Copied to
clipboard

## G = S3×C34order 486 = 2·35

### Direct product of C34 and S3

Aliases: S3×C34, C351C2, C3414C6, C3⋊(C33×C6), C3313(C3×C6), C323(C32×C6), SmallGroup(486,256)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C34
 Chief series C1 — C3 — C32 — C33 — C34 — C35 — S3×C34
 Lower central C3 — S3×C34
 Upper central C1 — C34

Generators and relations for S3×C34
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 3512 in 1968 conjugacy classes, 636 normal (6 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C3×S3, C3×C6, C33, C33, S3×C32, C32×C6, C34, C34, C34, S3×C33, C33×C6, C35, S3×C34
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C33, S3×C32, C32×C6, C34, S3×C33, C33×C6, S3×C34

Smallest permutation representation of S3×C34
On 162 points
Generators in S162
(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)
(1 47 18)(2 48 16)(3 46 17)(4 65 36)(5 66 34)(6 64 35)(7 60 39)(8 58 37)(9 59 38)(10 61 32)(11 62 33)(12 63 31)(13 71 45)(14 72 43)(15 70 44)(19 68 42)(20 69 40)(21 67 41)(22 80 54)(23 81 52)(24 79 53)(25 74 111)(26 75 109)(27 73 110)(28 157 131)(29 158 132)(30 159 130)(49 83 78)(50 84 76)(51 82 77)(55 106 155)(56 107 156)(57 108 154)(85 143 117)(86 144 115)(87 142 116)(88 137 120)(89 138 118)(90 136 119)(91 140 114)(92 141 112)(93 139 113)(94 152 126)(95 153 124)(96 151 125)(97 146 129)(98 147 127)(99 145 128)(100 149 123)(101 150 121)(102 148 122)(103 161 135)(104 162 133)(105 160 134)
(1 59 22)(2 60 23)(3 58 24)(4 51 15)(5 49 13)(6 50 14)(7 52 16)(8 53 17)(9 54 18)(10 111 21)(11 109 19)(12 110 20)(25 67 61)(26 68 62)(27 69 63)(28 153 120)(29 151 118)(30 152 119)(31 73 40)(32 74 41)(33 75 42)(34 78 45)(35 76 43)(36 77 44)(37 79 46)(38 80 47)(39 81 48)(55 102 144)(56 100 142)(57 101 143)(64 84 72)(65 82 70)(66 83 71)(85 154 121)(86 155 122)(87 156 123)(88 157 124)(89 158 125)(90 159 126)(91 160 127)(92 161 128)(93 162 129)(94 136 130)(95 137 131)(96 138 132)(97 139 133)(98 140 134)(99 141 135)(103 145 112)(104 146 113)(105 147 114)(106 148 115)(107 149 116)(108 150 117)
(1 74 4)(2 75 5)(3 73 6)(7 68 78)(8 69 76)(9 67 77)(10 70 80)(11 71 81)(12 72 79)(13 23 33)(14 24 31)(15 22 32)(16 26 34)(17 27 35)(18 25 36)(19 83 39)(20 84 37)(21 82 38)(28 91 101)(29 92 102)(30 93 100)(40 50 58)(41 51 59)(42 49 60)(43 53 63)(44 54 61)(45 52 62)(46 110 64)(47 111 65)(48 109 66)(55 118 128)(56 119 129)(57 120 127)(85 95 105)(86 96 103)(87 94 104)(88 98 108)(89 99 106)(90 97 107)(112 122 132)(113 123 130)(114 121 131)(115 125 135)(116 126 133)(117 124 134)(136 146 156)(137 147 154)(138 145 155)(139 149 159)(140 150 157)(141 148 158)(142 152 162)(143 153 160)(144 151 161)
(1 74 4)(2 75 5)(3 73 6)(7 68 78)(8 69 76)(9 67 77)(10 70 80)(11 71 81)(12 72 79)(13 23 33)(14 24 31)(15 22 32)(16 26 34)(17 27 35)(18 25 36)(19 83 39)(20 84 37)(21 82 38)(28 101 91)(29 102 92)(30 100 93)(40 50 58)(41 51 59)(42 49 60)(43 53 63)(44 54 61)(45 52 62)(46 110 64)(47 111 65)(48 109 66)(55 128 118)(56 129 119)(57 127 120)(85 105 95)(86 103 96)(87 104 94)(88 108 98)(89 106 99)(90 107 97)(112 132 122)(113 130 123)(114 131 121)(115 135 125)(116 133 126)(117 134 124)(136 156 146)(137 154 147)(138 155 145)(139 159 149)(140 157 150)(141 158 148)(142 162 152)(143 160 153)(144 161 151)
(1 145)(2 146)(3 147)(4 138)(5 136)(6 137)(7 139)(8 140)(9 141)(10 144)(11 142)(12 143)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 84)(29 82)(30 83)(31 85)(32 86)(33 87)(34 90)(35 88)(36 89)(37 91)(38 92)(39 93)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 111)(56 109)(57 110)(58 114)(59 112)(60 113)(61 115)(62 116)(63 117)(64 120)(65 118)(66 119)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)

