direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23.19D4, C2.D8⋊7C10, C22⋊C8⋊7C10, C4.Q8⋊10C10, C4⋊D4.7C10, (C2×C20).461D4, D4⋊C4⋊13C10, C23.18(C5×D4), C42⋊C2⋊5C10, (C22×C10).36D4, C10.128(C4○D8), C20.319(C4○D4), (C2×C20).938C23, (C2×C40).306C22, C22.103(D4×C10), C10.143(C8⋊C22), (D4×C10).197C22, (C22×C20).430C22, C10.97(C22.D4), (C5×C4.Q8)⋊25C2, (C5×C2.D8)⋊22C2, C2.15(C5×C4○D8), C4.31(C5×C4○D4), C4⋊C4.59(C2×C10), (C5×C22⋊C8)⋊24C2, (C2×C8).43(C2×C10), (C2×C4).107(C5×D4), C2.18(C5×C8⋊C22), (C5×D4⋊C4)⋊31C2, (C2×D4).20(C2×C10), (C2×C10).659(C2×D4), (C5×C4⋊D4).17C2, (C5×C42⋊C2)⋊26C2, (C5×C4⋊C4).382C22, (C22×C4).48(C2×C10), (C2×C4).113(C22×C10), C2.13(C5×C22.D4), SmallGroup(320,983)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.19D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >
Subgroups: 210 in 106 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C40, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C23.19D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C22×C20, D4×C10, D4×C10, C5×C22⋊C8, C5×D4⋊C4, C5×C4.Q8, C5×C2.D8, C5×C42⋊C2, C5×C4⋊D4, C5×C23.19D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C22.D4, C4○D8, C8⋊C22, C5×D4, C22×C10, C23.19D4, D4×C10, C5×C4○D4, C5×C22.D4, C5×C4○D8, C5×C8⋊C22, C5×C23.19D4
►On 160 points
Generators in S160
(1 58 24 50 16)(2 59 17 51 9)(3 60 18 52 10)(4 61 19 53 11)(5 62 20 54 12)(6 63 21 55 13)(7 64 22 56 14)(8 57 23 49 15)(25 85 75 33 67)(26 86 76 34 68)(27 87 77 35 69)(28 88 78 36 70)(29 81 79 37 71)(30 82 80 38 72)(31 83 73 39 65)(32 84 74 40 66)(41 156 118 148 110)(42 157 119 149 111)(43 158 120 150 112)(44 159 113 151 105)(45 160 114 152 106)(46 153 115 145 107)(47 154 116 146 108)(48 155 117 147 109)(89 123 142 104 134)(90 124 143 97 135)(91 125 144 98 136)(92 126 137 99 129)(93 127 138 100 130)(94 128 139 101 131)(95 121 140 102 132)(96 122 141 103 133)
(1 129)(2 106)(3 131)(4 108)(5 133)(6 110)(7 135)(8 112)(9 152)(10 101)(11 146)(12 103)(13 148)(14 97)(15 150)(16 99)(17 160)(18 128)(19 154)(20 122)(21 156)(22 124)(23 158)(24 126)(25 111)(26 136)(27 105)(28 130)(29 107)(30 132)(31 109)(32 134)(33 119)(34 144)(35 113)(36 138)(37 115)(38 140)(39 117)(40 142)(41 63)(42 85)(43 57)(44 87)(45 59)(46 81)(47 61)(48 83)(49 120)(50 137)(51 114)(52 139)(53 116)(54 141)(55 118)(56 143)(58 92)(60 94)(62 96)(64 90)(65 147)(66 104)(67 149)(68 98)(69 151)(70 100)(71 145)(72 102)(73 155)(74 123)(75 157)(76 125)(77 159)(78 127)(79 153)(80 121)(82 95)(84 89)(86 91)(88 93)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 78)(18 79)(19 80)(20 73)(21 74)(22 75)(23 76)(24 77)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(57 86)(58 87)(59 88)(60 81)(61 82)(62 83)(63 84)(64 85)(97 149)(98 150)(99 151)(100 152)(101 145)(102 146)(103 147)(104 148)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)(121 154)(122 155)(123 156)(124 157)(125 158)(126 159)(127 160)(128 153)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(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)
