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G = C5×C23.34D4order 320 = 26·5

Direct product of C5 and C23.34D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C23.34D4, (C22×C4)⋊7C20, (C22×C20)⋊24C4, (C23×C4).3C10, (C23×C20).6C2, C23.34(C5×D4), C23.26(C2×C20), C24.25(C2×C10), C22.31(D4×C10), C2.C422C10, (C22×C10).154D4, C23.54(C22×C10), C22.30(C22×C20), (C23×C10).85C22, C10.73(C42⋊C2), (C22×C20).490C22, (C22×C10).445C23, C10.85(C22.D4), (C2×C4).56(C2×C20), C2.6(C10×C22⋊C4), (C2×C20).457(C2×C4), (C2×C10).598(C2×D4), (C2×C22⋊C4).4C10, (C22×C4).3(C2×C10), C2.6(C5×C42⋊C2), C22.16(C5×C4○D4), (C10×C22⋊C4).10C2, C10.134(C2×C22⋊C4), (C5×C2.C42)⋊4C2, C22.16(C5×C22⋊C4), (C2×C10).206(C4○D4), C2.1(C5×C22.D4), (C22×C10).180(C2×C4), (C2×C10).318(C22×C4), (C2×C10).143(C22⋊C4), SmallGroup(320,882)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C23.34D4
C1C2C22C23C22×C10C22×C20C10×C22⋊C4 — C5×C23.34D4
C1C22 — C5×C23.34D4
C1C22×C10 — C5×C23.34D4

Generators and relations for C5×C23.34D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bc=cb, bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 354 in 218 conjugacy classes, 98 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C22, C22 [×10], C22 [×12], C5, C2×C4 [×4], C2×C4 [×24], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22×C4 [×10], C22×C4 [×4], C24, C20 [×8], C2×C10, C2×C10 [×10], C2×C10 [×12], C2.C42 [×4], C2×C22⋊C4 [×2], C23×C4, C2×C20 [×4], C2×C20 [×24], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.34D4, C5×C22⋊C4 [×4], C22×C20 [×10], C22×C20 [×4], C23×C10, C5×C2.C42 [×4], C10×C22⋊C4 [×2], C23×C20, C5×C23.34D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C2×C20 [×6], C5×D4 [×4], C22×C10, C23.34D4, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C5×C4○D4 [×4], C10×C22⋊C4, C5×C42⋊C2 [×2], C5×C22.D4 [×4], C5×C23.34D4

