metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.14D20, C24.10D10, (C2×C20).51D4, C2.6(C20⋊7D4), (C22×C10).65D4, (C22×C4).91D10, C10.58(C4⋊D4), C22.125(C2×D20), C5⋊4(C23.11D4), C10.38(C4.4D4), C2.6(C20.17D4), C22.98(C4○D20), (C23×C10).36C22, (C22×C20).60C22, C23.370(C22×D5), C10.10C42⋊15C2, C10.16(C42⋊2C2), C22.96(D4⋊2D5), (C22×C10).328C23, C10.73(C22.D4), C2.16(C22.D20), C2.8(C23.18D10), C2.14(C23.D10), (C22×Dic5).42C22, (C2×C4⋊Dic5)⋊12C2, (C2×C10).432(C2×D4), (C2×C4).30(C5⋊D4), (C2×C22⋊C4).15D5, (C10×C22⋊C4).16C2, C22.126(C2×C5⋊D4), (C2×C23.D5).15C2, (C2×C10).144(C4○D4), SmallGroup(320,580)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.14D20
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=c, f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >
Subgroups: 566 in 170 conjugacy classes, 59 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C23.11D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×C10, C10.10C42, C10.10C42, C2×C4⋊Dic5, C2×C23.D5, C10×C22⋊C4, C23.14D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, D20, C5⋊D4, C22×D5, C23.11D4, C2×D20, C4○D20, D4⋊2D5, C2×C5⋊D4, C23.D10, C22.D20, C20⋊7D4, C23.18D10, C20.17D4, C23.14D20
►On 160 points
Generators in S160
(1 11)(2 140)(3 13)(4 122)(5 15)(6 124)(7 17)(8 126)(9 19)(10 128)(12 130)(14 132)(16 134)(18 136)(20 138)(21 72)(22 118)(23 74)(24 120)(25 76)(26 102)(27 78)(28 104)(29 80)(30 106)(31 62)(32 108)(33 64)(34 110)(35 66)(36 112)(37 68)(38 114)(39 70)(40 116)(41 107)(42 63)(43 109)(44 65)(45 111)(46 67)(47 113)(48 69)(49 115)(50 71)(51 117)(52 73)(53 119)(54 75)(55 101)(56 77)(57 103)(58 79)(59 105)(60 61)(81 91)(82 151)(83 93)(84 153)(85 95)(86 155)(87 97)(88 157)(89 99)(90 159)(92 141)(94 143)(96 145)(98 147)(100 149)(121 131)(123 133)(125 135)(127 137)(129 139)(142 152)(144 154)(146 156)(148 158)(150 160)
(1 99)(2 100)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
(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)
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 136)(9 137)(10 138)(11 139)(12 140)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 101)(77 102)(78 103)(79 104)(80 105)(81 160)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)
(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)
(1 103 129 78)(2 102 130 77)(3 101 131 76)(4 120 132 75)(5 119 133 74)(6 118 134 73)(7 117 135 72)(8 116 136 71)(9 115 137 70)(10 114 138 69)(11 113 139 68)(12 112 140 67)(13 111 121 66)(14 110 122 65)(15 109 123 64)(16 108 124 63)(17 107 125 62)(18 106 126 61)(19 105 127 80)(20 104 128 79)(21 154 51 95)(22 153 52 94)(23 152 53 93)(24 151 54 92)(25 150 55 91)(26 149 56 90)(27 148 57 89)(28 147 58 88)(29 146 59 87)(30 145 60 86)(31 144 41 85)(32 143 42 84)(33 142 43 83)(34 141 44 82)(35 160 45 81)(36 159 46 100)(37 158 47 99)(38 157 48 98)(39 156 49 97)(40 155 50 96)
