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G = D5×C23×C4  order 320 = 26·5

Direct product of C23×C4 and D5

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

Aliases: D5×C23×C4, C20⋊3C24, C10.2C25, Dic5⋊3C24, C24.81D10, D10.22C24, C5⋊2(C24×C4), C10⋊2(C23×C4), C2.1(D5×C24), (C23×C20)⋊13C2, (C2×C20)⋊16C23, (D5×C24).7C2, (C2×C10).324C24, (C22×C20)⋊64C22, (C23×Dic5)⋊13C2, (C2×Dic5)⋊14C23, C22.52(C23×D5), C23.345(C22×D5), (C23×C10).114C22, (C22×C10).431C23, (C22×Dic5)⋊55C22, (C22×D5).298C23, (C23×D5).145C22, (C2×C10)⋊10(C22×C4), (C22×C10)⋊22(C2×C4), SmallGroup(320,1609)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — D5×C23×C4
Chief series
C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — D5×C24 — D5×C23×C4
Lower central
C5 — D5×C23×C4
Upper central
C1 — C23×C4

Generators and relations for D5×C23×C4
 G = < a,b,c,d,e,f | a2=b2=c2=d4=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 3614 in 1362 conjugacy classes, 799 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22×C4, C22×C4, C24, C24, Dic5, C20, D10, C2×C10, C23×C4, C23×C4, C25, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, C24×C4, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, C23×C10, D5×C22×C4, C23×Dic5, C23×C20, D5×C24, D5×C23×C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, C25, C4×D5, C22×D5, C24×C4, C2×C4×D5, C23×D5, D5×C22×C4, D5×C24, D5×C23×C4

Smallest permutation representation of D5×C23×C4
►On 160 points
Generators in S160
(1 89)(2 90)(3 86)(4 87)(5 88)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)

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

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

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

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

128 conjugacy classes

class 1 2A···2O2P···2AE4A···4P4Q···4AF5A5B10A···10AD20A···20AF
order12···22···24···44···45510···1020···20
size11···15···51···15···5222···22···2

128 irreducible representations

dim1111112222
type++++++++
imageC1C2C2C2C2C4D5D10D10C4×D5
kernelD5×C23×C4D5×C22×C4C23×Dic5C23×C20D5×C24C23×D5C23×C4C22×C4C24C23
# reps12811132228232

Matrix representation of D5×C23×C4 ►in GL5(𝔽41)

400000
01000
00100
000400
000040
,
400000
040000
004000
00010
00001
,
10000
040000
00100
000400
000040
,
90000
01000
00900
000400
000040
,
10000
01000
00100
000341
000400
,
400000
01000
004000
000134
000040

G:=sub<GL(5,GF(41))| [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],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,34,40,0,0,0,1,0],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,34,40] >;
 

D5×C23×C4 in GAP, Magma, Sage, TeX 

D_5\times C_2^3\times C_4
 
% in TeX
 
G:=Group("D5xC2^3xC4");
 
// GroupNames label
 
G:=SmallGroup(320,1609);
 
// by ID
 
G=gap.SmallGroup(320,1609);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^4=e^5=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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