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G = C2×D10⋊3Q8  order 320 = 26·5

Direct product of C2 and D10⋊3Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D10⋊3Q8, D10⋊4(C2×Q8), (C2×Q8)⋊28D10, (C22×D5)⋊7Q8, (C22×Q8)⋊5D5, C20.258(C2×D4), (C2×C20).214D4, C10⋊5(C22⋊Q8), C22.37(Q8×D5), C4⋊Dic5⋊79C22, (Q8×C10)⋊35C22, C10.53(C22×Q8), (C2×C10).305C24, (C2×C20).552C23, (C22×C4).384D10, C10.153(C22×D4), C10.D4⋊75C22, C23.341(C22×D5), C22.316(C23×D5), (C22×C20).285C22, (C22×C10).423C23, C22.39(Q8⋊2D5), (C2×Dic5).157C23, (C22×D5).251C23, (C23×D5).126C22, D10⋊C4.157C22, (C22×Dic5).164C22, (Q8×C2×C10)⋊4C2, C5⋊6(C2×C22⋊Q8), C2.35(C2×Q8×D5), C4.98(C2×C5⋊D4), (D5×C22×C4).9C2, (C2×C4⋊Dic5)⋊46C2, (C2×C10).98(C2×Q8), C10.127(C2×C4○D4), (C2×C10).588(C2×D4), C2.34(C2×Q8⋊2D5), (C2×C4×D5).332C22, C2.26(C22×C5⋊D4), (C2×C10.D4)⋊49C2, (C2×C4).202(C5⋊D4), (C2×C4).242(C22×D5), C22.116(C2×C5⋊D4), (C2×D10⋊C4).29C2, (C2×C10).200(C4○D4), SmallGroup(320,1485)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C2×D10⋊3Q8
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×D10⋊3Q8
Lower central
C5 — C2×C10 — C2×D10⋊3Q8

Subgroups: 1038 in 322 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×16], C5, C2×C4 [×10], C2×C4 [×24], Q8 [×8], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×11], C2×Q8 [×4], C2×Q8 [×4], C24, Dic5 [×6], C20 [×4], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×6], C2×C20 [×10], C2×C20 [×4], C5×Q8 [×8], C22×D5 [×6], C22×D5 [×4], C22×C10, C2×C22⋊Q8, C10.D4 [×8], C4⋊Dic5 [×4], D10⋊C4 [×8], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C23×D5, C2×C10.D4 [×2], C2×C4⋊Dic5, C2×D10⋊C4 [×2], D10⋊3Q8 [×8], D5×C22×C4, Q8×C2×C10, C2×D10⋊3Q8

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C5⋊D4 [×4], C22×D5 [×7], C2×C22⋊Q8, Q8×D5 [×2], Q8⋊2D5 [×2], C2×C5⋊D4 [×6], C23×D5, D10⋊3Q8 [×4], C2×Q8×D5, C2×Q8⋊2D5, C22×C5⋊D4, C2×D10⋊3Q8

Generators and relations
 G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=d-1 >

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

(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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

4000000
0400000
001000
000100
0000400
0000040
,
170000
34340000
000700
00353500
000010
000001
,
40340000
010000
0040000
0036100
0000400
0000040
,
2410000
40170000
00234000
00361800
000001
0000400
,
100000
010000
001000
000100
0000032
0000320

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,34,0,0,0,0,7,34,0,0,0,0,0,0,0,35,0,0,0,0,7,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,34,1,0,0,0,0,0,0,40,36,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[24,40,0,0,0,0,1,17,0,0,0,0,0,0,23,36,0,0,0,0,40,18,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0] >;
 

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10N20A···20X
order12···2222244444444444444445510···1020···20
size11···110101010222244441010101020202020222···24···4

68 irreducible representations

dim1111111222222244
type++++++++-+++-+
imageC1C2C2C2C2C2C2D4Q8D5C4○D4D10D10C5⋊D4Q8×D5Q8⋊2D5
kernelC2×D10⋊3Q8C2×C10.D4C2×C4⋊Dic5C2×D10⋊C4D10⋊3Q8D5×C22×C4Q8×C2×C10C2×C20C22×D5C22×Q8C2×C10C22×C4C2×Q8C2×C4C22C22
# reps12128114424681644

In GAP, Magma, Sage, TeX 

C_2\times D_{10}\rtimes_3Q_8
 
% in TeX
 
G:=Group("C2xD10:3Q8");
 
// GroupNames label
 
G:=SmallGroup(320,1485);
 
// by ID
 
G=gap.SmallGroup(320,1485);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,136,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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