Copied to
clipboard

G = C2×SD16⋊3D5order 320 = 26·5

Direct product of C2 and SD16⋊3D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×SD16⋊3D5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4⋊2D5 — C2×SD16⋊3D5
 Lower central C5 — C10 — C20 — C2×SD16⋊3D5
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C2×SD163D5
G = < a,b,c,d,e | a2=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b3, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 990 in 266 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×12], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23 [×3], D5 [×4], C10, C10 [×2], C10 [×2], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×12], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×4], C22×C8, C2×D8, C2×SD16, C2×SD16, C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C52C8 [×2], C40 [×2], Dic10 [×2], Dic10, C4×D5 [×4], C4×D5 [×4], D20 [×2], D20 [×5], C2×Dic5, C2×Dic5 [×5], C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5, C22×C10, C2×C4○D8, C8×D5 [×4], C40⋊C2 [×4], C2×C52C8, D4⋊D5 [×4], C5⋊Q16 [×4], C2×C40, C5×SD16 [×4], C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, D42D5 [×4], D42D5 [×2], Q82D5 [×4], Q82D5 [×2], C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, D5×C2×C8, C2×C40⋊C2, SD163D5 [×8], C2×D4⋊D5, C2×C5⋊Q16, C10×SD16, C2×D42D5, C2×Q82D5, C2×SD163D5
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C4○D8 [×2], C22×D4, C22×D5 [×7], C2×C4○D8, D4×D5 [×2], C23×D5, SD163D5 [×2], C2×D4×D5, C2×SD163D5

Smallest permutation representation of C2×SD163D5
On 160 points
Generators in S160
(1 96)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 121)(17 157)(18 158)(19 159)(20 160)(21 153)(22 154)(23 155)(24 156)(25 87)(26 88)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 140)(34 141)(35 142)(36 143)(37 144)(38 137)(39 138)(40 139)(41 150)(42 151)(43 152)(44 145)(45 146)(46 147)(47 148)(48 149)(49 101)(50 102)(51 103)(52 104)(53 97)(54 98)(55 99)(56 100)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 105)(64 106)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(73 133)(74 134)(75 135)(76 136)(77 129)(78 130)(79 131)(80 132)
(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 66)(2 69)(3 72)(4 67)(5 70)(6 65)(7 68)(8 71)(9 42)(10 45)(11 48)(12 43)(13 46)(14 41)(15 44)(16 47)(17 139)(18 142)(19 137)(20 140)(21 143)(22 138)(23 141)(24 144)(25 57)(26 60)(27 63)(28 58)(29 61)(30 64)(31 59)(32 62)(33 160)(34 155)(35 158)(36 153)(37 156)(38 159)(39 154)(40 157)(49 77)(50 80)(51 75)(52 78)(53 73)(54 76)(55 79)(56 74)(81 105)(82 108)(83 111)(84 106)(85 109)(86 112)(87 107)(88 110)(89 119)(90 114)(91 117)(92 120)(93 115)(94 118)(95 113)(96 116)(97 133)(98 136)(99 131)(100 134)(101 129)(102 132)(103 135)(104 130)(121 148)(122 151)(123 146)(124 149)(125 152)(126 147)(127 150)(128 145)
(1 106 39 122 135)(2 107 40 123 136)(3 108 33 124 129)(4 109 34 125 130)(5 110 35 126 131)(6 111 36 127 132)(7 112 37 128 133)(8 105 38 121 134)(9 75 96 64 138)(10 76 89 57 139)(11 77 90 58 140)(12 78 91 59 141)(13 79 92 60 142)(14 80 93 61 143)(15 73 94 62 144)(16 74 95 63 137)(17 45 54 119 25)(18 46 55 120 26)(19 47 56 113 27)(20 48 49 114 28)(21 41 50 115 29)(22 42 51 116 30)(23 43 52 117 31)(24 44 53 118 32)(65 83 153 150 102)(66 84 154 151 103)(67 85 155 152 104)(68 86 156 145 97)(69 87 157 146 98)(70 88 158 147 99)(71 81 159 148 100)(72 82 160 149 101)
(1 102)(2 103)(3 104)(4 97)(5 98)(6 99)(7 100)(8 101)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 142)(18 143)(19 144)(20 137)(21 138)(22 139)(23 140)(24 141)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 153)(40 154)(41 64)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 95)(50 96)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(65 135)(66 136)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 128)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)(105 149)(106 150)(107 151)(108 152)(109 145)(110 146)(111 147)(112 148)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 10 10 20 20 2 2 4 4 5 5 5 5 20 20 2 2 2 2 2 2 10 10 10 10 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 C4○D8 D4×D5 D4×D5 SD16⋊3D5 kernel C2×SD16⋊3D5 D5×C2×C8 C2×C40⋊C2 SD16⋊3D5 C2×D4⋊D5 C2×C5⋊Q16 C10×SD16 C2×D4⋊2D5 C2×Q8⋊2D5 C4×D5 C2×Dic5 C22×D5 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C10 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 2 1 1 2 2 8 2 2 8 2 2 8

Matrix representation of C2×SD163D5 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 11 0 0 0 15 11 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 17 0 0 0 29 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 1 34
,
 40 0 0 0 0 0 9 23 0 0 0 9 32 0 0 0 0 0 7 40 0 0 0 7 34

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,15,0,0,0,11,11,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,29,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,40,34],[40,0,0,0,0,0,9,9,0,0,0,23,32,0,0,0,0,0,7,7,0,0,0,40,34] >;

C2×SD163D5 in GAP, Magma, Sage, TeX

C_2\times {\rm SD}_{16}\rtimes_3D_5
% in TeX

G:=Group("C2xSD16:3D5");
// GroupNames label

G:=SmallGroup(320,1433);
// by ID

G=gap.SmallGroup(320,1433);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,1123,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^3,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