Smallest permutation representation of C5×C23.34D4
On 160 points
Generators in S160
(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 67)(3 68)(4 69)(5 70)(6 152)(7 153)(8 154)(9 155)(10 151)(11 41)(12 42)(13 43)(14 44)(15 45)(16 144)(17 145)(18 141)(19 142)(20 143)(21 160)(22 156)(23 157)(24 158)(25 159)(26 39)(27 40)(28 36)(29 37)(30 38)(31 136)(32 137)(33 138)(34 139)(35 140)(46 53)(47 54)(48 55)(49 51)(50 52)(56 76)(57 77)(58 78)(59 79)(60 80)(61 75)(62 71)(63 72)(64 73)(65 74)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 126)(97 127)(98 128)(99 129)(100 130)(101 148)(102 149)(103 150)(104 146)(105 147)(111 124)(112 125)(113 121)(114 122)(115 123)(116 133)(117 134)(118 135)(119 131)(120 132)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 143)(7 144)(8 145)(9 141)(10 142)(16 153)(17 154)(18 155)(19 151)(20 152)(21 140)(22 136)(23 137)(24 138)(25 139)(26 51)(27 52)(28 53)(29 54)(30 55)(31 156)(32 157)(33 158)(34 159)(35 160)(36 46)(37 47)(38 48)(39 49)(40 50)(41 70)(42 66)(43 67)(44 68)(45 69)(56 93)(57 94)(58 95)(59 91)(60 92)(61 83)(62 84)(63 85)(64 81)(65 82)(71 109)(72 110)(73 106)(74 107)(75 108)(76 86)(77 87)(78 88)(79 89)(80 90)(96 133)(97 134)(98 135)(99 131)(100 132)(101 123)(102 124)(103 125)(104 121)(105 122)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)
(1 37)(2 38)(3 39)(4 40)(5 36)(6 156)(7 157)(8 158)(9 159)(10 160)(11 46)(12 47)(13 48)(14 49)(15 50)(16 137)(17 138)(18 139)(19 140)(20 136)(21 151)(22 152)(23 153)(24 154)(25 155)(26 68)(27 69)(28 70)(29 66)(30 67)(31 143)(32 144)(33 145)(34 141)(35 142)(41 53)(42 54)(43 55)(44 51)(45 52)(56 110)(57 106)(58 107)(59 108)(60 109)(61 89)(62 90)(63 86)(64 87)(65 88)(71 92)(72 93)(73 94)(74 95)(75 91)(76 85)(77 81)(78 82)(79 83)(80 84)(96 150)(97 146)(98 147)(99 148)(100 149)(101 129)(102 130)(103 126)(104 127)(105 128)(111 132)(112 133)(113 134)(114 135)(115 131)(116 125)(117 121)(118 122)(119 123)(120 124)
(1 97 29 121)(2 98 30 122)(3 99 26 123)(4 100 27 124)(5 96 28 125)(6 85 136 56)(7 81 137 57)(8 82 138 58)(9 83 139 59)(10 84 140 60)(11 133 53 103)(12 134 54 104)(13 135 55 105)(14 131 51 101)(15 132 52 102)(16 106 157 77)(17 107 158 78)(18 108 159 79)(19 109 160 80)(20 110 156 76)(21 92 142 62)(22 93 143 63)(23 94 144 64)(24 95 145 65)(25 91 141 61)(31 86 152 72)(32 87 153 73)(33 88 154 74)(34 89 155 75)(35 90 151 71)(36 150 70 116)(37 146 66 117)(38 147 67 118)(39 148 68 119)(40 149 69 120)(41 126 46 112)(42 127 47 113)(43 128 48 114)(44 129 49 115)(45 130 50 111)
(1 32 47 7)(2 33 48 8)(3 34 49 9)(4 35 50 10)(5 31 46 6)(11 156 36 143)(12 157 37 144)(13 158 38 145)(14 159 39 141)(15 160 40 142)(16 66 23 54)(17 67 24 55)(18 68 25 51)(19 69 21 52)(20 70 22 53)(26 155 44 139)(27 151 45 140)(28 152 41 136)(29 153 42 137)(30 154 43 138)(56 150 72 133)(57 146 73 134)(58 147 74 135)(59 148 75 131)(60 149 71 132)(61 129 79 123)(62 130 80 124)(63 126 76 125)(64 127 77 121)(65 128 78 122)(81 117 87 104)(82 118 88 105)(83 119 89 101)(84 120 90 102)(85 116 86 103)(91 115 108 99)(92 111 109 100)(93 112 110 96)(94 113 106 97)(95 114 107 98)

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

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

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

140 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P5A5B5C5D10A···10AB10AC···10AR20A···20AF20AG···20BL
order12···222224···44···4555510···1010···1020···2020···20
size11···122222···24···411111···12···22···24···4

140 irreducible representations

dim11111111112222
type+++++
imageC1C2C2C2C4C5C10C10C10C20D4C4○D4C5×D4C5×C4○D4
kernelC5×C23.34D4C5×C2.C42C10×C22⋊C4C23×C20C22×C20C23.34D4C2.C42C2×C22⋊C4C23×C4C22×C4C22×C10C2×C10C23C22
# reps142184168432481632

Matrix representation of C5×C23.34D4 in GL5(𝔽41)

10000
010000
001000
00010
00001
,
400000
040000
00100
00010
000040
,
10000
040000
004000
000400
000040
,
400000
01000
00100
000400
000040
,
90000
00100
040000
00001
000400
,
320000
00900
09000
000032
00090

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,1,0],[32,0,0,0,0,0,0,9,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,32,0] >;

C5×C23.34D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{34}D_4
% in TeX

G:=Group("C5xC2^3.34D4");
// GroupNames label

G:=SmallGroup(320,882);
// by ID

G=gap.SmallGroup(320,882);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,226]);
// Polycyclic

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

׿
×
𝔽