direct product, metabelian, supersoluble, monomial, A-group
Aliases: C7×C33⋊C2, C33⋊3C14, (C3×C21)⋊9S3, C21⋊3(C3⋊S3), C32⋊4(S3×C7), (C32×C21)⋊7C2, C3⋊(C7×C3⋊S3), SmallGroup(378,58)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C7×C33⋊C2 |
Generators and relations for C7×C33⋊C2
G = < a,b,c,d,e | a7=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 424 in 112 conjugacy classes, 58 normal (6 characteristic)
C1, C2, C3, S3, C7, C32, C14, C3⋊S3, C21, C33, S3×C7, C33⋊C2, C3×C21, C7×C3⋊S3, C32×C21, C7×C33⋊C2
Quotients: C1, C2, S3, C7, C14, C3⋊S3, S3×C7, C33⋊C2, C7×C3⋊S3, C7×C33⋊C2
►On 189 points
Generators in S189
(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)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)
(1 80 171)(2 81 172)(3 82 173)(4 83 174)(5 84 175)(6 78 169)(7 79 170)(8 30 180)(9 31 181)(10 32 182)(11 33 176)(12 34 177)(13 35 178)(14 29 179)(15 40 167)(16 41 168)(17 42 162)(18 36 163)(19 37 164)(20 38 165)(21 39 166)(22 127 98)(23 128 92)(24 129 93)(25 130 94)(26 131 95)(27 132 96)(28 133 97)(43 67 134)(44 68 135)(45 69 136)(46 70 137)(47 64 138)(48 65 139)(49 66 140)(50 100 145)(51 101 146)(52 102 147)(53 103 141)(54 104 142)(55 105 143)(56 99 144)(57 155 126)(58 156 120)(59 157 121)(60 158 122)(61 159 123)(62 160 124)(63 161 125)(71 116 108)(72 117 109)(73 118 110)(74 119 111)(75 113 112)(76 114 106)(77 115 107)(85 185 149)(86 186 150)(87 187 151)(88 188 152)(89 189 153)(90 183 154)(91 184 148)
(1 61 44)(2 62 45)(3 63 46)(4 57 47)(5 58 48)(6 59 49)(7 60 43)(8 27 107)(9 28 108)(10 22 109)(11 23 110)(12 24 111)(13 25 112)(14 26 106)(15 147 185)(16 141 186)(17 142 187)(18 143 188)(19 144 189)(20 145 183)(21 146 184)(29 131 76)(30 132 77)(31 133 71)(32 127 72)(33 128 73)(34 129 74)(35 130 75)(36 55 152)(37 56 153)(38 50 154)(39 51 148)(40 52 149)(41 53 150)(42 54 151)(64 83 155)(65 84 156)(66 78 157)(67 79 158)(68 80 159)(69 81 160)(70 82 161)(85 167 102)(86 168 103)(87 162 104)(88 163 105)(89 164 99)(90 165 100)(91 166 101)(92 118 176)(93 119 177)(94 113 178)(95 114 179)(96 115 180)(97 116 181)(98 117 182)(120 139 175)(121 140 169)(122 134 170)(123 135 171)(124 136 172)(125 137 173)(126 138 174)
(1 151 96)(2 152 97)(3 153 98)(4 154 92)(5 148 93)(6 149 94)(7 150 95)(8 68 104)(9 69 105)(10 70 99)(11 64 100)(12 65 101)(13 66 102)(14 67 103)(15 75 121)(16 76 122)(17 77 123)(18 71 124)(19 72 125)(20 73 126)(21 74 120)(22 82 89)(23 83 90)(24 84 91)(25 78 85)(26 79 86)(27 80 87)(28 81 88)(29 134 141)(30 135 142)(31 136 143)(32 137 144)(33 138 145)(34 139 146)(35 140 147)(36 116 62)(37 117 63)(38 118 57)(39 119 58)(40 113 59)(41 114 60)(42 115 61)(43 53 179)(44 54 180)(45 55 181)(46 56 182)(47 50 176)(48 51 177)(49 52 178)(106 158 168)(107 159 162)(108 160 163)(109 161 164)(110 155 165)(111 156 166)(112 157 167)(127 173 189)(128 174 183)(129 175 184)(130 169 185)(131 170 186)(132 171 187)(133 172 188)
(8 17)(9 18)(10 19)(11 20)(12 21)(13 15)(14 16)(22 189)(23 183)(24 184)(25 185)(26 186)(27 187)(28 188)(29 168)(30 162)(31 163)(32 164)(33 165)(34 166)(35 167)(36 181)(37 182)(38 176)(39 177)(40 178)(41 179)(42 180)(43 60)(44 61)(45 62)(46 63)(47 57)(48 58)(49 59)(50 118)(51 119)(52 113)(53 114)(54 115)(55 116)(56 117)(64 126)(65 120)(66 121)(67 122)(68 123)(69 124)(70 125)(71 105)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 169)(79 170)(80 171)(81 172)(82 173)(83 174)(84 175)(85 130)(86 131)(87 132)(88 133)(89 127)(90 128)(91 129)(92 154)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(106 141)(107 142)(108 143)(109 144)(110 145)(111 146)(112 147)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)