(2 26)(3 7)(4 32)(6 30)(8 28)(9 68)(10 14)(11 66)(13 72)(15 70)(17 76)(18 22)(19 74)(21 80)(23 78)(25 29)(33 37)(34 51)(36 49)(38 55)(40 53)(41 43)(42 90)(44 96)(45 47)(46 94)(48 92)(52 56)(57 88)(59 86)(60 64)(61 84)(63 82)(67 71)(75 79)(81 85)(89 91)(93 95)(97 149)(98 104)(99 147)(100 102)(101 145)(103 151)(105 133)(106 108)(107 131)(109 129)(110 112)(111 135)(113 141)(114 116)(115 139)(117 137)(118 120)(119 143)(121 127)(122 159)(123 125)(124 157)(126 155)(128 153)(130 132)(134 136)(138 140)(142 144)(146 152)(148 150)(154 160)(156 158)
G:=sub<Sym(160)| (1,58,24,50,16)(2,59,17,51,9)(3,60,18,52,10)(4,61,19,53,11)(5,62,20,54,12)(6,63,21,55,13)(7,64,22,56,14)(8,57,23,49,15)(25,85,75,33,67)(26,86,76,34,68)(27,87,77,35,69)(28,88,78,36,70)(29,81,79,37,71)(30,82,80,38,72)(31,83,73,39,65)(32,84,74,40,66)(41,156,118,148,110)(42,157,119,149,111)(43,158,120,150,112)(44,159,113,151,105)(45,160,114,152,106)(46,153,115,145,107)(47,154,116,146,108)(48,155,117,147,109)(89,123,142,104,134)(90,124,143,97,135)(91,125,144,98,136)(92,126,137,99,129)(93,127,138,100,130)(94,128,139,101,131)(95,121,140,102,132)(96,122,141,103,133), (1,129)(2,106)(3,131)(4,108)(5,133)(6,110)(7,135)(8,112)(9,152)(10,101)(11,146)(12,103)(13,148)(14,97)(15,150)(16,99)(17,160)(18,128)(19,154)(20,122)(21,156)(22,124)(23,158)(24,126)(25,111)(26,136)(27,105)(28,130)(29,107)(30,132)(31,109)(32,134)(33,119)(34,144)(35,113)(36,138)(37,115)(38,140)(39,117)(40,142)(41,63)(42,85)(43,57)(44,87)(45,59)(46,81)(47,61)(48,83)(49,120)(50,137)(51,114)(52,139)(53,116)(54,141)(55,118)(56,143)(58,92)(60,94)(62,96)(64,90)(65,147)(66,104)(67,149)(68,98)(69,151)(70,100)(71,145)(72,102)(73,155)(74,123)(75,157)(76,125)(77,159)(78,127)(79,153)(80,121)(82,95)(84,89)(86,91)(88,93), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(97,149)(98,150)(99,151)(100,152)(101,145)(102,146)(103,147)(104,148)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)(121,154)(122,155)(123,156)(124,157)(125,158)(126,159)(127,160)(128,153), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (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), (2,26)(3,7)(4,32)(6,30)(8,28)(9,68)(10,14)(11,66)(13,72)(15,70)(17,76)(18,22)(19,74)(21,80)(23,78)(25,29)(33,37)(34,51)(36,49)(38,55)(40,53)(41,43)(42,90)(44,96)(45,47)(46,94)(48,92)(52,56)(57,88)(59,86)(60,64)(61,84)(63,82)(67,71)(75,79)(81,85)(89,91)(93,95)(97,149)(98,104)(99,147)(100,102)(101,145)(103,151)(105,133)(106,108)(107,131)(109,129)(110,112)(111,135)(113,141)(114,116)(115,139)(117,137)(118,120)(119,143)(121,127)(122,159)(123,125)(124,157)(126,155)(128,153)(130,132)(134,136)(138,140)(142,144)(146,152)(148,150)(154,160)(156,158)>;
