direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×C32.29He3, (C3×C6).19He3, C33.6(C3×C6), C32⋊C9.12C6, C6.4(He3.C3), (C32×C6).6C32, C32.34(C2×He3), C6.3(He3⋊C3), C6.3(C3.He3), (C2×C32⋊C9).3C3, C3.7(C2×He3.C3), C3.5(C2×He3⋊C3), C3.5(C2×C3.He3), SmallGroup(486,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32.29He3
G = < a,b,c,d,e,f | a2=b3=c3=e3=1, d3=b, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ede-1=cd=dc, ce=ec, cf=fc, fdf-1=bde-1, fef-1=b-1e >
Smallest permutation representation of C2×C32.29He3
►On 162 points
Generators in S162
(1 106)(2 107)(3 108)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 82)(18 83)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 91)(27 92)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(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)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(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)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)(145 148 151)(146 149 152)(147 150 153)(154 157 160)(155 158 161)(156 159 162)
(1 43 33)(2 44 34)(3 45 35)(4 37 36)(5 38 28)(6 39 29)(7 40 30)(8 41 31)(9 42 32)(10 161 26)(11 162 27)(12 154 19)(13 155 20)(14 156 21)(15 157 22)(16 158 23)(17 159 24)(18 160 25)(46 56 69)(47 57 70)(48 58 71)(49 59 72)(50 60 64)(51 61 65)(52 62 66)(53 63 67)(54 55 68)(73 93 86)(74 94 87)(75 95 88)(76 96 89)(77 97 90)(78 98 82)(79 99 83)(80 91 84)(81 92 85)(100 118 117)(101 119 109)(102 120 110)(103 121 111)(104 122 112)(105 123 113)(106 124 114)(107 125 115)(108 126 116)(127 137 150)(128 138 151)(129 139 152)(130 140 153)(131 141 145)(132 142 146)(133 143 147)(134 144 148)(135 136 149)
(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)
(2 34 44)(3 45 35)(5 28 38)(6 39 29)(8 31 41)(9 42 32)(10 158 20)(11 17 14)(12 25 157)(13 161 23)(15 19 160)(16 155 26)(18 22 154)(21 27 24)(46 49 52)(47 64 63)(48 61 68)(50 67 57)(51 55 71)(53 70 60)(54 58 65)(56 59 62)(66 69 72)(73 83 96)(74 91 90)(75 81 78)(76 86 99)(77 94 84)(79 89 93)(80 97 87)(82 88 85)(92 98 95)(101 109 119)(102 120 110)(104 112 122)(105 123 113)(107 115 125)(108 126 116)(127 130 133)(128 145 144)(129 142 149)(131 148 138)(132 136 152)(134 151 141)(135 139 146)(137 140 143)(147 150 153)(156 162 159)
(1 85 62 43 81 66 33 92 52)(2 99 67 44 83 53 34 79 63)(3 90 55 45 77 68 35 97 54)(4 88 56 37 75 69 36 95 46)(5 93 70 38 86 47 28 73 57)(6 84 58 39 80 71 29 91 48)(7 82 59 40 78 72 30 98 49)(8 96 64 41 89 50 31 76 60)(9 87 61 42 74 65 32 94 51)(10 139 120 161 152 110 26 129 102)(11 143 124 162 147 114 27 133 106)(12 128 109 154 138 101 19 151 119)(13 142 123 155 146 113 20 132 105)(14 137 118 156 150 117 21 127 100)(15 131 112 157 141 104 22 145 122)(16 136 126 158 149 116 23 135 108)(17 140 121 159 153 111 24 130 103)(18 134 115 160 144 107 25 148 125)
G:=sub<Sym(162)| (1,106)(2,107)(3,108)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,82)(18,83)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,91)(27,92)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(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)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(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), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144)(145,148,151)(146,149,152)(147,150,153)(154,157,160)(155,158,161)(156,159,162), (1,43,33)(2,44,34)(3,45,35)(4,37,36)(5,38,28)(6,39,29)(7,40,30)(8,41,31)(9,42,32)(10,161,26)(11,162,27)(12,154,19)(13,155,20)(14,156,21)(15,157,22)(16,158,23)(17,159,24)(18,160,25)(46,56,69)(47,57,70)(48,58,71)(49,59,72)(50,60,64)(51,61,65)(52,62,66)(53,63,67)(54,55,68)(73,93,86)(74,94,87)(75,95,88)(76,96,89)(77,97,90)(78,98,82)(79,99,83)(80,91,84)(81,92,85)(100,118,117)(101,119,109)(102,120,110)(103,121,111)(104,122,112)(105,123,113)(106,124,114)(107,125,115)(108,126,116)(