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## G = D5×C25order 320 = 26·5

### Direct product of C25 and D5

Aliases: D5×C25, C5⋊C26, C10⋊C25, (C24×C10)⋊5C2, (C2×C10)⋊4C24, (C23×C10)⋊20C22, (C22×C10)⋊10C23, SmallGroup(320,1639)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C25
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C23×D5 — D5×C24 — D5×C25
 Lower central C5 — D5×C25
 Upper central C1 — C25

Generators and relations for D5×C25
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f5=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 15454 in 5650 conjugacy classes, 3199 normal (5 characteristic)
C1, C2, C2, C22, C22, C5, C23, C23, D5, C10, C24, C24, D10, C2×C10, C25, C25, C22×D5, C22×C10, C26, C23×D5, C23×C10, D5×C24, C24×C10, D5×C25
Quotients: C1, C2, C22, C23, D5, C24, D10, C25, C22×D5, C26, C23×D5, D5×C24, D5×C25

Smallest permutation representation of D5×C25
On 160 points
Generators in S160
(1 159)(2 160)(3 156)(4 157)(5 158)(6 151)(7 152)(8 153)(9 154)(10 155)(11 146)(12 147)(13 148)(14 149)(15 150)(16 141)(17 142)(18 143)(19 144)(20 145)(21 136)(22 137)(23 138)(24 139)(25 140)(26 131)(27 132)(28 133)(29 134)(30 135)(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)(51 106)(52 107)(53 108)(54 109)(55 110)(56 101)(57 102)(58 103)(59 104)(60 105)(61 96)(62 97)(63 98)(64 99)(65 100)(66 91)(67 92)(68 93)(69 94)(70 95)(71 86)(72 87)(73 88)(74 89)(75 90)(76 81)(77 82)(78 83)(79 84)(80 85)
(1 59)(2 60)(3 56)(4 57)(5 58)(6 51)(7 52)(8 53)(9 54)(10 55)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 76)(22 77)(23 78)(24 79)(25 80)(26 71)(27 72)(28 73)(29 74)(30 75)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)(81 136)(82 137)(83 138)(84 139)(85 140)(86 131)(87 132)(88 133)(89 134)(90 135)(91 126)(92 127)(93 128)(94 129)(95 130)(96 121)(97 122)(98 123)(99 124)(100 125)(101 156)(102 157)(103 158)(104 159)(105 160)(106 151)(107 152)(108 153)(109 154)(110 155)(111 146)(112 147)(113 148)(114 149)(115 150)(116 141)(117 142)(118 143)(119 144)(120 145)
(1 29)(2 30)(3 26)(4 27)(5 28)(6 21)(7 22)(8 23)(9 24)(10 25)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)(81 106)(82 107)(83 108)(84 109)(85 110)(86 101)(87 102)(88 103)(89 104)(90 105)(91 116)(92 117)(93 118)(94 119)(95 120)(96 111)(97 112)(98 113)(99 114)(100 115)(121 146)(122 147)(123 148)(124 149)(125 150)(126 141)(127 142)(128 143)(129 144)(130 145)(131 156)(132 157)(133 158)(134 159)(135 160)(136 151)(137 152)(138 153)(139 154)(140 155)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 116)(112 117)(113 118)(114 119)(115 120)(121 126)(122 127)(123 128)(124 129)(125 130)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)(151 156)(152 157)(153 158)(154 159)(155 160)
(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)
(1 113)(2 112)(3 111)(4 115)(5 114)(6 116)(7 120)(8 119)(9 118)(10 117)(11 101)(12 105)(13 104)(14 103)(15 102)(16 106)(17 110)(18 109)(19 108)(20 107)(21 91)(22 95)(23 94)(24 93)(25 92)(26 96)(27 100)(28 99)(29 98)(30 97)(31 81)(32 85)(33 84)(34 83)(35 82)(36 86)(37 90)(38 89)(39 88)(40 87)(41 151)(42 155)(43 154)(44 153)(45 152)(46 156)(47 160)(48 159)(49 158)(50 157)(51 141)(52 145)(53 144)(54 143)(55 142)(56 146)(57 150)(58 149)(59 148)(60 147)(61 131)(62 135)(63 134)(64 133)(65 132)(66 136)(67 140)(68 139)(69 138)(70 137)(71 121)(72 125)(73 124)(74 123)(75 122)(76 126)(77 130)(78 129)(79 128)(80 127)

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

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

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

128 conjugacy classes

 class 1 2A ··· 2AE 2AF ··· 2BK 5A 5B 10A ··· 10BJ order 1 2 ··· 2 2 ··· 2 5 5 10 ··· 10 size 1 1 ··· 1 5 ··· 5 2 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D5 D10 kernel D5×C25 D5×C24 C24×C10 C25 C24 # reps 1 62 1 2 62

Matrix representation of D5×C25 in GL6(𝔽11)

 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 10 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 10 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 1 0 0 0 0 6 10
,
 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10 10 0 0 0 0 0 1

G:=sub<GL(6,GF(11))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[10,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,6,0,0,0,0,1,10],[1,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,10,1] >;

D5×C25 in GAP, Magma, Sage, TeX

D_5\times C_2^5
% in TeX

G:=Group("D5xC2^5");
// GroupNames label

G:=SmallGroup(320,1639);
// by ID

G=gap.SmallGroup(320,1639);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,12550]);
// Polycyclic

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

׿
×
𝔽