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## G = C2×D10.13D4order 320 = 26·5

### Direct product of C2 and D10.13D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×D10.13D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×D10.13D4
 Lower central C5 — C2×C10 — C2×D10.13D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×D10.13D4
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=ece-1=b5c, ede-1=d-1 >

Subgroups: 1374 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 [×6], C2×C4 [×22], D4 [×8], C23, C23 [×18], D5 [×6], C10 [×3], C10 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×10], C2×D4 [×8], C24 [×2], Dic5 [×4], C20 [×6], D10 [×4], D10 [×22], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×3], C2×C4⋊C4, C2×C4⋊C4, C22.D4 [×8], C23×C4, C22×D4, C4×D5 [×8], D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×6], C2×C20 [×6], C22×D5 [×8], C22×D5 [×10], C22×C10, C2×C22.D4, C10.D4 [×4], D10⋊C4 [×12], C5×C4⋊C4 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C22×C20 [×3], C23×D5 [×2], D10.13D4 [×8], C2×C10.D4, C2×D10⋊C4 [×3], C10×C4⋊C4, D5×C22×C4, C22×D20, C2×D10.13D4
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], Q82D5 [×2], C23×D5, D10.13D4 [×4], C2×C4○D20, C2×D4×D5, C2×Q82D5, C2×D10.13D4

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

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 10 10 10 20 20 2 2 2 2 4 4 4 4 10 10 10 10 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 C4○D20 D4×D5 Q8⋊2D5 kernel C2×D10.13D4 D10.13D4 C2×C10.D4 C2×D10⋊C4 C10×C4⋊C4 D5×C22×C4 C22×D20 C22×D5 C2×C4⋊C4 C2×C10 C4⋊C4 C22×C4 C22 C22 C22 # reps 1 8 1 3 1 1 1 4 2 8 8 6 16 4 4

Matrix representation of C2×D10.13D4 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 34 7 0 0 0 34 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 11 32 0 0 0 27 30 0 0 0 0 0 10 10 0 0 0 27 31
,
 40 0 0 0 0 0 17 1 0 0 0 40 24 0 0 0 0 0 32 0 0 0 0 18 9
,
 40 0 0 0 0 0 30 9 0 0 0 32 11 0 0 0 0 0 31 31 0 0 0 6 10

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,34,34,0,0,0,7,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,11,27,0,0,0,32,30,0,0,0,0,0,10,27,0,0,0,10,31],[40,0,0,0,0,0,17,40,0,0,0,1,24,0,0,0,0,0,32,18,0,0,0,0,9],[40,0,0,0,0,0,30,32,0,0,0,9,11,0,0,0,0,0,31,6,0,0,0,31,10] >;

C2×D10.13D4 in GAP, Magma, Sage, TeX

C_2\times D_{10}._{13}D_4
% in TeX

G:=Group("C2xD10.13D4");
// GroupNames label

G:=SmallGroup(320,1177);
// by ID

G=gap.SmallGroup(320,1177);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,675,297,136,12550]);
// 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=e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
// generators/relations

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