127,137,150)(128,138,151)(129,139,152)(130,140,153)(131,141,145)(132,142,146)(133,143,147)(134,144,148)(135,136,149), (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), (2,34,44)(3,45,35)(5,28,38)(6,39,29)(8,31,41)(9,42,32)(10,158,20)(11,17,14)(12,25,157)(13,161,23)(15,19,160)(16,155,26)(18,22,154)(21,27,24)(46,49,52)(47,64,63)(48,61,68)(50,67,57)(51,55,71)(53,70,60)(54,58,65)(56,59,62)(66,69,72)(73,83,96)(74,91,90)(75,81,78)(76,86,99)(77,94,84)(79,89,93)(80,97,87)(82,88,85)(92,98,95)(101,109,119)(102,120,110)(104,112,122)(105,123,113)(107,115,125)(108,126,116)(127,130,133)(128,145,144)(129,142,149)(131,148,138)(132,136,152)(134,151,141)(135,139,146)(137,140,143)(147,150,153)(156,162,159), (1,85,62,43,81,66,33,92,52)(2,99,67,44,83,53,34,79,63)(3,90,55,45,77,68,35,97,54)(4,88,56,37,75,69,36,95,46)(5,93,70,38,86,47,28,73,57)(6,84,58,39,80,71,29,91,48)(7,82,59,40,78,72,30,98,49)(8,96,64,41,89,50,31,76,60)(9,87,61,42,74,65,32,94,51)(10,139,120,161,152,110,26,129,102)(11,143,124,162,147,114,27,133,106)(12,128,109,154,138,101,19,151,119)(13,142,123,155,146,113,20,132,105)(14,137,118,156,150,117,21,127,100)(15,131,112,157,141,104,22,145,122)(16,136,126,158,149,116,23,135,108)(17,140,121,159,153,111,24,130,103)(18,134,115,160,144,107,25,148,125)>;
G:=Group( (1,106)(2,107)(3,108)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,82)(18,83)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,91)(27,92)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(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)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(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), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144)(145,148,151)(146,149,152)(147,150,153)(154,157,160)(155,158,161)(156,159,162), (1,43,33)(2,44,34)(3,45,35)(4,37,36)(5,38,28)(6,39,29)(7,40,30)(8,41,31)(9,42,32)(10,161,26)(11,162,27)(12,154,19)(13,155,20)(14,156,21)(15,157,22)(16,158,23)(17,159,24)(18,160,25)(46,56,69)(47,57,70)(48,58,71)(49,59,72)(50,60,64)(51,61,65)(52,62,66)(53,63,67)(54,55,68)(73,93,86)(74,94,87)(75,95,88)(76,96,89)(77,97,90)(78,98,82)(79,99,83)(80,91,84)(81,92,85)(100,118,117)(101,119,109)(102,120,110)(103,121,111)(104,122,112)(105,123,113)(106,124,114)(107,125,115)(108,126,116)(127,137,150)(128,138,151)(129,139,152)(130,140,153)(131,141,145)(132,142,146)(133,143,147)(134,144,148)(135,136,149), (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), (2,34,44)(3,45,35)(5,28,38)(6,39,29)(8,31,41)(9,42,32)(10,158,20)(11,17,14)(12,25,157)(13,161,23)(15,19,160)(16,155,26)(18,22,154)(21,27,24)(46,49,52)(47,64,63)(48,61,68)(50,67,57)(51,55,71)(53,70,60)(54,58,65)(56,59,62)(66,69,72)(73,83,96)(74,91,90)(75,81,78)(76,86,99)(77,94,84)(79,89,93)(80,97,87)(82,88,85)(92,98,95)(101,109,119)(102,120,110)(104,112,122)(105,123,113)(107,115,125)(108,126,116)(127,130,133)(128,145,144)(129,142,149)(131,148,138)(132,136,152)(134,151,141)(135,139,146)(137,140,143)(147,150,153)(156,162,159), (1,85,62,43,81,66,33,92,52)(2,99,67,44,83,53,34,79,63)(3,90,55,45,77,68,35,97,54)(4,88,56,37,75,69,36,95,46)(5,93,70,38,86,47,28,73,57)(6,84,58,39,80,71,29,91,48)(7,82,59,40,78,72,30,98,49)(8,96,64,41,89,50,31,76,60)(9,87,61,42,74,65,32,94,51)(10,139,120,161,152,110,26,129,102)(11,143,124,162,147,114,27,133,106)(12,128,109,154,138,101,19,151,119)(13,142,123,155,146,113,20,132,105)(14,137,118,156,150,117,21,127,100)(15,131,112,157,141,104,22,145,122)(16,136,126,158,149,116,23,135,108)(17,140,121,159,153,111,24,130,103)(18,134,115,160,144,107,25,148,125) );
