direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C23.D7, C42.44D4, (C2×C42)⋊4C4, (C2×C6)⋊1Dic7, (C2×C14)⋊10C12, C21⋊8(C22⋊C4), C42.36(C2×C4), (C6×Dic7)⋊8C2, (C2×Dic7)⋊8C6, C14.27(C3×D4), (C2×C6).35D14, C2.5(C6×Dic7), C22.7(C6×D7), (C22×C6).1D7, C23.2(C3×D7), C14.23(C2×C12), C6.27(C7⋊D4), (C22×C42).4C2, C6.15(C2×Dic7), C22⋊2(C3×Dic7), (C2×C42).36C22, (C22×C14).10C6, C7⋊5(C3×C22⋊C4), C2.3(C3×C7⋊D4), (C2×C14).24(C2×C6), SmallGroup(336,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.D7
G = < a,b,c,d,e,f | a3=b2=c2=d2=e7=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 176 in 68 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C7, C2×C4, C23, C12, C2×C6, C2×C6, C2×C6, C14, C14, C14, C22⋊C4, C21, C2×C12, C22×C6, Dic7, C2×C14, C2×C14, C2×C14, C42, C42, C42, C3×C22⋊C4, C2×Dic7, C22×C14, C3×Dic7, C2×C42, C2×C42, C2×C42, C23.D7, C6×Dic7, C22×C42, C3×C23.D7
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, D7, C22⋊C4, C2×C12, C3×D4, Dic7, D14, C3×D7, C3×C22⋊C4, C2×Dic7, C7⋊D4, C3×Dic7, C6×D7, C23.D7, C6×Dic7, C3×C7⋊D4, C3×C23.D7
►On 168 points
Generators in S168
(1 57 29)(2 58 30)(3 59 31)(4 60 32)(5 61 33)(6 62 34)(7 63 35)(8 64 36)(9 65 37)(10 66 38)(11 67 39)(12 68 40)(13 69 41)(14 70 42)(15 71 43)(16 72 44)(17 73 45)(18 74 46)(19 75 47)(20 76 48)(21 77 49)(22 78 50)(23 79 51)(24 80 52)(25 81 53)(26 82 54)(27 83 55)(28 84 56)(85 141 113)(86 142 114)(87 143 115)(88 144 116)(89 145 117)(90 146 118)(91 147 119)(92 148 120)(93 149 121)(94 150 122)(95 151 123)(96 152 124)(97 153 125)(98 154 126)(99 155 127)(100 156 128)(101 157 129)(102 158 130)(103 159 131)(104 160 132)(105 161 133)(106 162 134)(107 163 135)(108 164 136)(109 165 137)(110 166 138)(111 167 139)(112 168 140)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)(141 162)(142 163)(143 164)(144 165)(145 166)(146 167)(147 168)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)
(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)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)
(1 103 15 89)(2 102 16 88)(3 101 17 87)(4 100 18 86)(5 99 19 85)(6 105 20 91)(7 104 21 90)(8 110 22 96)(9 109 23 95)(10 108 24 94)(11 107 25 93)(12 106 26 92)(13 112 27 98)(14 111 28 97)(29 131 43 117)(30 130 44 116)(31 129 45 115)(32 128 46 114)(33 127 47 113)(34 133 48 119)(35 132 49 118)(36 138 50 124)(37 137 51 123)(38 136 52 122)(39 135 53 121)(40 134 54 120)(41 140 55 126)(42 139 56 125)(57 159 71 145)(58 158 72 144)(59 157 73 143)(60 156 74 142)(61 155 75 141)(62 161 76 147)(63 160 77 146)(64 166 78 152)(65 165 79 151)(66 164 80 150)(67 163 81 149)(68 162 82 148)(69 168 83 154)(70 167 84 153)
