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G = C2×C23.23D10  order 320 = 26·5

Direct product of C2 and C23.23D10

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

Aliases: C2×C23.23D10, C24.70D10, (C23×C4)⋊4D5, (C23×C20)⋊4C2, (C22×C4)⋊43D10, (C2×C20).704C23, (C2×C10).287C24, (C22×C20)⋊56C22, (C22×C10).205D4, C10.133(C22×D4), C23.91(C5⋊D4), C23.D5⋊55C22, D10⋊C4⋊41C22, C22.82(C4○D20), C10.D4⋊44C22, C10⋊4(C22.D4), (C23×D5).74C22, C23.233(C22×D5), C22.302(C23×D5), (C23×C10).109C22, (C22×C10).416C23, (C2×Dic5).149C23, (C22×D5).125C23, (C22×Dic5).161C22, C2.70(C2×C4○D20), C10.62(C2×C4○D4), C2.6(C22×C5⋊D4), C5⋊5(C2×C22.D4), (C2×C10).574(C2×D4), (C2×C23.D5)⋊22C2, (C2×D10⋊C4)⋊13C2, (C2×C10.D4)⋊18C2, (C2×C4).657(C22×D5), (C22×C5⋊D4).13C2, C22.103(C2×C5⋊D4), (C2×C10).113(C4○D4), (C2×C5⋊D4).144C22, SmallGroup(320,1461)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C2×C23.23D10
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×C5⋊D4 — C2×C23.23D10
Lower central
C5 — C2×C10 — C2×C23.23D10

Subgroups: 1118 in 342 conjugacy classes, 127 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×10], C22, C22 [×10], C22 [×22], C5, C2×C4 [×4], C2×C4 [×24], D4 [×8], C23, C23 [×6], C23 [×12], D5 [×2], C10, C10 [×6], C10 [×4], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4 [×6], C22×C4 [×7], C2×D4 [×8], C24, C24, Dic5 [×6], C20 [×4], D10 [×10], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C2×Dic5 [×6], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×4], C2×C20 [×12], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22.D4, C10.D4 [×8], D10⋊C4 [×8], C23.D5 [×4], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×6], C22×C20 [×4], C23×D5, C23×C10, C2×C10.D4 [×2], C2×D10⋊C4 [×2], C23.23D10 [×8], C2×C23.D5, C22×C5⋊D4, C23×C20, C2×C23.23D10

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], C5⋊D4 [×4], C22×D5 [×7], C2×C22.D4, C4○D20 [×4], C2×C5⋊D4 [×6], C23×D5, C23.23D10 [×4], C2×C4○D20 [×2], C22×C5⋊D4, C2×C23.23D10

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
01000
00100
00010
00001
,
400000
040000
004000
000165
0003125
,
10000
040000
004000
000400
000040
,
10000
01000
00100
000400
000040
,
10000
0173800
033800
0002037
000821
,
400000
024300
0401700
0002027
000821

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,16,31,0,0,0,5,25],[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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,17,3,0,0,0,38,38,0,0,0,0,0,20,8,0,0,0,37,21],[40,0,0,0,0,0,24,40,0,0,0,3,17,0,0,0,0,0,20,8,0,0,0,27,21] >;
 

92 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4N5A5B10A···10AD20A···20AF
order12···22222224···44···45510···1020···20
size11···1222220202···220···20222···22···2

92 irreducible representations

dim11111112222222
type+++++++++++
imageC1C2C2C2C2C2C2D4D5C4○D4D10D10C5⋊D4C4○D20
kernelC2×C23.23D10C2×C10.D4C2×D10⋊C4C23.23D10C2×C23.D5C22×C5⋊D4C23×C20C22×C10C23×C4C2×C10C22×C4C24C23C22
# reps12281114281221632

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{23}D_{10}
 
% in TeX
 
G:=Group("C2xC2^3.23D10");
 
// GroupNames label
 
G:=SmallGroup(320,1461);
 
// by ID
 
G=gap.SmallGroup(320,1461);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,136,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=d,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,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^9>;
 
// generators/relations
 

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