G:=sub<Sym(189)| (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)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189), (1,80,171)(2,81,172)(3,82,173)(4,83,174)(5,84,175)(6,78,169)(7,79,170)(8,30,180)(9,31,181)(10,32,182)(11,33,176)(12,34,177)(13,35,178)(14,29,179)(15,40,167)(16,41,168)(17,42,162)(18,36,163)(19,37,164)(20,38,165)(21,39,166)(22,127,98)(23,128,92)(24,129,93)(25,130,94)(26,131,95)(27,132,96)(28,133,97)(43,67,134)(44,68,135)(45,69,136)(46,70,137)(47,64,138)(48,65,139)(49,66,140)(50,100,145)(51,101,146)(52,102,147)(53,103,141)(54,104,142)(55,105,143)(56,99,144)(57,155,126)(58,156,120)(59,157,121)(60,158,122)(61,159,123)(62,160,124)(63,161,125)(71,116,108)(72,117,109)(73,118,110)(74,119,111)(75,113,112)(76,114,106)(77,115,107)(85,185,149)(86,186,150)(87,187,151)(88,188,152)(89,189,153)(90,183,154)(91,184,148), (1,61,44)(2,62,45)(3,63,46)(4,57,47)(5,58,48)(6,59,49)(7,60,43)(8,27,107)(9,28,108)(10,22,109)(11,23,110)(12,24,111)(13,25,112)(14,26,106)(15,147,185)(16,141,186)(17,142,187)(18,143,188)(19,144,189)(20,145,183)(21,146,184)(29,131,76)(30,132,77)(31,133,71)(32,127,72)(33,128,73)(34,129,74)(35,130,75)(36,55,152)(37,56,153)(38,50,154)(39,51,148)(40,52,149)(41,53,150)(42,54,151)(64,83,155)(65,84,156)(66,78,157)(67,79,158)(68,80,159)(69,81,160)(70,82,161)(85,167,102)(86,168,103)(87,162,104)(88,163,105)(89,164,99)(90,165,100)(91,166,101)(92,118,176)(93,119,177)(94,113,178)(95,114,179)(96,115,180)(97,116,181)(98,117,182)(120,139,175)(121,140,169)(122,134,170)(123,135,171)(124,136,172)(125,137,173)(126,138,174), (1,151,96)(2,152,97)(3,153,98)(4,154,92)(5,148,93)(6,149,94)(7,150,95)(8,68,104)(9,69,105)(10,70,99)(11,64,100)(12,65,101)(13,66,102)(14,67,103)(15,75,121)(16,76,122)(17,77,123)(18,71,124)(19,72,125)(20,73,126)(21,74,120)(22,82,89)(23,83,90)(24,84,91)(25,78,85)(26,79,86)(27,80,87)(28,81,88)(29,134,141)(30,135,142)(31,136,143)(32,137,144)(33,138,145)(34,139,146)(35,140,147)(36,116,62)(37,117,63)(38,118,57)(39,119,58)(40,113,59)(41,114,60)(42,115,61)(43,53,179)(44,54,180)(45,55,181)(46,56,182)(47,50,176)(48,51,177)(49,52,178)(106,158,168)(107,159,162)(108,160,163)(109,161,164)(110,155,165)(111,156,166)(112,157,167)(127,173,189)(128,174,183)(129,175,184)(130,169,185)(131,170,186)(132,171,187)(133,172,188), (8,17)(9,18)(10,19)(11,20)(12,21)(13,15)(14,16)(22,189)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,168)(30,162)(31,163)(32,164)(33,165)(34,166)(35,167)(36,181)(37,182)(38,176)(39,177)(40,178)(41,179)(42,180)(43,60)(44,61)(45,62)(46,63)(47,57)(48,58)(49,59)(50,118)(51,119)(52,113)(53,114)(54,115)(55,116)(56,117)(64,126)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,105)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,130)(86,131)(87,132)(88,133)(89,127)(90,128)(91,129)(92,154)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)(112,147)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)>;
G:=Group( (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)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189), (1,80,171)(2,81,172)(3,82,173)(4,83,174)(5,84,175)(6,78,169)(7,79,170)(8,30,180)(9,31,181)(10,32,182)(11,33,176)(12,34,177)(13,35,178)(14,29,179)(15,40,167)(16,41,168)(17,42,162)(18,36,163)(19,37,164)(20,38,165)(21,39,166)(22,127,98)(23,128,92)(24,129,93)(25,130,94)(26,131,95)(27,132,96)(28,133,97)(43,67,134)(44,68,135)(45,69,136)(46,70,137)(47,64,138)(48,65,139)(49,66,140)(50,100,145)(51,101,146)(52,102,147)(53,103,141)(54,104,142)(55,105,143)(56,99,144)(57,155,126)(58,156,120)(59,157,121)(60,158,122)(61,159,123)(62,160,124)(63,161,125)(71,116,108)(72,117,109)(73,118,110)(74,119,111)(75,113,112)(76,114,106)(77,115