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G = C24.65D10order 320 = 26·5

5th non-split extension by C24 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.65D10, C23.28D20, (C2×C20)⋊35D4, (C23×C4)⋊1D5, (C23×C20)⋊1C2, C10.115(C4×D4), C23.38(C4×D5), C10.70C22≀C2, C2.5(C207D4), C22.61(C2×D20), C10.80(C4⋊D4), (C22×C10).193D4, (C22×C4).408D10, C2.2(C242D5), C23.83(C5⋊D4), C55(C23.23D4), C222(D10⋊C4), C22.64(C4○D20), (C23×D5).24C22, C23.304(C22×D5), C10.10C4225C2, (C22×C20).485C22, (C23×C10).100C22, (C22×C10).364C23, C10.69(C22.D4), C2.5(C23.23D10), (C22×Dic5).67C22, (C2×C5⋊D4)⋊12C4, C2.29(C4×C5⋊D4), (C2×C4)⋊15(C5⋊D4), (C22×D5)⋊7(C2×C4), (C2×C23.D5)⋊7C2, C22.150(C2×C4×D5), (C2×C10)⋊8(C22⋊C4), (C2×Dic5)⋊11(C2×C4), (C2×C10).550(C2×D4), (C2×D10⋊C4)⋊11C2, (C22×C5⋊D4).7C2, C22.88(C2×C5⋊D4), (C2×C10).92(C4○D4), C2.36(C2×D10⋊C4), C10.106(C2×C22⋊C4), (C2×C10).244(C22×C4), (C22×C10).168(C2×C4), SmallGroup(320,840)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.65D10
C1C5C10C2×C10C22×C10C23×D5C22×C5⋊D4 — C24.65D10
C5C2×C10 — C24.65D10
C1C23C23×C4

Generators and relations for C24.65D10
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 1022 in 286 conjugacy classes, 87 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×8], C22 [×3], C22 [×8], C22 [×22], C5, C2×C4 [×4], C2×C4 [×22], D4 [×8], C23, C23 [×6], C23 [×12], D5 [×2], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×9], C2×D4 [×8], C24, C24, Dic5 [×4], C20 [×4], D10 [×10], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4 [×3], C23×C4, C22×D4, C2×Dic5 [×2], C2×Dic5 [×8], C5⋊D4 [×8], C2×C20 [×4], C2×C20 [×12], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.23D4, D10⋊C4 [×4], C23.D5 [×2], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C22×C20 [×6], C23×D5, C23×C10, C10.10C42 [×2], C2×D10⋊C4 [×2], C2×C23.D5, C22×C5⋊D4, C23×C20, C24.65D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C4×D5 [×2], D20 [×2], C5⋊D4 [×6], C22×D5, C23.23D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C4○D20 [×2], C2×C5⋊D4 [×3], C2×D10⋊C4, C4×C5⋊D4 [×2], C23.23D10, C207D4 [×2], C242D5, C24.65D10

Smallest permutation representation of C24.65D10
On 160 points
Generators in S160
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 118)(22 119)(23 120)(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)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 121)(53 122)(54 123)(55 124)(56 125)(57 126)(58 127)(59 128)(60 129)(81 148)(82 149)(83 150)(84 151)(85 152)(86 153)(87 154)(88 155)(89 156)(90 157)(91 158)(92 159)(93 160)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 80)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 100)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(121 158)(122 159)(123 160)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)
(1 99)(2 100)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 137)(22 138)(23 139)(24 140)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 111)(42 112)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 155)(62 156)(63 157)(64 158)(65 159)(66 160)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)
(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)
(1 109 110 20)(2 19 111 108)(3 107 112 18)(4 17 113 106)(5 105 114 16)(6 15 115 104)(7 103 116 14)(8 13 117 102)(9 101 118 12)(10 11 119 120)(21 157 80 140)(22 139 61 156)(23 155 62 138)(24 137 63 154)(25 153 64 136)(26 135 65 152)(27 151 66 134)(28 133 67 150)(29 149 68 132)(30 131 69 148)(31 147 70 130)(32 129 71 146)(33 145 72 128)(34 127 73 144)(35 143 74 126)(36 125 75 142)(37 141 76 124)(38 123 77 160)(39 159 78 122)(40 121 79 158)(41 58 100 97)(42 96 81 57)(43 56 82 95)(44 94 83 55)(45 54 84 93)(46 92 85 53)(47 52 86 91)(48 90 87 51)(49 50 88 89)(59 60 98 99)

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

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

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

92 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4N5A5B10A···10AD20A···20AF
order12···22222224···44···45510···1020···20
size11···1222220202···220···20222···22···2

92 irreducible representations

dim111111122222222222
type++++++++++++
imageC1C2C2C2C2C2C4D4D4D5C4○D4D10D10C5⋊D4C4×D5D20C5⋊D4C4○D20
kernelC24.65D10C10.10C42C2×D10⋊C4C2×C23.D5C22×C5⋊D4C23×C20C2×C5⋊D4C2×C20C22×C10C23×C4C2×C10C22×C4C24C2×C4C23C23C23C22
# reps12211184424421688816

Matrix representation of C24.65D10 in GL6(𝔽41)

2360000
35180000
0023600
00351800
0000181
0000523
,
100000
010000
0040000
0004000
000010
000001
,
4000000
0400000
0040000
0004000
0000400
0000040
,
0400000
1350000
000900
00321300
00002511
00001439
,
0400000
4000000
000900
009000
0000211
00003739

G:=sub<GL(6,GF(41))| [23,35,0,0,0,0,6,18,0,0,0,0,0,0,23,35,0,0,0,0,6,18,0,0,0,0,0,0,18,5,0,0,0,0,1,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,1,0,0,0,0,40,35,0,0,0,0,0,0,0,32,0,0,0,0,9,13,0,0,0,0,0,0,25,14,0,0,0,0,11,39],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,2,37,0,0,0,0,11,39] >;

C24.65D10 in GAP, Magma, Sage, TeX

C_2^4._{65}D_{10}
% in TeX

G:=Group("C2^4.65D10");
// GroupNames label

G:=SmallGroup(320,840);
// by ID

G=gap.SmallGroup(320,840);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,58,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations

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