Copied to
clipboard

## G = C2×C32.27He3order 486 = 2·35

### Direct product of C2 and C32.27He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×C32.27He3
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32.27He3 — C2×C32.27He3
 Lower central C1 — C32 — C33 — C2×C32.27He3
 Upper central C1 — C3×C6 — C32×C6 — C2×C32.27He3

Generators and relations for C2×C32.27He3
G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=1, f3=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bc-1, fbf-1=bd-1, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bc-1d-1e >

Subgroups: 288 in 72 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C18, C3×C6, C3×C6, C3×C9, He3, C33, C33, C3×C18, C2×He3, C32×C6, C32×C6, C32⋊C9, C32⋊C9, C3×He3, C2×C32⋊C9, C2×C32⋊C9, C6×He3, C32.27He3, C2×C32.27He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C2×He3, C3≀C3, He3.C3, He3⋊C3, C2×C3≀C3, C2×He3.C3, C2×He3⋊C3, C32.27He3, C2×C32.27He3

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

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

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

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

70 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3P 6A ··· 6H 6I ··· 6P 9A ··· 9R 18A ··· 18R 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 1 1 3 3 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 C3≀C3 He3.C3 He3⋊C3 C2×C3≀C3 C2×He3.C3 C2×He3⋊C3 kernel C2×C32.27He3 C32.27He3 C2×C32⋊C9 C6×He3 C32⋊C9 C3×He3 C3×C6 C32 C6 C6 C6 C3 C3 C3 # reps 1 1 6 2 6 2 2 2 6 12 6 6 12 6

Matrix representation of C2×C32.27He3 in GL6(𝔽19)

 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7
,
 6 0 0 0 0 0 0 6 0 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C2×C32.27He3 in GAP, Magma, Sage, TeX

C_2\times C_3^2._{27}{\rm He}_3
% in TeX

G:=Group("C2xC3^2.27He3");
// GroupNames label

G:=SmallGroup(486,66);
// by ID

G=gap.SmallGroup(486,66);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,338,873,735]);
// Polycyclic

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

׿
×
𝔽