,107)(85,185,149)(86,186,150)(87,187,151)(88,188,152)(89,189,153)(90,183,154)(91,184,148), (1,61,44)(2,62,45)(3,63,46)(4,57,47)(5,58,48)(6,59,49)(7,60,43)(8,27,107)(9,28,108)(10,22,109)(11,23,110)(12,24,111)(13,25,112)(14,26,106)(15,147,185)(16,141,186)(17,142,187)(18,143,188)(19,144,189)(20,145,183)(21,146,184)(29,131,76)(30,132,77)(31,133,71)(32,127,72)(33,128,73)(34,129,74)(35,130,75)(36,55,152)(37,56,153)(38,50,154)(39,51,148)(40,52,149)(41,53,150)(42,54,151)(64,83,155)(65,84,156)(66,78,157)(67,79,158)(68,80,159)(69,81,160)(70,82,161)(85,167,102)(86,168,103)(87,162,104)(88,163,105)(89,164,99)(90,165,100)(91,166,101)(92,118,176)(93,119,177)(94,113,178)(95,114,179)(96,115,180)(97,116,181)(98,117,182)(120,139,175)(121,140,169)(122,134,170)(123,135,171)(124,136,172)(125,137,173)(126,138,174), (1,151,96)(2,152,97)(3,153,98)(4,154,92)(5,148,93)(6,149,94)(7,150,95)(8,68,104)(9,69,105)(10,70,99)(11,64,100)(12,65,101)(13,66,102)(14,67,103)(15,75,121)(16,76,122)(17,77,123)(18,71,124)(19,72,125)(20,73,126)(21,74,120)(22,82,89)(23,83,90)(24,84,91)(25,78,85)(26,79,86)(27,80,87)(28,81,88)(29,134,141)(30,135,142)(31,136,143)(32,137,144)(33,138,145)(34,139,146)(35,140,147)(36,116,62)(37,117,63)(38,118,57)(39,119,58)(40,113,59)(41,114,60)(42,115,61)(43,53,179)(44,54,180)(45,55,181)(46,56,182)(47,50,176)(48,51,177)(49,52,178)(106,158,168)(107,159,162)(108,160,163)(109,161,164)(110,155,165)(111,156,166)(112,157,167)(127,173,189)(128,174,183)(129,175,184)(130,169,185)(131,170,186)(132,171,187)(133,172,188), (8,17)(9,18)(10,19)(11,20)(12,21)(13,15)(14,16)(22,189)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,168)(30,162)(31,163)(32,164)(33,165)(34,166)(35,167)(36,181)(37,182)(38,176)(39,177)(40,178)(41,179)(42,180)(43,60)(44,61)(45,62)(46,63)(47,57)(48,58)(49,59)(50,118)(51,119)(52,113)(53,114)(54,115)(55,116)(56,117)(64,126)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,105)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,130)(86,131)(87,132)(88,133)(89,127)(90,128)(91,129)(92,154)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)(112,147)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157) );
G=PermutationGroup([[(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),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189)], [(1,80,171),(2,81,172),(3,82,173),(4,83,174),(5,84,175),(6,78,169),(7,79,170),(8,30,180),(9,31,181),(10,32,182),(11,33,176),(12,34,177),(13,35,178),(14,29,179),(15,40,167),(16,41,168),(17,42,162),(18,36,163),(19,37,164),(20,38,165),(21,39,166),(22,127,98),(23,128,92),(24,129,93),(25,130,94),(26,131,95),(27,132,96),(28,133,97),(43,67,134),(44,68,135),(45,69,136),(46,70,137),(47,64,138),(48,65,139),(49,66,140),(50,100,145),(51,101,146),(52,102,147),(53,103,141),(54,104,142),(55,105,143),(56,99,144),(57,155,126),(58,156,120),(59,157,121),(60,158,122),(61,159,123),(62,160,124),(63,161,125),(71,116,108),(72,117,109),(73,118,110),(74,119,111),(75,113,112),(76,114,106),(77,115,107),(85,185,149),(86,186,150),(87,187,151),(88,188,152),(89,189,153),(90,183,154),(91,184,148)], [(1,61,44),(2,62,45),(3,63,46),(4,57,47),(5,58,48),(6,59,49),(7,60,43),(8,27,107),(9,28,108),(10,22,109),(11,23,110),(12,24,111),(13,25,112),(14,26,106),(15,147,185),(16,141,186),(17,142,187),(18,143,188),(19,144,189),(20,145,183),(21,146,184),(29,131,76),(30,132,77),(31,133,71),(32,127,72),(33,128,73),(34,129,74),(35,130,75),(36,55,152),(37,56,153),(38,50,154),(39,51,148),(40,52,149),(41,53,150),(42,54,151),(64,83,155),(65,84,156),(66,78,157),(67,79,158),(68,80,159),(69,81,160),(70,82,161),(85,167,102),(86,168,103),(87,162,104),(88,163,105),(89,164,99