G:=Group( (1,58,24,50,16)(2,59,17,51,9)(3,60,18,52,10)(4,61,19,53,11)(5,62,20,54,12)(6,63,21,55,13)(7,64,22,56,14)(8,57,23,49,15)(25,85,75,33,67)(26,86,76,34,68)(27,87,77,35,69)(28,88,78,36,70)(29,81,79,37,71)(30,82,80,38,72)(31,83,73,39,65)(32,84,74,40,66)(41,156,118,148,110)(42,157,119,149,111)(43,158,120,150,112)(44,159,113,151,105)(45,160,114,152,106)(46,153,115,145,107)(47,154,116,146,108)(48,155,117,147,109)(89,123,142,104,134)(90,124,143,97,135)(91,125,144,98,136)(92,126,137,99,129)(93,127,138,100,130)(94,128,139,101,131)(95,121,140,102,132)(96,122,141,103,133), (1,129)(2,106)(3,131)(4,108)(5,133)(6,110)(7,135)(8,112)(9,152)(10,101)(11,146)(12,103)(13,148)(14,97)(15,150)(16,99)(17,160)(18,128)(19,154)(20,122)(21,156)(22,124)(23,158)(24,126)(25,111)(26,136)(27,105)(28,130)(29,107)(30,132)(31,109)(32,134)(33,119)(34,144)(35,113)(36,138)(37,115)(38,140)(39,117)(40,142)(41,63)(42,85)(43,57)(44,87)(45,59)(46,81)(47,61)(48,83)(49,120)(50,137)(51,114)(52,139)(53,116)(54,141)(55,118)(56,143)(58,92)(60,94)(62,96)(64,90)(65,147)(66,104)(67,149)(68,98)(69,151)(70,100)(71,145)(72,102)(73,155)(74,123)(75,157)(76,125)(77,159)(78,127)(79,153)(80,121)(82,95)(84,89)(86,91)(88,93), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(97,149)(98,150)(99,151)(100,152)(101,145)(102,146)(103,147)(104,148)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)(121,154)(122,155)(123,156)(124,157)(125,158)(126,159)(127,160)(128,153), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (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), (2,26)(3,7)(4,32)(6,30)(8,28)(9,68)(10,14)(11,66)(13,72)(15,70)(17,76)(18,22)(19,74)(21,80)(23,78)(25,29)(33,37)(34,51)(36,49)(38,55)(40,53)(41,43)(42,90)(44,96)(45,47)(46,94)(48,92)(52,56)(57,88)(59,86)(60,64)(61,84)(63,82)(67,71)(75,79)(81,85)(89,91)(93,95)(97,149)(98,104)(99,147)(100,102)(101,145)(103,151)(105,133)(106,108)(107,131)(109,129)(110,112)(111,135)(113,141)(114,116)(115,139)(117,137)(118,120)(119,143)(121,127)(122,159)(123,125)(124,157)(126,155)(128,153)(130,132)(134,136)(138,140)(142,144)(146,152)(148,150)(154,160)(156,158) );
G=PermutationGroup([[(1,58,24,50,16),(2,59,17,51,9),(3,60,18,52,10),(4,61,19,53,11),(5,62,20,54,12),(6,63,21,55,13),(7,64,22,56,14),(8,57,23,49,15),(25,85,75,33,67),(26,86,76,34,68),(27,87,77,35,69),(28,88,78,36,70),(29,81,79,37,71),(30,82,80,38,72),(31,83,73,39,65),(32,84,74,40,66),(41,156,118,148,110),(42,157,119,149,111),(43,158,120,150,112),(44,159,113,151,105),(45,160,114,152,106),(46,153,115,145,107),(47,154,116,146,108),(48,155,117,147,109),(89,123,142,104,134),(90,124,143,97,135),(91,125,144,98,136),(92,126,137,99,129),(93,127,138,100,130),(94,128,139,101,131),(95,121,140,102,132),(96,122,141,103,133)], [(1,129),(2,106),(3,131),(4,108),(5,133),(6,110),(7,135),(8,112),(9,152),(10,101),(11,146),(12,103),(13,148),(14,97),(15,150),(16,99),(17,160),(18,128),(19,154),(20,122),(21,156),(22,124),(23,158),(24,126),(25,111),(26,136),(27,105),(28,130),(29,107),(30,132),(31,109),(32,134),(33,119),(34,144),(35,113),(36,138),(37,115),(38,140),(39,117),(40,142),(41,63),(42,85),(43,57),(44,87),(45,59),(46,81),(47,61),(48,83),(49,120),(50,137),(51,114),(52,139),(53,116),(54,141),(55,118