G:=sub<Sym(160)| (1,11)(2,140)(3,13)(4,122)(5,15)(6,124)(7,17)(8,126)(9,19)(10,128)(12,130)(14,132)(16,134)(18,136)(20,138)(21,72)(22,118)(23,74)(24,120)(25,76)(26,102)(27,78)(28,104)(29,80)(30,106)(31,62)(32,108)(33,64)(34,110)(35,66)(36,112)(37,68)(38,114)(39,70)(40,116)(41,107)(42,63)(43,109)(44,65)(45,111)(46,67)(47,113)(48,69)(49,115)(50,71)(51,117)(52,73)(53,119)(54,75)(55,101)(56,77)(57,103)(58,79)(59,105)(60,61)(81,91)(82,151)(83,93)(84,153)(85,95)(86,155)(87,97)(88,157)(89,99)(90,159)(92,141)(94,143)(96,145)(98,147)(100,149)(121,131)(123,133)(125,135)(127,137)(129,139)(142,152)(144,154)(146,156)(148,158)(150,160), (1,99)(2,100)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (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), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,101)(77,102)(78,103)(79,104)(80,105)(81,160)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159), (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), (1,103,129,78)(2,102,130,77)(3,101,131,76)(4,120,132,75)(5,119,133,74)(6,118,134,73)(7,117,135,72)(8,116,136,71)(9,115,137,70)(10,114,138,69)(11,113,139,68)(12,112,140,67)(13,111,121,66)(14,110,122,65)(15,109,123,64)(16,108,124,63)(17,107,125,62)(18,106,126,61)(19,105,127,80)(20,104,128,79)(21,154,51,95)(22,153,52,94)(23,152,53,93)(24,151,54,92)(25,150,55,91)(26,149,56,90)(27,148,57,89)(28,147,58,88)(29,146,59,87)(30,145,60,86)(31,144,41,85)(32,143,42,84)(33,142,43,83)(34,141,44,82)(35,160,45,81)(36,159,46,100)(37,158,47,99)(38,157,48,98)(39,156,49,97)(40,155,50,96)>;
G:=Group( (1,11)(2,140)(3,13)(4,122)(5,15)(6,124)(7,17)(8,126)(9,19)(10,128)(12,130)(14,132)(16,134)(18,136)(20,138)(21,72)(22,118)(23,74)(24,120)(25,76)(26,102)(27,78)(28,104)(29,80)(30,106)(31,62)(32,108)(33,64)(34,110)(35,66)(36,112)(37,68)(38,114)(39,70)(40,116)(41,107)(42,63)(43,109)(44,65)(45,111)(46,67)(47,113)(48,69)(49,115)(50,71)(51,117)(52,73)(53,119)(54,75)(55,101)(56,77)(57,103)(58,79)(59,105)(60,61)(81,91)(82,151)(83,93)(84,153)(85,95)(86,155)(87,97)(88,157)(89,99)(90,159)(92,141)(94,143)(96,145)(98,147)(100,149)(121,131)(123,133)(125,135)(127,137)(129,139)(142,152)(144,154)(146,156)(148,158)(150,160), (1,99)(2,100)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (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), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,101)(77,102)(78,103)(79,104)(80,105)(81,160)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159), (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), (1,103,129,78)(2,102,130,77)(3,101,131,76)(4,120,132,75)(5,119,133,74)(6,118,134,73)(7,117,135,72)(8,116,136,71)(9,115,137,70)(10,114,138,69)(11,113,139,68)(12,112,140,67)(13,111,121,66)(14,110,122,65)(15,109,123,64)(16,108,124,63)(17,107,125,62)(18,106,126,61)(19,105,127,80)(20,104,128,79)(21,154,51,95)(22,153,52,94)(23,152,53,93)(24,151,54,92)(25,150,55,91)(26,149,56,90)(27,148,57,89)(28,147,58,88)(29,146,59,87)(30,145,60,86)(31,144,41,85)(32,143,42,84)(33,142,43,83)(34,141,44,82)(35,160,45,81)(36,159,46,100)(37,158,47,99)(38,157,48,98)(39,156,49,97)(40,155,50,96) );
