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## G = C23.49D20order 320 = 26·5

### 20th non-split extension by C23 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C23.49D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C2×C4○D20 — C23.49D20
 Lower central C5 — C10 — C20 — C23.49D20
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C23.49D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, dad-1=ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >

Subgroups: 670 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×7], Q8 [×3], C23, C23, D5 [×2], C10 [×3], C10 [×2], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C40 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5 [×5], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5, C22×C10, C23.36D4, C4⋊Dic5 [×2], C4⋊Dic5, C2×C40 [×2], C5×M4(2) [×2], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C22×Dic5, C2×C5⋊D4, C22×C20, C20.44D4 [×2], D205C4 [×2], C2×C4⋊Dic5, C10×M4(2), C2×C4○D20, C23.49D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C8⋊C22, C8.C22, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.36D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C8⋊D10, C8.D10, C2×D10⋊C4, C23.49D20

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 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 2 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 20 20 2 2 2 2 20 ··· 20 2 2 4 4 4 4 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D5 D10 D10 C4×D5 D20 C5⋊D4 D20 C8⋊C22 C8.C22 C8⋊D10 C8.D10 kernel C23.49D20 C20.44D4 D20⋊5C4 C2×C4⋊Dic5 C10×M4(2) C2×C4○D20 C4○D20 C2×C20 C22×C10 C2×M4(2) C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C10 C10 C2 C2 # reps 1 2 2 1 1 1 8 3 1 2 4 2 8 4 8 4 1 1 4 4

Matrix representation of C23.49D20 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 18 28 23 34 0 0 13 9 7 15 0 0 34 1 23 13 0 0 40 0 28 32
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 16 33 14 39 0 0 8 1 2 0 0 0 7 26 25 8 0 0 15 25 33 40
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 26 22 39 14 0 0 40 15 0 2 0 0 19 31 8 25 0 0 0 22 40 33

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,13,34,40,0,0,28,9,1,0,0,0,23,7,23,28,0,0,34,15,13,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,8,7,15,0,0,33,1,26,25,0,0,14,2,25,33,0,0,39,0,8,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,26,40,19,0,0,0,22,15,31,22,0,0,39,0,8,40,0,0,14,2,25,33] >;`

C23.49D20 in GAP, Magma, Sage, TeX

`C_2^3._{49}D_{20}`
`% in TeX`

`G:=Group("C2^3.49D20");`
`// GroupNames label`

`G:=SmallGroup(320,760);`
`// by ID`

`G=gap.SmallGroup(320,760);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,387,142,1123,136,12550]);`
`// 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,d*a*d^-1=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`

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