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G = C2×D10.13D4  order 320 = 26·5

Direct product of C2 and D10.13D4

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

Aliases: C2×D10.13D4, C4⋊C4⋊37D10, D10.71(C2×D4), (C2×C10).49C24, (C22×D20).9C2, C22.132(D4×D5), C10.42(C22×D4), (C2×C20).616C23, (C22×D5).132D4, (C22×C4).318D10, D10⋊C4⋊64C22, C22.83(C23×D5), (C2×D20).213C22, C22.76(C4○D20), C10.D4⋊51C22, C10⋊3(C22.D4), (C22×D5).12C23, C23.327(C22×D5), (C22×C10).398C23, (C22×C20).359C22, C22.35(Q8⋊2D5), (C2×Dic5).197C23, (C23×D5).111C22, (C22×Dic5).235C22, C2.14(C2×D4×D5), (C2×C4⋊C4)⋊14D5, (C10×C4⋊C4)⋊11C2, (D5×C22×C4)⋊20C2, (C2×C4×D5)⋊68C22, (C5×C4⋊C4)⋊45C22, C10.19(C2×C4○D4), C2.21(C2×C4○D20), C2.6(C2×Q8⋊2D5), C5⋊3(C2×C22.D4), (C2×C10).388(C2×D4), (C2×D10⋊C4)⋊33C2, (C2×C10.D4)⋊23C2, (C2×C4).140(C22×D5), (C2×C10).106(C4○D4), SmallGroup(320,1177)

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

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], Q8⋊2D5 [×2], C23×D5, D10.13D4 [×4], C2×C4○D20, C2×D4×D5, C2×Q8⋊2D5, C2×D10.13D4

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=ece-1=b5c, ede-1=d-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 84)(2 83)(3 82)(4 81)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 99)(12 98)(13 97)(14 96)(15 95)(16 94)(17 93)(18 92)(19 91)(20 100)(21 73)(22 72)(23 71)(24 80)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 70)(40 69)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 60)(50 59)(101 153)(102 152)(103 151)(104 160)(105 159)(106 158)(107 157)(108 156)(109 155)(110 154)(111 148)(112 147)(113 146)(114 145)(115 144)(116 143)(117 142)(118 141)(119 150)(120 149)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 140)(130 139)

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

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
040000
004000
000400
000040
,
10000
034700
034100
000400
000040
,
400000
0113200
0273000
0001010
0002731
,
400000
017100
0402400
000320
000189
,
400000
030900
0321100
0003131
000610

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] >;
 

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B10A···10N20A···20X
order12···2222222444444444444445510···1020···20
size11···110101010202022224444101010102020222···24···4

68 irreducible representations

dim111111122222244
type+++++++++++++
imageC1C2C2C2C2C2C2D4D5C4○D4D10D10C4○D20D4×D5Q8⋊2D5
kernelC2×D10.13D4D10.13D4C2×C10.D4C2×D10⋊C4C10×C4⋊C4D5×C22×C4C22×D20C22×D5C2×C4⋊C4C2×C10C4⋊C4C22×C4C22C22C22
# reps1813111428861644

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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