G:=sub<Sym(168)| (1,57,29)(2,58,30)(3,59,31)(4,60,32)(5,61,33)(6,62,34)(7,63,35)(8,64,36)(9,65,37)(10,66,38)(11,67,39)(12,68,40)(13,69,41)(14,70,42)(15,71,43)(16,72,44)(17,73,45)(18,74,46)(19,75,47)(20,76,48)(21,77,49)(22,78,50)(23,79,51)(24,80,52)(25,81,53)(26,82,54)(27,83,55)(28,84,56)(85,141,113)(86,142,114)(87,143,115)(88,144,116)(89,145,117)(90,146,118)(91,147,119)(92,148,120)(93,149,121)(94,150,122)(95,151,123)(96,152,124)(97,153,125)(98,154,126)(99,155,127)(100,156,128)(101,157,129)(102,158,130)(103,159,131)(104,160,132)(105,161,133)(106,162,134)(107,163,135)(108,164,136)(109,165,137)(110,166,138)(111,167,139)(112,168,140), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168), (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)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168), (1,103,15,89)(2,102,16,88)(3,101,17,87)(4,100,18,86)(5,99,19,85)(6,105,20,91)(7,104,21,90)(8,110,22,96)(9,109,23,95)(10,108,24,94)(11,107,25,93)(12,106,26,92)(13,112,27,98)(14,111,28,97)(29,131,43,117)(30,130,44,116)(31,129,45,115)(32,128,46,114)(33,127,47,113)(34,133,48,119)(35,132,49,118)(36,138,50,124)(37,137,51,123)(38,136,52,122)(39,135,53,121)(40,134,54,120)(41,140,55,126)(42,139,56,125)(57,159,71,145)(58,158,72,144)(59,157,73,143)(60,156,74,142)(61,155,75,141)(62,161,76,147)(63,160,77,146)(64,166,78,152)(65,165,79,151)(66,164,80,150)(67,163,81,149)(68,162,82,148)(69,168,83,154)(70,167,84,153)>;
G:=Group( (1,57,29)(2,58,30)(3,59,31)(4,60,32)(5,61,33)(6,62,34)(7,63,35)(8,64,36)(9,65,37)(10,66,38)(11,67,39)(12,68,40)(13,69,41)(14,70,42)(15,71,43)(16,72,44)(17,73,45)(18,74,46)(19,75,47)(20,76,48)(21,77,49)(22,78,50)(23,79,51)(24,80,52)(25,81,53)(26,82,54)(27,83,55)(28,84,56)(85,141,113)(86,142,114)(87,143,115)(88,144,116)(89,145,117)(90,146,118)(91,147,119)(92,148,120)(93,149,121)(94,150,122)(95,151,123)(96,152,124)(97,153,125)(98,154,126)(99,155,127)(100,156,128)(101,157,129)(102,158,130)(103,159,131)(104,160,132)(105,161,133)(106,162,134)(107,163,135)(108,164,136)(109,165,137)(110,166,138)(111,167,139)(112,168,140), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168), (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)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168), (1,103,15,89)(2,102,16,88)(3,101,17,87)(4,100,18,86)(5,99,19,85)(6,105,20,91)(7,104,21,90)(8,110,22,96)(9,109,23,95)(10,108,24,94)(11,107,25,93)(12,106,26,92)(13,112,27,98)(14,111,28,97)(29,131,43,117)(30,130,44,116)(31,129,45,115)(32,128,46,114)(33,127,47,113)(34,133,48,119)(35,132,49,118)(36,138,50,124)(37,137,51,123)(38,136,52,122)(39,135,53,121)(40,134,54,120)(41,140,55,126)(42,139,56,125)(57,159,71,145)(58,158,72,144)(59,157,73,143)(60,156,74,142)(61,155,75,141)(62,161,76,147)(63,160,77,146)(64,166,78,152)(65,165,79,151)(66,164,80,150)(67,163,81,149)(68,162,82,148)(69,168,83,154)(70,167,84,153) );