G=PermutationGroup([[(1,11),(2,140),(3,13),(4,122),(5,15),(6,124),(7,17),(8,126),(9,19),(10,128),(12,130),(14,132),(16,134),(18,136),(20,138),(21,72),(22,118),(23,74),(24,120),(25,76),(26,102),(27,78),(28,104),(29,80),(30,106),(31,62),(32,108),(33,64),(34,110),(35,66),(36,112),(37,68),(38,114),(39,70),(40,116),(41,107),(42,63),(43,109),(44,65),(45,111),(46,67),(47,113),(48,69),(49,115),(50,71),(51,117),(52,73),(53,119),(54,75),(55,101),(56,77),(57,103),(58,79),(59,105),(60,61),(81,91),(82,151),(83,93),(84,153),(85,95),(86,155),(87,97),(88,157),(89,99),(90,159),(92,141),(94,143),(96,145),(98,147),(100,149),(121,131),(123,133),(125,135),(127,137),(129,139),(142,152),(144,154),(146,156),(148,158),(150,160)], [(1,99),(2,100),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(33,119),(34,120),(35,101),(36,102),(37,103),(38,104),(39,105),(40,106),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,157),(129,158),(130,159),(131,160),(132,141),(133,142),(134,143),(135,144),(136,145),(137,146),(138,147),(139,148),(140,149)], [(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)], [(1,129),(2,130),(3,131),(4,132),(5,133),(6,134),(7,135),(8,136),(9,137),(10,138),(11,139),(12,140),(13,121),(14,122),(15,123),(16,124),(17,125),(18,126),(19,127),(20,128),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49),(40,50),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,101),(77,102),(78,103),(79,104),(80,105),(81,160),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159)], [(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)], [(1,103,129,78),(2,102,130,77),(3,101,131,76),(4,120,132,75),(5,119,133,74),(6,118,134,73),(7,117,135,72),(8,116,136,71),(9,115,137,70),(10,114,138,69),(11,113,139,68),(12,112,140,67),(13,111,121,66),(14,110,122,65),(15,109,123,64),(16,108,124,63),(17,107,125,62),(18,106,126,61),(19,105,127,80),(20,104,128,79),(21,154,51,95),(22,153,52,94),(23,152,53,93),(24,151,54,92),(25,150,55,91),(26,149,56,90),(27,148,57,89),(28,147,58,88),(29,146,59,87),(30,145,60,86),(31,144,41,85),(32,143,42,84),(33,142,43,83),(34,141,44,82),(35,160,45,81),(36,159,46,100),(37,158,47,99),(38,157,48,98),(39,156,49,97),(40,155,50,96)]])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | C4○D20 | D4⋊2D5 |
| kernel | C23.14D20 | C10.10C42 | C2×C4⋊Dic5 | C2×C23.D5 | C10×C22⋊C4 | C2×C20 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 2 | 10 | 4 | 2 | 8 | 8 | 8 | 8 |
Matrix representation of C23.14D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 38 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 16 | 0 | 0 |
| 0 | 0 | 36 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 6 | 28 | 0 | 0 | 0 | 0 |
| 9 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 23 | 0 | 0 |
| 0 | 0 | 27 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [1,23,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[10,38,0,0,0,0,0,37,0,0,0,0,0,0,14,36,0,0,0,0,16,9,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[6,9,0,0,0,0,28,35,0,0,0,0,0,0,35,27,0,0,0,0,23,6,0,0,0,0,0,0,9,0,0,0,0,0,0,32] >;
C23.14D20 in GAP, Magma, Sage, TeX
C_2^3._{14}D_{20} % in TeX
G:=Group("C2^3.14D20"); // GroupNames label
G:=SmallGroup(320,580);
// by ID
G=gap.SmallGroup(320,580);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,387,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=c,f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,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^-1=c*e^9>;
// generators/relations