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G = C23.49D20order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.49D20, C4○D2012C4, D2028(C2×C4), C20.419(C2×D4), (C2×C8).190D10, (C2×C20).175D4, (C2×C4).154D20, D205C440C2, Dic1026(C2×C4), C2.5(C8⋊D10), (C2×M4(2))⋊13D5, C22.58(C2×D20), C20.44D440C2, C20.75(C22⋊C4), C10.21(C8⋊C22), (C10×M4(2))⋊21C2, (C2×C40).320C22, (C2×C20).774C23, C20.175(C22×C4), C2.5(C8.D10), (C22×C4).141D10, (C22×C10).102D4, C55(C23.36D4), C4.39(D10⋊C4), (C2×D20).207C22, C10.21(C8.C22), C4⋊Dic5.285C22, C22.3(D10⋊C4), (C22×C20).190C22, (C2×Dic10).227C22, C4.74(C2×C4×D5), (C2×C4).54(C4×D5), (C2×C4⋊Dic5)⋊33C2, C4.112(C2×C5⋊D4), (C2×C20).283(C2×C4), (C2×C4○D20).13C2, (C2×C10).164(C2×D4), (C2×C4).78(C5⋊D4), C2.32(C2×D10⋊C4), C10.101(C2×C22⋊C4), (C2×C4).723(C22×D5), (C2×C10).86(C22⋊C4), SmallGroup(320,760)

Series: Derived Chief Lower central Upper central

C1C20 — C23.49D20
C1C5C10C20C2×C20C2×D20C2×C4○D20 — C23.49D20
C5C10C20 — C23.49D20
C1C22C22×C4C2×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 2A2B2C2D2E2F2G4A4B4C4D4E···4J5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order1222222244444···455888810···101010101020···202020202040···40
size1111222020222220···202244442···244442···244444···4

62 irreducible representations

dim11111112222222224444
type++++++++++++++-+-
imageC1C2C2C2C2C2C4D4D4D5D10D10C4×D5D20C5⋊D4D20C8⋊C22C8.C22C8⋊D10C8.D10
kernelC23.49D20C20.44D4D205C4C2×C4⋊Dic5C10×M4(2)C2×C4○D20C4○D20C2×C20C22×C10C2×M4(2)C2×C8C22×C4C2×C4C2×C4C2×C4C23C10C10C2C2
# reps12211183124284841144

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

4000000
0400000
0018282334
00139715
003412313
004002832
,
4000000
0400000
0040000
0004000
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
010000
100000
0016331439
008120
00726258
0015253340
,
010000
4000000
0026223914
00401502
001931825
000224033

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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