G=PermutationGroup([[(1,57,29),(2,58,30),(3,59,31),(4,60,32),(5,61,33),(6,62,34),(7,63,35),(8,64,36),(9,65,37),(10,66,38),(11,67,39),(12,68,40),(13,69,41),(14,70,42),(15,71,43),(16,72,44),(17,73,45),(18,74,46),(19,75,47),(20,76,48),(21,77,49),(22,78,50),(23,79,51),(24,80,52),(25,81,53),(26,82,54),(27,83,55),(28,84,56),(85,141,113),(86,142,114),(87,143,115),(88,144,116),(89,145,117),(90,146,118),(91,147,119),(92,148,120),(93,149,121),(94,150,122),(95,151,123),(96,152,124),(97,153,125),(98,154,126),(99,155,127),(100,156,128),(101,157,129),(102,158,130),(103,159,131),(104,160,132),(105,161,133),(106,162,134),(107,163,135),(108,164,136),(109,165,137),(110,166,138),(111,167,139),(112,168,140)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(91,112),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133),(141,162),(142,163),(143,164),(144,165),(145,166),(146,167),(147,168),(148,155),(149,156),(150,157),(151,158),(152,159),(153,160),(154,161)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(133,140),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(161,168)], [(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),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168)], [(1,103,15,89),(2,102,16,88),(3,101,17,87),(4,100,18,86),(5,99,19,85),(6,105,20,91),(7,104,21,90),(8,110,22,96),(9,109,23,95),(10,108,24,94),(11,107,25,93),(12,106,26,92),(13,112,27,98),(14,111,28,97),(29,131,43,117),(30,130,44,116),(31,129,45,115),(32,128,46,114),(33,127,47,113),(34,133,48,119),(35,132,49,118),(36,138,50,124),(37,137,51,123),(38,136,52,122),(39,135,53,121),(40,134,54,120),(41,140,55,126),(42,139,56,125),(57,159,71,145),(58,158,72,144),(59,157,73,143),(60,156,74,142),(61,155,75,141),(62,161,76,147),(63,160,77,146),(64,166,78,152),(65,165,79,151),(66,164,80,150),(67,163,81,149),(68,162,82,148),(69,168,83,154),(70,167,84,153)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 7A | 7B | 7C | 12A | ··· | 12H | 14A | ··· | 14U | 21A | ··· | 21F | 42A | ··· | 42AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | ··· | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 14 | 14 | 14 | 14 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | D7 | C3×D4 | Dic7 | D14 | C3×D7 | C7⋊D4 | C3×Dic7 | C6×D7 | C3×C7⋊D4 |
| kernel | C3×C23.D7 | C6×Dic7 | C22×C42 | C23.D7 | C2×C42 | C2×Dic7 | C22×C14 | C2×C14 | C42 | C22×C6 | C14 | C2×C6 | C2×C6 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 3 | 4 | 6 | 3 | 6 | 12 | 12 | 6 | 24 |
Matrix representation of C3×C23.D7 ►in GL3(𝔽337) generated by
| 1 | 0 | 0 |
| 0 | 208 | 0 |
| 0 | 0 | 208 |
| 1 | 0 | 0 |
| 0 | 1 | 274 |
| 0 | 0 | 336 |
| 336 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 336 | 0 |
| 0 | 0 | 336 |
| 1 | 0 | 0 |
| 0 | 79 | 33 |
| 0 | 0 | 64 |
| 148 | 0 | 0 |
| 0 | 171 | 254 |
| 0 | 198 | 166 |
G:=sub<GL(3,GF(337))| [1,0,0,0,208,0,0,0,208],[1,0,0,0,1,0,0,274,336],[336,0,0,0,1,0,0,0,1],[1,0,0,0,336,0,0,0,336],[1,0,0,0,79,0,0,33,64],[148,0,0,0,171,198,0,254,166] >;
C3×C23.D7 in GAP, Magma, Sage, TeX
C_3\times C_2^3.D_7
% in TeX
G:=Group("C3xC2^3.D7"); // GroupNames label
G:=SmallGroup(336,73);
// by ID
G=gap.SmallGroup(336,73);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,313,10373]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^7=1,f^2=c,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*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations