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G = C2×D10.12D4  order 320 = 26·5

Direct product of C2 and D10.12D4

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

Aliases: C2×D10.12D4, C24.25D10, C22⋊C4⋊38D10, D10.70(C2×D4), (C2×C10).32C24, C4⋊Dic5⋊51C22, C10.35(C22×D4), C22.126(D4×D5), (C2×C20).572C23, (C22×D5).131D4, (C22×C4).313D10, C23.D5⋊45C22, D10⋊C4⋊45C22, C23.79(C22×D5), C22.71(C23×D5), C22.72(C4○D20), C10.D4⋊59C22, C10⋊1(C22.D4), (C23×C10).58C22, C22.67(D4⋊2D5), (C22×C10).124C23, (C22×C20).352C22, (C2×Dic5).188C23, (C22×D5).161C23, (C23×D5).107C22, (C22×Dic5).227C22, C2.9(C2×D4×D5), (C2×C4×D5)⋊65C22, (D5×C22×C4)⋊17C2, (C2×C4⋊Dic5)⋊19C2, (C2×C22⋊C4)⋊11D5, C2.14(C2×C4○D20), C10.12(C2×C4○D4), C2.9(C2×D4⋊2D5), (C10×C22⋊C4)⋊16C2, C5⋊1(C2×C22.D4), (C2×C10).381(C2×D4), (C2×C23.D5)⋊16C2, (C2×D10⋊C4)⋊17C2, (C2×C10.D4)⋊36C2, (C5×C22⋊C4)⋊51C22, (C2×C4).258(C22×D5), (C2×C5⋊D4).97C22, (C22×C5⋊D4).10C2, (C2×C10).101(C4○D4), SmallGroup(320,1160)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C2×D10.12D4
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×D10.12D4
Lower central
C5 — C2×C10 — C2×D10.12D4

Subgroups: 1214 in 342 conjugacy classes, 119 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×10], C22, C22 [×6], C22 [×26], C5, C2×C4 [×4], C2×C4 [×24], D4 [×8], C23, C23 [×2], C23 [×16], D5 [×4], C10 [×3], C10 [×4], C10 [×2], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×11], C2×D4 [×8], C24, C24, Dic5 [×6], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×4], C2×C20 [×4], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×2], C22×C10 [×6], C2×C22.D4, C10.D4 [×4], C4⋊Dic5 [×4], D10⋊C4 [×4], C23.D5 [×4], C5×C22⋊C4 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5 [×3], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, D10.12D4 [×8], C2×C10.D4, C2×C4⋊Dic5, C2×D10⋊C4, C2×C23.D5, C10×C22⋊C4, D5×C22×C4, C22×C5⋊D4, C2×D10.12D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C2×C22.D4, C4○D20 [×2], D4×D5 [×2], D4⋊2D5 [×2], C23×D5, D10.12D4 [×4], C2×C4○D20, C2×D4×D5, C2×D4⋊2D5, C2×D10.12D4

Generators and relations
 G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=b5d-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 80)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 126)(32 127)(33 128)(34 129)(35 130)(36 121)(37 122)(38 123)(39 124)(40 125)(41 116)(42 117)(43 118)(44 119)(45 120)(46 111)(47 112)(48 113)(49 114)(50 115)(51 146)(52 147)(53 148)(54 149)(55 150)(56 141)(57 142)(58 143)(59 144)(60 145)(61 136)(62 137)(63 138)(64 139)(65 140)(66 131)(67 132)(68 133)(69 134)(70 135)(81 156)(82 157)(83 158)(84 159)(85 160)(86 151)(87 152)(88 153)(89 154)(90 155)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
0000040
,
4000000
0400000
00403400
007700
0000400
0000040
,
100000
0400000
001700
0004000
000010
00004040
,
0320000
3200000
0040000
0004000
00003223
000099
,
900000
090000
0040000
0004000
0000320
000099

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,7,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,0,0,0,23,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,0,0,0,0,9] >;
 

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B10A···10N10O···10V20A···20P
order12···2222222444444444444445510···1010···1020···20
size11···144101010102222441010101020202020222···24···44···4

68 irreducible representations

dim111111111222222244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10C4○D20D4×D5D4⋊2D5
kernelC2×D10.12D4D10.12D4C2×C10.D4C2×C4⋊Dic5C2×D10⋊C4C2×C23.D5C10×C22⋊C4D5×C22×C4C22×C5⋊D4C22×D5C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C22C22C22
# reps1811111114288421644

In GAP, Magma, Sage, TeX 

C_2\times D_{10}._{12}D_4
 
% in TeX
 
G:=Group("C2xD10.12D4");
 
// GroupNames label
 
G:=SmallGroup(320,1160);
 
// by ID
 
G=gap.SmallGroup(320,1160);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,675,297,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^4=1,e^2=b^5,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=b^5*d^-1>;
 
// generators/relations
 

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