),(56,143),(58,92),(60,94),(62,96),(64,90),(65,147),(66,104),(67,149),(68,98),(69,151),(70,100),(71,145),(72,102),(73,155),(74,123),(75,157),(76,125),(77,159),(78,127),(79,153),(80,121),(82,95),(84,89),(86,91),(88,93)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,70),(10,71),(11,72),(12,65),(13,66),(14,67),(15,68),(16,69),(17,78),(18,79),(19,80),(20,73),(21,74),(22,75),(23,76),(24,77),(33,56),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(57,86),(58,87),(59,88),(60,81),(61,82),(62,83),(63,84),(64,85),(97,149),(98,150),(99,151),(100,152),(101,145),(102,146),(103,147),(104,148),(105,129),(106,130),(107,131),(108,132),(109,133),(110,134),(111,135),(112,136),(113,137),(114,138),(115,139),(116,140),(117,141),(118,142),(119,143),(120,144),(121,154),(122,155),(123,156),(124,157),(125,158),(126,159),(127,160),(128,153)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160)], [(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)], [(2,26),(3,7),(4,32),(6,30),(8,28),(9,68),(10,14),(11,66),(13,72),(15,70),(17,76),(18,22),(19,74),(21,80),(23,78),(25,29),(33,37),(34,51),(36,49),(38,55),(40,53),(41,43),(42,90),(44,96),(45,47),(46,94),(48,92),(52,56),(57,88),(59,86),(60,64),(61,84),(63,82),(67,71),(75,79),(81,85),(89,91),(93,95),(97,149),(98,104),(99,147),(100,102),(101,145),(103,151),(105,133),(106,108),(107,131),(109,129),(110,112),(111,135),(113,141),(114,116),(115,139),(117,137),(118,120),(119,143),(121,127),(122,159),(123,125),(124,157),(126,155),(128,153),(130,132),(134,136),(138,140),(142,144),(146,152),(148,150),(154,160),(156,158)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 10Q | 10R | 10S | 10T | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | 20AH | 20AI | 20AJ | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D4 | C4○D8 | C5×D4 | C5×D4 | C5×C4○D4 | C5×C4○D8 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C23.19D4 | C5×C22⋊C8 | C5×D4⋊C4 | C5×C4.Q8 | C5×C2.D8 | C5×C42⋊C2 | C5×C4⋊D4 | C23.19D4 | C22⋊C8 | D4⋊C4 | C4.Q8 | C2.D8 | C42⋊C2 | C4⋊D4 | C2×C20 | C22×C10 | C20 | C10 | C2×C4 | C23 | C4 | C2 | C10 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 1 | 1 | 4 | 4 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×C23.19D4 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 9 | 0 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 29 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 21 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[0,32,0,0,9,0,0,0,0,0,0,16,0,0,18,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,29,12,0,0,0,0,0,21,0,0,2,0],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C5×C23.19D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{19}D_4 % in TeX
G:=Group("C5xC2^3.19D4"); // GroupNames label
G:=SmallGroup(320,983);
// by ID
G=gap.SmallGroup(320,983);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1128,1766,226,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*d*e^3>;
// generators/relations