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## G = C2×Dic5⋊D4order 320 = 26·5

### Direct product of C2 and Dic5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×Dic5⋊D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×C5⋊D4 — C2×Dic5⋊D4
 Lower central C5 — C2×C10 — C2×Dic5⋊D4
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×Dic5⋊D4
G = < a,b,c,d,e | a2=b10=d4=e2=1, c2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede=d-1 >

Subgroups: 1486 in 426 conjugacy classes, 135 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×10], C22, C22 [×10], C22 [×32], C5, C2×C4 [×2], C2×C4 [×24], D4 [×24], C23, C23 [×8], C23 [×18], D5 [×2], C10 [×3], C10 [×4], C10 [×6], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×11], C2×D4 [×4], C2×D4 [×20], C24 [×2], C24, Dic5 [×4], Dic5 [×4], C20 [×2], D10 [×10], C2×C10, C2×C10 [×10], C2×C10 [×22], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4, C22×D4 [×2], C2×Dic5 [×10], C2×Dic5 [×12], C5⋊D4 [×16], C2×C20 [×2], C2×C20 [×2], C5×D4 [×8], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×8], C22×C10 [×10], C2×C4⋊D4, C10.D4 [×4], D10⋊C4 [×4], C23.D5 [×4], C22×Dic5 [×3], C22×Dic5 [×4], C22×Dic5 [×4], C2×C5⋊D4 [×8], C2×C5⋊D4 [×8], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×D5, C23×C10 [×2], C2×C10.D4, C2×D10⋊C4, Dic5⋊D4 [×8], C2×C23.D5, C23×Dic5, C22×C5⋊D4 [×2], D4×C2×C10, C2×Dic5⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C4○D4 [×2], C24, D10 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C5⋊D4 [×4], C22×D5 [×7], C2×C4⋊D4, D4×D5 [×2], D42D5 [×2], C2×C5⋊D4 [×6], C23×D5, Dic5⋊D4 [×4], C2×D4×D5, C2×D42D5, C22×C5⋊D4, C2×Dic5⋊D4

Smallest permutation representation of C2×Dic5⋊D4
On 160 points
Generators in S160
(1 107)(2 108)(3 109)(4 110)(5 101)(6 102)(7 103)(8 104)(9 105)(10 106)(11 78)(12 79)(13 80)(14 71)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 91)(29 92)(30 93)(31 128)(32 129)(33 130)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 111)(49 112)(50 113)(51 148)(52 149)(53 150)(54 141)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 131)(69 132)(70 133)(81 154)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 151)(89 152)(90 153)
(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 47 6 42)(2 46 7 41)(3 45 8 50)(4 44 9 49)(5 43 10 48)(11 137 16 132)(12 136 17 131)(13 135 18 140)(14 134 19 139)(15 133 20 138)(21 36 26 31)(22 35 27 40)(23 34 28 39)(24 33 29 38)(25 32 30 37)(51 86 56 81)(52 85 57 90)(53 84 58 89)(54 83 59 88)(55 82 60 87)(61 76 66 71)(62 75 67 80)(63 74 68 79)(64 73 69 78)(65 72 70 77)(91 126 96 121)(92 125 97 130)(93 124 98 129)(94 123 99 128)(95 122 100 127)(101 116 106 111)(102 115 107 120)(103 114 108 119)(104 113 109 118)(105 112 110 117)(141 156 146 151)(142 155 147 160)(143 154 148 159)(144 153 149 158)(145 152 150 157)
(1 87 27 80)(2 88 28 71)(3 89 29 72)(4 90 30 73)(5 81 21 74)(6 82 22 75)(7 83 23 76)(8 84 24 77)(9 85 25 78)(10 86 26 79)(11 105 158 98)(12 106 159 99)(13 107 160 100)(14 108 151 91)(15 109 152 92)(16 110 153 93)(17 101 154 94)(18 102 155 95)(19 103 156 96)(20 104 157 97)(31 68 48 51)(32 69 49 52)(33 70 50 53)(34 61 41 54)(35 62 42 55)(36 63 43 56)(37 64 44 57)(38 65 45 58)(39 66 46 59)(40 67 47 60)(111 148 128 131)(112 149 129 132)(113 150 130 133)(114 141 121 134)(115 142 122 135)(116 143 123 136)(117 144 124 137)(118 145 125 138)(119 146 126 139)(120 147 127 140)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 121)(18 122)(19 123)(20 124)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(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)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C ··· 4J 4K 4L 5A 5B 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 4 4 ··· 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 20 20 4 4 10 ··· 10 20 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 D4×D5 D4⋊2D5 kernel C2×Dic5⋊D4 C2×C10.D4 C2×D10⋊C4 Dic5⋊D4 C2×C23.D5 C23×Dic5 C22×C5⋊D4 D4×C2×C10 C2×Dic5 C22×C10 C22×D4 C2×C10 C22×C4 C2×D4 C24 C23 C22 C22 # reps 1 1 1 8 1 1 2 1 4 4 2 4 2 8 4 16 4 4

Matrix representation of C2×Dic5⋊D4 in GL6(𝔽41)

 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
,
 0 1 0 0 0 0 40 7 0 0 0 0 0 0 0 1 0 0 0 0 40 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 38 38 0 0 0 0 17 3 0 0 0 0 0 0 38 38 0 0 0 0 17 3 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 17 1 0 0 0 0 40 24 0 0 0 0 0 0 17 1 0 0 0 0 40 24 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 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 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(41))| [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],[0,40,0,0,0,0,1,7,0,0,0,0,0,0,0,40,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[38,17,0,0,0,0,38,3,0,0,0,0,0,0,38,17,0,0,0,0,38,3,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[17,40,0,0,0,0,1,24,0,0,0,0,0,0,17,40,0,0,0,0,1,24,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×Dic5⋊D4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5\rtimes D_4
% in TeX

G:=Group("C2xDic5:D4");
// GroupNames label

G:=SmallGroup(320,1474);
// by ID

G=gap.SmallGroup(320,1474);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,297,12550]);
// Polycyclic

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

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