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## G = C2×D4.8D10order 320 = 26·5

### Direct product of C2 and D4.8D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D4.8D10
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C2×D4.8D10
 Lower central C5 — C10 — C20 — C2×D4.8D10
 Upper central C1 — C2×C4 — C22×C4 — C2×C4○D4

Generators and relations for C2×D4.8D10
G = < a,b,c,d,e | a2=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d9 >

Subgroups: 830 in 266 conjugacy classes, 111 normal (35 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×4], C2×C8 [×6], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×4], C4○D4 [×8], Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×6], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4, C2×C4○D4, C52C8 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×D5, C22×C10, C22×C10, C2×C4○D8, C2×C52C8 [×2], C2×C52C8 [×4], D4⋊D5 [×4], D4.D5 [×4], Q8⋊D5 [×4], C5⋊Q16 [×4], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C22×C52C8, C2×D4⋊D5, C2×D4.D5, C2×Q8⋊D5, C2×C5⋊Q16, D4.8D10 [×8], C2×C4○D20, C10×C4○D4, C2×D4.8D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C4○D8 [×2], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C4○D8, C2×C5⋊D4 [×6], C23×D5, D4.8D10 [×2], C22×C5⋊D4, C2×D4.8D10

Smallest permutation representation of C2×D4.8D10
On 160 points
Generators in S160
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 61)(37 62)(38 63)(39 64)(40 65)(41 152)(42 153)(43 154)(44 155)(45 156)(46 157)(47 158)(48 159)(49 160)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 151)(101 136)(102 137)(103 138)(104 139)(105 140)(106 121)(107 122)(108 123)(109 124)(110 125)(111 126)(112 127)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 134)(120 135)
(1 113 11 103)(2 114 12 104)(3 115 13 105)(4 116 14 106)(5 117 15 107)(6 118 16 108)(7 119 17 109)(8 120 18 110)(9 101 19 111)(10 102 20 112)(21 154 31 144)(22 155 32 145)(23 156 33 146)(24 157 34 147)(25 158 35 148)(26 159 36 149)(27 160 37 150)(28 141 38 151)(29 142 39 152)(30 143 40 153)(41 74 51 64)(42 75 52 65)(43 76 53 66)(44 77 54 67)(45 78 55 68)(46 79 56 69)(47 80 57 70)(48 61 58 71)(49 62 59 72)(50 63 60 73)(81 139 91 129)(82 140 92 130)(83 121 93 131)(84 122 94 132)(85 123 95 133)(86 124 96 134)(87 125 97 135)(88 126 98 136)(89 127 99 137)(90 128 100 138)
(1 148)(2 149)(3 150)(4 151)(5 152)(6 153)(7 154)(8 155)(9 156)(10 157)(11 158)(12 159)(13 160)(14 141)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 119)(22 120)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 89)(57 90)(58 91)(59 92)(60 93)(61 129)(62 130)(63 131)(64 132)(65 133)(66 134)(67 135)(68 136)(69 137)(70 138)(71 139)(72 140)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 151 31 141)(22 160 32 150)(23 149 33 159)(24 158 34 148)(25 147 35 157)(26 156 36 146)(27 145 37 155)(28 154 38 144)(29 143 39 153)(30 152 40 142)(41 65 51 75)(42 74 52 64)(43 63 53 73)(44 72 54 62)(45 61 55 71)(46 70 56 80)(47 79 57 69)(48 68 58 78)(49 77 59 67)(50 66 60 76)(81 98 91 88)(82 87 92 97)(83 96 93 86)(84 85 94 95)(89 90 99 100)(101 104 111 114)(102 113 112 103)(105 120 115 110)(106 109 116 119)(107 118 117 108)(121 124 131 134)(122 133 132 123)(125 140 135 130)(126 129 136 139)(127 138 137 128)

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

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

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

68 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 ··· 8H 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T 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 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 4 20 20 1 1 1 1 2 2 4 4 20 20 2 2 10 ··· 10 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D10 C4○D8 C5⋊D4 C5⋊D4 D4.8D10 kernel C2×D4.8D10 C22×C5⋊2C8 C2×D4⋊D5 C2×D4.D5 C2×Q8⋊D5 C2×C5⋊Q16 D4.8D10 C2×C4○D20 C10×C4○D4 C2×C20 C22×C10 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C10 C2×C4 C23 C2 # reps 1 1 1 1 1 1 8 1 1 3 1 2 2 2 2 8 8 12 4 8

Matrix representation of C2×D4.8D10 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
,
 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 40 39 0 0 0 0 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 4 0 0 0 0 0 1 0 0 0 0 0 0 17 17 0 0 0 0 12 24
,
 1 35 0 0 0 0 6 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 35 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 21 40 0 0 0 0 0 0 9 0 0 0 0 0 32 32

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,4,1,0,0,0,0,0,0,17,12,0,0,0,0,17,24],[1,6,0,0,0,0,35,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,35,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40,0,0,0,0,0,0,9,32,0,0,0,0,0,32] >;

C2×D4.8D10 in GAP, Magma, Sage, TeX

C_2\times D_4._8D_{10}
% in TeX

G:=Group("C2xD4.8D10");
// GroupNames label

G:=SmallGroup(320,1493);
// by ID

G=gap.SmallGroup(320,1493);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,1684,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^9>;
// generators/relations

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