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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C23.34D4
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C20 — C10×C22⋊C4 — C5×C23.34D4
 Lower central C1 — C22 — C5×C23.34D4
 Upper central C1 — C22×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, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C24, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C23×C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C23.34D4, C5×C22⋊C4, C22×C20, C22×C20, C23×C10, C5×C2.C42, C10×C22⋊C4, C23×C20, C5×C23.34D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C4○D4, C20, C2×C10, C2×C22⋊C4, C42⋊C2, C22.D4, C2×C20, C5×D4, C22×C10, C23.34D4, C5×C22⋊C4, C22×C20, D4×C10, C5×C4○D4, C10×C22⋊C4, C5×C42⋊C2, C5×C22.D4, 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)(6 142)(7 143)(8 144)(9 145)(10 141)(11 36)(12 37)(13 38)(14 39)(15 40)(16 154)(17 155)(18 151)(19 152)(20 153)(21 160)(22 156)(23 157)(24 158)(25 159)(26 51)(27 52)(28 53)(29 54)(30 55)(31 128)(32 129)(33 130)(34 126)(35 127)(41 60)(42 56)(43 57)(44 58)(45 59)(46 79)(47 80)(48 76)(49 77)(50 78)(61 75)(62 71)(63 72)(64 73)(65 74)(81 100)(82 96)(83 97)(84 98)(85 99)(86 137)(87 138)(88 139)(89 140)(90 136)(91 108)(92 109)(93 110)(94 106)(95 107)(101 148)(102 149)(103 150)(104 146)(105 147)(111 132)(112 133)(113 134)(114 135)(115 131)(116 125)(117 121)(118 122)(119 123)(120 124)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 151)(7 152)(8 153)(9 154)(10 155)(11 45)(12 41)(13 42)(14 43)(15 44)(16 145)(17 141)(18 142)(19 143)(20 144)(21 127)(22 128)(23 129)(24 130)(25 126)(31 156)(32 157)(33 158)(34 159)(35 160)(36 59)(37 60)(38 56)(39 57)(40 58)(46 83)(47 84)(48 85)(49 81)(50 82)(51 67)(52 68)(53 69)(54 70)(55 66)(61 91)(62 92)(63 93)(64 94)(65 95)(71 109)(72 110)(73 106)(74 107)(75 108)(76 99)(77 100)(78 96)(79 97)(80 98)(86 123)(87 124)(88 125)(89 121)(90 122)(101 131)(102 132)(103 133)(104 134)(105 135)(111 149)(112 150)(113 146)(114 147)(115 148)(116 139)(117 140)(118 136)(119 137)(120 138)
(1 37)(2 38)(3 39)(4 40)(5 36)(6 127)(7 128)(8 129)(9 130)(10 126)(11 70)(12 66)(13 67)(14 68)(15 69)(16 158)(17 159)(18 160)(19 156)(20 157)(21 151)(22 152)(23 153)(24 154)(25 155)(26 56)(27 57)(28 58)(29 59)(30 60)(31 143)(32 144)(33 145)(34 141)(35 142)(41 55)(42 51)(43 52)(44 53)(45 54)(46 108)(47 109)(48 110)(49 106)(50 107)(61 97)(62 98)(63 99)(64 100)(65 96)(71 84)(72 85)(73 81)(74 82)(75 83)(76 93)(77 94)(78 95)(79 91)(80 92)(86 148)(87 149)(88 150)(89 146)(90 147)(101 137)(102 138)(103 139)(104 140)(105 136)(111 124)(112 125)(113 121)(114 122)(115 123)(116 133)(117 134)(118 135)(119 131)(120 132)
(1 89 12 134)(2 90 13 135)(3 86 14 131)(4 87 15 132)(5 88 11 133)(6 109 160 80)(7 110 156 76)(8 106 157 77)(9 107 158 78)(10 108 159 79)(16 95 130 50)(17 91 126 46)(18 92 127 47)(19 93 128 48)(20 94 129 49)(21 84 142 62)(22 85 143 63)(23 81 144 64)(24 82 145 65)(25 83 141 61)(26 122 42 105)(27 123 43 101)(28 124 44 102)(29 125 45 103)(30 121 41 104)(31 99 152 72)(32 100 153 73)(33 96 154 74)(34 97 155 75)(35 98 151 71)(36 150 70 116)(37 146 66 117)(38 147 67 118)(39 148 68 119)(40 149 69 120)(51 136 56 114)(52 137 57 115)(53 138 58 111)(54 139 59 112)(55 140 60 113)
(1 32 60 20)(2 33 56 16)(3 34 57 17)(4 35 58 18)(5 31 59 19)(6 69 21 44)(7 70 22 45)(8 66 23 41)(9 67 24 42)(10 68 25 43)(11 152 54 128)(12 153 55 129)(13 154 51 130)(14 155 52 126)(15 151 53 127)(26 158 38 145)(27 159 39 141)(28 160 40 142)(29 156 36 143)(30 157 37 144)(46 148 75 123)(47 149 71 124)(48 150 72 125)(49 146 73 121)(50 147 74 122)(61 137 79 131)(62 138 80 132)(63 139 76 133)(64 140 77 134)(65 136 78 135)(81 113 106 89)(82 114 107 90)(83 115 108 86)(84 111 109 87)(85 112 110 88)(91 119 97 101)(92 120 98 102)(93 116 99 103)(94 117 100 104)(95 118 96 105)

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

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

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

140 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 5A 5B 5C 5D 10A ··· 10AB 10AC ··· 10AR 20A ··· 20AF 20AG ··· 20BL order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 D4 C4○D4 C5×D4 C5×C4○D4 kernel C5×C23.34D4 C5×C2.C42 C10×C22⋊C4 C23×C20 C22×C20 C23.34D4 C2.C42 C2×C22⋊C4 C23×C4 C22×C4 C22×C10 C2×C10 C23 C22 # reps 1 4 2 1 8 4 16 8 4 32 4 8 16 32

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

 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 1 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 40 0
,
 32 0 0 0 0 0 0 9 0 0 0 9 0 0 0 0 0 0 0 32 0 0 0 9 0

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

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