(41,92)(42,151)(43,94)(44,153)(45,96)(46,155)(47,98)(48,157)(49,100)(50,159)(51,102)(52,121)(53,104)(54,123)(55,106)(56,125)(57,108)(58,127)(59,110)(60,129)(61,112)(62,131)(63,114)(64,133)(65,116)(66,135)(67,118)(68,137)(69,120)(70,139)(71,82)(72,141)(73,84)(74,143)(75,86)(76,145)(77,88)(78,147)(79,90)(80,149), (1,76)(2,77)(3,78)(4,79)(5,80)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(81,159)(82,160)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158), (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), (1,55,56,40)(2,39,57,54)(3,53,58,38)(4,37,59,52)(5,51,60,36)(6,35,61,50)(7,49,62,34)(8,33,63,48)(9,47,64,32)(10,31,65,46)(11,45,66,30)(12,29,67,44)(13,43,68,28)(14,27,69,42)(15,41,70,26)(16,25,71,80)(17,79,72,24)(18,23,73,78)(19,77,74,22)(20,21,75,76)(81,112,139,130)(82,129,140,111)(83,110,141,128)(84,127,142,109)(85,108,143,126)(86,125,144,107)(87,106,145,124)(88,123,146,105)(89,104,147,122)(90,121,148,103)(91,102,149,160)(92,159,150,101)(93,100,151,158)(94,157,152,99)(95,98,153,156)(96,155,154,97)(113,120,131,138)(114,137,132,119)(115,118,133,136)(116,135,134,117)>;
G:=Group( (1,107)(2,126)(3,109)(4,128)(5,111)(6,130)(7,113)(8,132)(9,115)(10,134)(11,117)(12,136)(13,119)(14,138)(15,81)(16,140)(17,83)(18,142)(19,85)(20,144)(21,87)(22,146)(23,89)(24,148)(25,91)(26,150)(27,93)(28,152)(29,95)(30,154)(31,97)(32,156)(33,99)(34,158)(35,101)(36,160)(37,103)(38,122)(39,105)(40,124)(41,92)(42,151)(43,94)(44,153)(45,96)(46,155)(47,98)(48,157)(49,100)(50,159)(51,102)(52,121)(53,104)(54,123)(55,106)(56,125)(57,108)(58,127)(59,110)(60,129)(61,112)(62,131)(63,114)(64,133)(65,116)(66,135)(67,118)(68,137)(69,120)(70,139)(71,82)(72,141)(73,84)(74,143)(75,86)(76,145)(77,88)(78,147)(79,90)(80,149), (1,76)(2,77)(3,78)(4,79)(5,80)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(81,159)(82,160)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158), (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), (1,55,56,40)(2,39,57,54)(3,53,58,38)(4,37,59,52)(5,51,60,36)(6,35,61,50)(7,49,62,34)(8,33,63,48)(9,47,64,32)(10,31,65,46)(11,45,66,30)(12,29,67,44)(13,43,68,28)(14,27,69,42)(15,41,70,26)(16,25,71,80)(17,79,72,24)(18,23,73,78)(19,77,74,22)(20,21,75,76)(81,112,139,130)(82,129,140,111)(83,110,141,128)(84,127,142,109)(85,108,143,126)(86,125,144,107)(87,106,145,124)(88,123,146,105)(89,104,147,122)(90,121,148,103)(91,102,149,160)(92,159,150,101)(93,100,151,158)(94,157,152,99)(95,98,153,156)(96,155,154,97)(113,120,131,138)(114,137,132,119)(115,118,133,136)(116,135,134,117) );
G=PermutationGroup([[(1,107),(2,126),(3,109),(4,128),(5,111),(6,130),(7,113),(8,132),(9,115),(10,134),(11,117),(12,136),(13,119),(14,138),(15,81),(16,140),(17,83),(18,142),(19,85),(20,144),(21,87),(22,146),(23,89),(24,148),(25,91),(26,150),(27,93),(28,152),(29,95),(30,154),(31,97),(32,156),(33,99),(34,158),(35,101),(36,160),(37,103),(38,122),(39,105),(40,124),(41,92),(42,151),(43,94),(44,153),(45,96),(46,155),(47,98),(48,157),(49,100),(50,159),(51,102),(52,121),(53,104),(54,123),(55,106),(56,125),(57,108),(58,127),(59,110),(60,129),(61,112),(62,131),(63,114),(64,133),(65,116),(66,135),(67,118),(68,137),(69,120),(70,139),(71,82),(72,141),(73,84),(74,143),(75,86),(76,145),(77,88),(78,147),(79,90),(80,149)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(81,159),(82,160),(83,121),(84,122),(85,123),(86,124),(87,125),(88,126),(89,127),(90,128),(91,129),(92,130),(93,131),(94,132),(95,133),(96,134),(97,135),(98,136),(99,137),(100,138),(101,139),(102,140),(103,141),(104,142),(105,143),(106,144),(107,145),(108,146),(109,147),(110,148),(111,149),(112,150),(113,151),(114,152),(115,153),(116,154),(117,155),(118,156),(119,157),(120,158)], [(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)], [(1,55,56,40),(2,39,57,54),(3,53,58,38),(4,37,59,52),(5,51,60,36),(6,35,61,50),(7,49,62,34),(8,33,63,48),(9,47,64,32),(10,31,65,46),(11,45,66,30),(12,29,67,44),(13,43,68,28),(14,27,69,42),(15,41,70,26),(16,25,71,80),(17,79,72,24),(18,23,73,78),(19,77,74,22),(20,21,75,76),(81,112,139,130),(82,129,140,111),(83,110,141,128),(84,127,142,109),(85,108,143,126),(86,125,144,107),(87,106,145,124),(88,123,146,105),(89,104,147,122),(90,121,148,103),(91,102,149,160),(92,159,150,101),(93,100,151,158),(94,157,152,99),(95,98,153,156),(96,155,154,97),(113,120,131,138),(114,137,132,119),(115,118,133,136),(116,135,134,117)]])
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