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## G = C2×C32.19He3order 486 = 2·35

### Direct product of C2 and C32.19He3

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C32.19He3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32.19He3 — C2×C32.19He3
 Lower central C1 — C3 — C32 — C2×C32.19He3
 Upper central C1 — C3×C6 — C32×C6 — C2×C32.19He3

Generators and relations for C2×C32.19He3
G = < a,b,c,d,e,f | a2=b3=c3=e3=1, d3=c, f3=b, 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, fdf-1=cde-1, fef-1=c-1e >

Subgroups: 180 in 72 conjugacy classes, 36 normal (14 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C33, C3×C18, C3×C18, C32×C6, C32⋊C9, C32×C9, C2×C32⋊C9, C32×C18, C32.19He3, C2×C32.19He3
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, He3, 3- 1+2, C3×C18, C2×He3, C2×3- 1+2, C32⋊C9, He3.C3, C2×C32⋊C9, C2×He3.C3, C32.19He3, C2×C32.19He3

Smallest permutation representation of C2×C32.19He3
On 162 points
Generators in S162
(1 115)(2 116)(3 117)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 88)(11 89)(12 90)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 97)(20 98)(21 99)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 108)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(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 44 32)(2 45 33)(3 37 34)(4 38 35)(5 39 36)(6 40 28)(7 41 29)(8 42 30)(9 43 31)(10 26 162)(11 27 154)(12 19 155)(13 20 156)(14 21 157)(15 22 158)(16 23 159)(17 24 160)(18 25 161)(46 56 67)(47 57 68)(48 58 69)(49 59 70)(50 60 71)(51 61 72)(52 62 64)(53 63 65)(54 55 66)(73 89 96)(74 90 97)(75 82 98)(76 83 99)(77 84 91)(78 85 92)(79 86 93)(80 87 94)(81 88 95)(100 112 122)(101 113 123)(102 114 124)(103 115 125)(104 116 126)(105 117 118)(106 109 119)(107 110 120)(108 111 121)(127 137 148)(128 138 149)(129 139 150)(130 140 151)(131 141 152)(132 142 153)(133 143 145)(134 144 146)(135 136 147)
(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 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)
(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(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 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(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)(154 160 157)(155 161 158)(156 162 159)
(1 92 70 44 78 49 32 85 59)(2 99 71 45 76 50 33 83 60)(3 97 72 37 74 51 34 90 61)(4 95 64 38 81 52 35 88 62)(5 93 65 39 79 53 36 86 63)(6 91 66 40 77 54 28 84 55)(7 98 67 41 75 46 29 82 56)(8 96 68 42 73 47 30 89 57)(9 94 69 43 80 48 31 87 58)(10 143 109 26 145 119 162 133 106)(11 138 113 27 149 123 154 128 101)(12 142 117 19 153 118 155 132 105)(13 137 112 20 148 122 156 127 100)(14 141 116 21 152 126 157 131 104)(15 136 111 22 147 121 158 135 108)(16 140 115 23 151 125 159 130 103)(17 144 110 24 146 120 160 134 107)(18 139 114 25 150 124 161 129 102)

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

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

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

102 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 6A ··· 6H 6I ··· 6N 9A ··· 9R 9S ··· 9AJ 18A ··· 18R 18S ··· 18AJ order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 C9 C18 He3 3- 1+2 C2×He3 C2×3- 1+2 He3.C3 C2×He3.C3 kernel C2×C32.19He3 C32.19He3 C2×C32⋊C9 C32×C18 C32⋊C9 C32×C9 C3×C18 C3×C9 C3×C6 C3×C6 C32 C32 C6 C3 # reps 1 1 6 2 6 2 18 18 2 4 2 4 18 18

Matrix representation of C2×C32.19He3 in GL4(𝔽19) generated by

 1 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 7 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 11 0 0 0 0 4 0 0 0 0 4 0 0 8 0 9
,
 1 0 0 0 0 1 0 0 0 9 7 0 0 16 0 11
,
 6 0 0 0 0 9 6 0 0 9 10 1 0 13 17 0
G:=sub<GL(4,GF(19))| [1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[7,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[11,0,0,0,0,4,0,8,0,0,4,0,0,0,0,9],[1,0,0,0,0,1,9,16,0,0,7,0,0,0,0,11],[6,0,0,0,0,9,9,13,0,6,10,17,0,0,1,0] >;

C2×C32.19He3 in GAP, Magma, Sage, TeX

C_2\times C_3^2._{19}{\rm He}_3
% in TeX

G:=Group("C2xC3^2.19He3");
// GroupNames label

G:=SmallGroup(486,74);
// by ID

G=gap.SmallGroup(486,74);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,500,2169]);
// Polycyclic

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

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