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G = C4○D4⋊Dic5order 320 = 26·5

1st semidirect product of C4○D4 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D41Dic5, Q84(C2×Dic5), D44(C2×Dic5), (C2×C20).476D4, C20.212(C2×D4), (C2×D4).201D10, Q8⋊Dic540C2, D4⋊Dic540C2, (C2×Q8).170D10, C2.5(D4⋊D10), C20.142(C22×C4), (C2×C20).479C23, (C22×C4).161D10, (C22×C10).111D4, C23.65(C5⋊D4), C56(C23.36D4), C4.22(C23.D5), C4.13(C22×Dic5), C10.124(C8⋊C22), C20.137(C22⋊C4), C2.5(D4.9D10), (D4×C10).242C22, C4⋊Dic5.354C22, (Q8×C10).205C22, C22.2(C23.D5), C10.124(C8.C22), (C22×C20).205C22, (C5×C4○D4)⋊7C4, (C5×D4)⋊27(C2×C4), (C5×Q8)⋊25(C2×C4), (C2×C4○D4).1D5, C4.94(C2×C5⋊D4), (C2×C4⋊Dic5)⋊36C2, (C10×C4○D4).1C2, (C2×C20).295(C2×C4), (C2×C10).565(C2×D4), (C2×C4.Dic5)⋊19C2, (C2×C4).28(C2×Dic5), C22.96(C2×C5⋊D4), C2.17(C2×C23.D5), (C2×C4).261(C5⋊D4), C10.122(C2×C22⋊C4), (C2×C4).564(C22×D5), (C2×C10).88(C22⋊C4), (C2×C52C8).177C22, SmallGroup(320,859)

Series: Derived Chief Lower central Upper central

C1C20 — C4○D4⋊Dic5
C1C5C10C2×C10C2×C20C4⋊Dic5C2×C4⋊Dic5 — C4○D4⋊Dic5
C5C10C20 — C4○D4⋊Dic5
C1C22C22×C4C2×C4○D4

Generators and relations for C4○D4⋊Dic5
 G = < a,b,c,d,e | a4=c2=d10=1, b2=a2, e2=d5, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, ebe-1=abc, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 446 in 162 conjugacy classes, 71 normal (31 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 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10 [×3], C10 [×4], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×6], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C52C8 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C23.36D4, C2×C52C8 [×2], C4.Dic5 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C22×Dic5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], D4⋊Dic5 [×2], Q8⋊Dic5 [×2], C2×C4.Dic5, C2×C4⋊Dic5, C10×C4○D4, C4○D4⋊Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C8⋊C22, C8.C22, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.36D4, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4⋊D10, D4.9D10, C2×C23.D5, C4○D4⋊Dic5

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F10G···10R20A···20H20I···20T
order12222222444444444455888810···1010···1020···2020···20
size111122442222442020202022202020202···24···42···24···4

62 irreducible representations

dim11111112222222224444
type++++++++++++-+-+-
imageC1C2C2C2C2C2C4D4D4D5D10D10D10Dic5C5⋊D4C5⋊D4C8⋊C22C8.C22D4⋊D10D4.9D10
kernelC4○D4⋊Dic5D4⋊Dic5Q8⋊Dic5C2×C4.Dic5C2×C4⋊Dic5C10×C4○D4C5×C4○D4C2×C20C22×C10C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C2×C4C23C10C10C2C2
# reps122111831222281241144

Matrix representation of C4○D4⋊Dic5 in GL6(𝔽41)

100000
010000
009000
00333200
000090
00003332
,
4000000
0400000
0000320
000089
0032000
008900
,
100000
010000
0000320
000089
009000
00333200
,
2300000
0250000
0025000
00312300
0000250
00003123
,
0400000
100000
0029141227
003812329
0012271227
00329329

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,33,0,0,0,0,0,32,0,0,0,0,0,0,9,33,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,8,0,0,0,0,0,9,0,0,32,8,0,0,0,0,0,9,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,33,0,0,0,0,0,32,0,0,32,8,0,0,0,0,0,9,0,0],[23,0,0,0,0,0,0,25,0,0,0,0,0,0,25,31,0,0,0,0,0,23,0,0,0,0,0,0,25,31,0,0,0,0,0,23],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,29,38,12,3,0,0,14,12,27,29,0,0,12,3,12,3,0,0,27,29,27,29] >;

C4○D4⋊Dic5 in GAP, Magma, Sage, TeX

C_4\circ D_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C4oD4:Dic5");
// GroupNames label

G:=SmallGroup(320,859);
// by ID

G=gap.SmallGroup(320,859);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^2=d^10=1,b^2=a^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c=a^2*b,b*d=d*b,e*b*e^-1=a*b*c,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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