),(90,165,100),(91,166,101),(92,118,176),(93,119,177),(94,113,178),(95,114,179),(96,115,180),(97,116,181),(98,117,182),(120,139,175),(121,140,169),(122,134,170),(123,135,171),(124,136,172),(125,137,173),(126,138,174)], [(1,151,96),(2,152,97),(3,153,98),(4,154,92),(5,148,93),(6,149,94),(7,150,95),(8,68,104),(9,69,105),(10,70,99),(11,64,100),(12,65,101),(13,66,102),(14,67,103),(15,75,121),(16,76,122),(17,77,123),(18,71,124),(19,72,125),(20,73,126),(21,74,120),(22,82,89),(23,83,90),(24,84,91),(25,78,85),(26,79,86),(27,80,87),(28,81,88),(29,134,141),(30,135,142),(31,136,143),(32,137,144),(33,138,145),(34,139,146),(35,140,147),(36,116,62),(37,117,63),(38,118,57),(39,119,58),(40,113,59),(41,114,60),(42,115,61),(43,53,179),(44,54,180),(45,55,181),(46,56,182),(47,50,176),(48,51,177),(49,52,178),(106,158,168),(107,159,162),(108,160,163),(109,161,164),(110,155,165),(111,156,166),(112,157,167),(127,173,189),(128,174,183),(129,175,184),(130,169,185),(131,170,186),(132,171,187),(133,172,188)], [(8,17),(9,18),(10,19),(11,20),(12,21),(13,15),(14,16),(22,189),(23,183),(24,184),(25,185),(26,186),(27,187),(28,188),(29,168),(30,162),(31,163),(32,164),(33,165),(34,166),(35,167),(36,181),(37,182),(38,176),(39,177),(40,178),(41,179),(42,180),(43,60),(44,61),(45,62),(46,63),(47,57),(48,58),(49,59),(50,118),(51,119),(52,113),(53,114),(54,115),(55,116),(56,117),(64,126),(65,120),(66,121),(67,122),(68,123),(69,124),(70,125),(71,105),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,169),(79,170),(80,171),(81,172),(82,173),(83,174),(84,175),(85,130),(86,131),(87,132),(88,133),(89,127),(90,128),(91,129),(92,154),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(106,141),(107,142),(108,143),(109,144),(110,145),(111,146),(112,147),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157)]])
105 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3M | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21BZ |
| order | 1 | 2 | 3 | ··· | 3 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 |
| size | 1 | 27 | 2 | ··· | 2 | 1 | ··· | 1 | 27 | ··· | 27 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | S3 | S3×C7 |
| kernel | C7×C33⋊C2 | C32×C21 | C33⋊C2 | C33 | C3×C21 | C32 |
| # reps | 1 | 1 | 6 | 6 | 13 | 78 |
Matrix representation of C7×C33⋊C2 ►in GL6(𝔽43)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 0 | 0 |
| 0 | 0 | 0 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 42 | 1 | 0 | 0 | 0 | 0 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 3 | 0 | 0 |
| 0 | 0 | 42 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 0 | 0 | 0 | 0 | 1 | 42 |
| 42 | 1 | 0 | 0 | 0 | 0 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 3 | 0 | 0 |
| 0 | 0 | 42 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 42 | 1 | 0 | 0 | 0 | 0 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 1 |
| 0 | 0 | 0 | 0 | 42 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(43))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,21,0,0,0,0,0,0,21,0,0,0,0,0,0,35,0,0,0,0,0,0,35],[42,42,0,0,0,0,1,0,0,0,0,0,0,0,41,42,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,42,42],[42,42,0,0,0,0,1,0,0,0,0,0,0,0,41,42,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,42,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,42,0,0,0,0,1,0],[0,42,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C7×C33⋊C2 in GAP, Magma, Sage, TeX
C_7\times C_3^3\rtimes C_2
% in TeX
G:=Group("C7xC3^3:C2"); // GroupNames label
G:=SmallGroup(378,58);
// by ID
G=gap.SmallGroup(378,58);
# by ID
G:=PCGroup([5,-2,-7,-3,-3,-3,422,1683,6304]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations