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G = C2×C23.23D10order 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×C10).287C24, (C2×C20).704C23, (C22×C20)⋊56C22, C10.133(C22×D4), (C22×C10).205D4, C23.91(C5⋊D4), C23.D555C22, D10⋊C441C22, C22.82(C4○D20), C10.D444C22, C104(C22.D4), (C23×D5).74C22, C23.233(C22×D5), C22.302(C23×D5), (C22×C10).416C23, (C23×C10).109C22, (C2×Dic5).149C23, (C22×D5).125C23, (C22×Dic5).161C22, C2.70(C2×C4○D20), C10.62(C2×C4○D4), C2.6(C22×C5⋊D4), C55(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

C1C2×C10 — C2×C23.23D10
C1C5C10C2×C10C22×D5C23×D5C22×C5⋊D4 — C2×C23.23D10
C5C2×C10 — C2×C23.23D10
C1C23C23×C4

Generators and relations for C2×C23.23D10
 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 >

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

Smallest permutation representation of C2×C23.23D10
On 160 points
Generators in S160
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 141)(19 142)(20 143)(21 57)(22 58)(23 59)(24 60)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(61 132)(62 133)(63 134)(64 135)(65 136)(66 137)(67 138)(68 139)(69 140)(70 121)(71 122)(72 123)(73 124)(74 125)(75 126)(76 127)(77 128)(78 129)(79 130)(80 131)(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)
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 97)(15 98)(16 99)(17 100)(18 81)(19 82)(20 83)(21 136)(22 137)(23 138)(24 139)(25 140)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 128)(34 129)(35 130)(36 131)(37 132)(38 133)(39 134)(40 135)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 150)(22 151)(23 152)(24 153)(25 154)(26 155)(27 156)(28 157)(29 158)(30 159)(31 160)(32 141)(33 142)(34 143)(35 144)(36 145)(37 146)(38 147)(39 148)(40 149)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 81)(77 82)(78 83)(79 84)(80 85)(101 127)(102 128)(103 129)(104 130)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)
(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 41 50)(2 49 42 9)(3 8 43 48)(4 47 44 7)(5 6 45 46)(11 20 51 60)(12 59 52 19)(13 18 53 58)(14 57 54 17)(15 16 55 56)(21 38 160 157)(22 156 141 37)(23 36 142 155)(24 154 143 35)(25 34 144 153)(26 152 145 33)(27 32 146 151)(28 150 147 31)(29 30 148 149)(39 40 158 159)(61 76 96 91)(62 90 97 75)(63 74 98 89)(64 88 99 73)(65 72 100 87)(66 86 81 71)(67 70 82 85)(68 84 83 69)(77 80 92 95)(78 94 93 79)(101 122 137 106)(102 105 138 121)(103 140 139 104)(107 136 123 120)(108 119 124 135)(109 134 125 118)(110 117 126 133)(111 132 127 116)(112 115 128 131)(113 130 129 114)

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

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

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

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

Matrix representation of C2×C23.23D10 in 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] >;

C2×C23.23D10 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

׿
×
𝔽