G:=sub<Sym(162)| (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), (1,47,18)(2,48,16)(3,46,17)(4,65,36)(5,66,34)(6,64,35)(7,60,39)(8,58,37)(9,59,38)(10,61,32)(11,62,33)(12,63,31)(13,71,45)(14,72,43)(15,70,44)(19,68,42)(20,69,40)(21,67,41)(22,80,54)(23,81,52)(24,79,53)(25,74,111)(26,75,109)(27,73,110)(28,157,131)(29,158,132)(30,159,130)(49,83,78)(50,84,76)(51,82,77)(55,106,155)(56,107,156)(57,108,154)(85,143,117)(86,144,115)(87,142,116)(88,137,120)(89,138,118)(90,136,119)(91,140,114)(92,141,112)(93,139,113)(94,152,126)(95,153,124)(96,151,125)(97,146,129)(98,147,127)(99,145,128)(100,149,123)(101,150,121)(102,148,122)(103,161,135)(104,162,133)(105,160,134), (1,59,22)(2,60,23)(3,58,24)(4,51,15)(5,49,13)(6,50,14)(7,52,16)(8,53,17)(9,54,18)(10,111,21)(11,109,19)(12,110,20)(25,67,61)(26,68,62)(27,69,63)(28,153,120)(29,151,118)(30,152,119)(31,73,40)(32,74,41)(33,75,42)(34,78,45)(35,76,43)(36,77,44)(37,79,46)(38,80,47)(39,81,48)(55,102,144)(56,100,142)(57,101,143)(64,84,72)(65,82,70)(66,83,71)(85,154,121)(86,155,122)(87,156,123)(88,157,124)(89,158,125)(90,159,126)(91,160,127)(92,161,128)(93,162,129)(94,136,130)(95,137,131)(96,138,132)(97,139,133)(98,140,134)(99,141,135)(103,145,112)(104,146,113)(105,147,114)(106,148,115)(107,149,116)(108,150,117), (1,74,4)(2,75,5)(3,73,6)(7,68,78)(8,69,76)(9,67,77)(10,70,80)(11,71,81)(12,72,79)(13,23,33)(14,24,31)(15,22,32)(16,26,34)(17,27,35)(18,25,36)(19,83,39)(20,84,37)(21,82,38)(28,91,101)(29,92,102)(30,93,100)(40,50,58)(41,51,59)(42,49,60)(43,53,63)(44,54,61)(45,52,62)(46,110,64)(47,111,65)(48,109,66)(55,118,128)(56,119,129)(57,120,127)(85,95,105)(86,96,103)(87,94,104)(88,98,108)(89,99,106)(90,97,107)(112,122,132)(113,123,130)(114,121,131)(115,125,135)(116,126,133)(117,124,134)(136,146,156)(137,147,154)(138,145,155)(139,149,159)(140,150,157)(141,148,158)(142,152,162)(143,153,160)(144,151,161), (1,74,4)(2,75,5)(3,73,6)(7,68,78)(8,69,76)(9,67,77)(10,70,80)(11,71,81)(12,72,79)(13,23,33)(14,24,31)(15,22,32)(16,26,34)(17,27,35)(18,25,36)(19,83,39)(20,84,37)(21,82,38)(28,101,91)(29,102,92)(30,100,93)(40,50,58)(41,51,59)(42,49,60)(43,53,63)(44,54,61)(45,52,62)(46,110,64)(47,111,65)(48,109,66)(55,128,118)(56,129,119)(57,127,120)(85,105,95)(86,103,96)(87,104,94)(88,108,98)(89,106,99)(90,107,97)(112,132,122)(113,130,123)(114,131,121)(115,135,125)(116,133,126)(117,134,124)(136,156,146)(137,154,147)(138,155,145)(139,159,149)(140,157,150)(141,158,148)(142,162,152)(143,160,153)(144,161,151), (1,145)(2,146)(3,147)(4,138)(5,136)(6,137)(7,139)(8,140)(9,141)(10,144)(11,142)(12,143)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,84)(29,82)(30,83)(31,85)(32,86)(33,87)(34,90)(35,88)(36,89)(37,91)(38,92)(39,93)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,111)(56,109)(57,110)(58,114)(59,112)(60,113)(61,115)(62,116)(63,117)(64,120)(65,118)(66,119)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162)>;