G=PermutationGroup([[(1,106),(2,107),(3,108),(4,100),(5,101),(6,102),(7,103),(8,104),(9,105),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,82),(18,83),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,91),(27,92),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(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),(55,136),(56,137),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(64,145),(65,146),(66,147),(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)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,94,97),(92,95,98),(93,96,99),(100,103,106),(101,104,107),(102,105,108),(109,112,115),(110,113,116),(111,114,117),(118,121,124),(119,122,125),(120,123,126),(127,130,133),(128,131,134),(129,132,135),(136,139,142),(137,140,143),(138,141,144),(145,148,151),(146,149,152),(147,150,153),(154,157,160),(155,158,161),(156,159,162)], [(1,43,33),(2,44,34),(3,45,35),(4,37,36),(5,38,28),(6,39,29),(7,40,30),(8,41,31),(9,42,32),(10,161,26),(11,162,27),(12,154,19),(13,155,20),(14,156,21),(15,157,22),(16,158,23),(17,159,24),(18,160,25),(46,56,69),(47,57,70),(48,58,71),(49,59,72),(50,60,64),(51,61,65),(52,62,66),(53,63,67),(54,55,68),(73,93,86),(74,94,87),(75,95,88),(76,96,89),(77,97,90),(78,98,82),(79,99,83),(80,91,84),(81,92,85),(100,118,117),(101,119,109),(102,120,110),(103,121,111),(104,122,112),(105,123,113),(106,124,114),(107,125,115),(108,126,116),(127,137,150),(128,138,151),(129,139,152),(130,140,153),(131,141,145),(132,142,146),(133,143,147),(134,144,148),(135,136,149)], [(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)], [(2,34,44),(3,45,35),(5,28,38),(6,39,29),(8,31,41),(9,42,32),(10,158,20),(11,17,14),(12,25,157),(13,161,23),(15,19,160),(16,155,26),(18,22,154),(21,27,24),(46,49,52),(47,64,63),(48,61,68),(50,67,57),(51,55,71),(53,70,60),(54,58,65),(56,59,62),(66,69,72),(73,83,96),(74,91,90),(75,81,78),(76,86,99),(77,94,84),(79,89,93),(80,97,87),(82,88,85),(92,98,95),(101,109,119),(102,120,110),(104,112,122),(105,123,113),(107,115,125),(108,126,116),(127,130,133),(128,145,144),(129,142,149),(131,148,138),(132,136,152),(134,151,141),(135,139,146),(137,140,143),(147,150,153),(156,162,159)], [(1,85,62,43,81,66,33,92,52),(2,99,67,44,83,53,34,79,63),(3,90,55,45,77,68,35,97,54),(4,88,56,37,75,69,36,95,46),(5,93,70,38,86,47,28,73,57),(6,84,58,39,80,71,29,91,48),(7,82,59,40,78,72,30,98,49),(8,96,64,41,89,50,31,76,60),(9,87,61,42,74,65,32,94,51),(10,139,120,161,152,110,26,129,102),(11,143,124,162,147,114,27,133,106),(12,128,109,154,138,101,19,151,119),(13,142,123,155,146,113,20,132,105),(14,137,118,156,150,117,21,127,100),(15,131,112,157,141,104,22,145,122),(16,136,126,158,149,116,23,135,108),(17,140,121,159,153,111,24,130,103),(18,134,115,160,144,107,25,148,125)]])
70 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | 3J | 6A | ··· | 6H | 6I | 6J | 9A | ··· | 9X | 18A | ··· | 18X |
| order | 1 | 2 | 3 | ··· | 3 | 3 | 3 | 6 | ··· | 6 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 9 | 9 | 1 | ··· | 1 | 9 | 9 | 9 | ··· | 9 | 9 | ··· | 9 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | He3 | C2×He3 | He3.C3 | He3⋊C3 | C3.He3 | C2×He3.C3 | C2×He3⋊C3 | C2×C3.He3 |
| kernel | C2×C32.29He3 | C32.29He3 | C2×C32⋊C9 | C32⋊C9 | C3×C6 | C32 | C6 | C6 | C6 | C3 | C3 | C3 |
| # reps | 1 | 1 | 8 | 8 | 2 | 2 | 6 | 6 | 12 | 6 | 6 | 12 |
Matrix representation of C2×C32.29He3 ►in GL7(𝔽19)
| 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(7,GF(19))| [18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,5] >;
C2×C32.29He3 in GAP, Magma, Sage, TeX
C_2\times C_3^2._{29}{\rm He}_3 % in TeX
G:=Group("C2xC3^2.29He3"); // GroupNames label
G:=SmallGroup(486,68);
// by ID
G=gap.SmallGroup(486,68);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,224,176,873,735]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=e^3=1,d^3=b,f^3=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,b*d=d*b,b*e=e*b,b*f=f*b,e*d*e^-1=c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=b*d*e^-1,f*e*f^-1=b^-1*e>;
// generators/relations
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