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), (1,47,18)(2,48,16)(3,46,17)(4,65,36)(5,66,34)(6,64,35)(7,60,39)(8,58,37)(9,59,38)(10,61,32)(11,62,33)(12,63,31)(13,71,45)(14,72,43)(15,70,44)(19,68,42)(20,69,40)(21,67,41)(22,80,54)(23,81,52)(24,79,53)(25,74,111)(26,75,109)(27,73,110)(28,157,131)(29,158,132)(30,159,130)(49,83,78)(50,84,76)(51,82,77)(55,106,155)(56,107,156)(57,108,154)(85,143,117)(86,144,115)(87,142,116)(88,137,120)(89,138,118)(90,136,119)(91,140,114)(92,141,112)(93,139,113)(94,152,126)(95,153,124)(96,151,125)(97,146,129)(98,147,127)(99,145,128)(100,149,123)(101,150,121)(102,148,122)(103,161,135)(104,162,133)(105,160,134), (1,59,22)(2,60,23)(3,58,24)(4,51,15)(5,49,13)(6,50,14)(7,52,16)(8,53,17)(9,54,18)(10,111,21)(11,109,19)(12,110,20)(25,67,61)(26,68,62)(27,69,63)(28,153,120)(29,151,118)(30,152,119)(31,73,40)(32,74,41)(33,75,42)(34,78,45)(35,76,43)(36,77,44)(37,79,46)(38,80,47)(39,81,48)(55,102,144)(56,100,142)(57,101,143)(64,84,72)(65,82,70)(66,83,71)(85,154,121)(86,155,122)(87,156,123)(88,157,124)(89,158,125)(90,159,126)(91,160,127)(92,161,128)(93,162,129)(94,136,130)(95,137,131)(96,138,132)(97,139,133)(98,140,134)(99,141,135)(103,145,112)(104,146,113)(105,147,114)(106,148,115)(107,149,116)(108,150,117), (1,74,4)(2,75,5)(3,73,6)(7,68,78)(8,69,76)(9,67,77)(10,70,80)(11,71,81)(12,72,79)(13,23,33)(14,24,31)(15,22,32)(16,26,34)(17,27,35)(18,25,36)(19,83,39)(20,84,37)(21,82,38)(28,91,101)(29,92,102)(30,93,100)(40,50,58)(41,51,59)(42,49,60)(43,53,63)(44,54,61)(45,52,62)(46,110,64)(47,111,65)(48,109,66)(55,118,128)(56,119,129)(57,120,127)(85,95,105)(86,96,103)(87,94,104)(88,98,108)(89,99,106)(90,97,107)(112,122,132)(113,123,130)(114,121,131)(115,125,135)(116,126,133)(117,124,134)(136,146,156)(137,147,154)(138,145,155)(139,149,159)(140,150,157)(141,148,158)(142,152,162)(143,153,160)(144,151,161), (1,74,4)(2,75,5)(3,73,6)(7,68,78)(8,69,76)(9,67,77)(10,70,80)(11,71,81)(12,72,79)(13,23,33)(14,24,31)(15,22,32)(16,26,34)(17,27,35)(18,25,36)(19,83,39)(20,84,37)(21,82,38)(28,101,91)(29,102,92)(30,100,93)(40,50,58)(41,51,59)(42,49,60)(43,53,63)(44,54,61)(45,52,62)(46,110,64)(47,111,65)(48,109,66)(55,128,118)(56,129,119)(57,127,120)(85,105,95)(86,103,96)(87,104,94)(88,108,98)(89,106,99)(90,107,97)(112,132,122)(113,130,123)(114,131,121)(115,135,125)(116,133,126)(117,134,124)(136,156,146)(137,154,147)(138,155,145)(139,159,149)(140,157,150)(141,158,148)(142,162,152)(143,160,153)(144,161,151), (1,145)(2,146)(3,147)(4,138)(5,136)(6,137)(7,139)(8,140)(9,141)(10,144)(11,142)(12,143)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,84)(29,82)(30,83)(31,85)(32,86)(33,87)(34,90)(35,88)(36,89)(37,91)(38,92)(39,93)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,111)(56,109)(57,110)(58,114)(59,112)(60,113)(61,115)(62,116)(63,117)(64,120)(65,118)(66,119)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162) );

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)], [(1,47,18),(2,48,16),(3,46,17),(4,65,36),(5,66,34),(6,64,35),(7,60,39),(8,58,37),(9,59,38),(10,61,32),(11,62,33),(12,63,31),(13,71,45),(14,72,43),(15,70,44),(19,68,42),(20,69,40),(21,67,41),(22,80,54),(23,81,52),(24,79,53),(25,74,111),(26,75,109),(27,73,110),(28,157,131),(29,158,132),(30,159,130),(49,83,78),(50,84,76),(51,82,77),(55,106,155),(56,107,156),(57,108,154),(85,143,117),(86,144,115),(87,142,116),(88,137,120),(89,138,118),(90,136,119),(91,140,114),(92,141,112),(93,139,113),(94,152,126),(95,153,124),(96,151,125),(97,146,129),(98,147,127),(99,145,128),(100,149,123),(101,150,121),(102,148,122),(103,161,135),(104,162,133),(105,160,134)], [(1,59,22),(2,60,23),(3,58,24),(4,51,15),(5,49,13),(6,50,14),(7,52,16),(8,53,17),(9,54,18),(10,111,21),(11,109,19),(12,110,20),(25,67,61),(26,68,62),(27,69,63),(28,153,120),(29,151,118),(30,152,119),(31,73,40),(32,74,41),(33,75,42),(34,78,45),(35,76,43),(36,77,44),(37,79,46),(38,80,47),(39,81,48),(55,102,144),(56,100,142),(57,101,143),(64,84,72),(65,82,70),(66,83,71),(85,154,121),(86,155,122),(87,156,123),(88,157,124),(89,158,125),(90,159,126),(91,160,127),(92,161,128),(93,162,129),(94,136,130),(95,137,131),(96,138,132),(97,139,133),(98,140,134),(99,141,135),(103,145,112),(104,146,113),(105,147,114),(106,148,115),(107,149,116),(108,150,117)], [(1,74,4),(2,75,5),(3,73,6),(7,68,78),(8,69,76),(9,67,77),(10,70,80),(11,71,81),(12,72,79),(13,23,33),(14,24,31),(15,22,32),(16,26,34),(17,27,35),(18,25,36),(19,83,39),(20,84,37),(21,82,38),(28,91,101),(29,92,102),(30,93,100),(40,50,58),(41,51,59),(42,49,60),(43,53,63),(44,54,61),(45,52,62),(46,110,64),(47,111,65),(48,109,66),(55,118,128),(56,119,129),(57,120,127),(85,95,105),(86,96,103),(87,94,104),(88,98,108),(89,99,106),(90,97,107),(112,122,132),(113,123,130),(114,121,131),(115,125,135),(116,126,133),(117,124,134),(136,146,156),(137,147,154),(138,145,155),(139,149,159),(140,150,157),(141,148,158),(142,152,162),(143,153,160),(144,151,161)], [(1,74,4),(2,75,5),(3,73,6),(7,68,78),(8,69,76),(9,67,77),(10,70,80),(11,71,81),(12,72,79),(13,23,33),(14,24,31),(15,22,32),(16,26,34),(17,27,35),(18,25,36),(19,83,39),(20,84,37),(21,82,38),(28,101,91),(29,102,92),(30,100,93),(40,50,58),(41,51,59),(42,49,60),(43,53,63),(44,54,61),(45,52,62),(46,110,64),(47,111,65),(48,109,66),(55,128,118),(56,129,119),(57,127,120),(85,105,95),(86,103,96),(87,104,94),(88,108,98),(89,106,99),(90,107,97),(112,132,122),(113,130,123),(114,131,121),(115,135,125),(116,133,126),(117,134,124),(136,156,146),(137,154,147),(138,155,145),(139,159,149),(140,157,150),(141,158,148),(142,162,152),(143,160,153),(144,161,151)], [(1,145),(2,146),(3,147),(4,138),(5,136),(6,137),(7,139),(8,140),(9,141),(10,144),(11,142),(12,143),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,84),(29,82),(30,83),(31,85),(32,86),(33,87),(34,90),(35,88),(36,89),(37,91),(38,92),(39,93),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,130),(50,131),(51,132),(52,133),(53,134),(54,135),(55,111),(56,109),(57,110),(58,114),(59,112),(60,113),(61,115),(62,116),(63,117),(64,120),(65,118),(66,119),(67,148),(68,149),(69,150),(70,151),(71,152),(72,153),(73,154),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,161),(81,162)]])

243 conjugacy classes

 class 1 2 3A ··· 3CB 3CC ··· 3FE 6A ··· 6CB order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 3 1 ··· 1 2 ··· 2 3 ··· 3

243 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 S3 C3×S3 kernel S3×C34 C35 S3×C33 C34 C34 C33 # reps 1 1 80 80 1 80

Matrix representation of S3×C34 in GL5(𝔽7)

 1 0 0 0 0 0 4 0 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 1
,
 2 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 2 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 2
,
 6 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(7))| [1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,2],[6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

S3×C34 in GAP, Magma, Sage, TeX

S_3\times C_3^4
% in TeX

G:=Group("S3xC3^4");
// GroupNames label

G:=SmallGroup(486,256);
// by ID

G=gap.SmallGroup(486,256);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,11669]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,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=e^-1>;
// generators/relations

׿
×
𝔽