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## G = C2×C34⋊C2order 324 = 22·34

### Direct product of C2 and C34⋊C2

Aliases: C2×C34⋊C2, C3321D6, C3410C22, C6⋊(C33⋊C2), (C33×C6)⋊4C2, (C32×C6)⋊9S3, (C3×C6)⋊4(C3⋊S3), C329(C2×C3⋊S3), C32(C2×C33⋊C2), SmallGroup(324,175)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — C2×C34⋊C2
 Chief series C1 — C3 — C32 — C33 — C34 — C34⋊C2 — C2×C34⋊C2
 Lower central C34 — C2×C34⋊C2
 Upper central C1 — C2

Generators and relations for C2×C34⋊C2
G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 7780 in 1060 conjugacy classes, 427 normal (5 characteristic)
C1, C2, C2 [×2], C3 [×40], C22, S3 [×80], C6 [×40], C32 [×130], D6 [×40], C3⋊S3 [×260], C3×C6 [×130], C33 [×40], C2×C3⋊S3 [×130], C33⋊C2 [×80], C32×C6 [×40], C34, C2×C33⋊C2 [×40], C34⋊C2 [×2], C33×C6, C2×C34⋊C2
Quotients: C1, C2 [×3], C22, S3 [×40], D6 [×40], C3⋊S3 [×130], C2×C3⋊S3 [×130], C33⋊C2 [×40], C2×C33⋊C2 [×40], C34⋊C2, C2×C34⋊C2

Smallest permutation representation of C2×C34⋊C2
On 162 points
Generators in S162
(1 86)(2 87)(3 85)(4 137)(5 138)(6 136)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(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)(31 112)(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)(55 111)(56 109)(57 110)(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 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 11 8)(2 12 9)(3 10 7)(4 63 60)(5 61 58)(6 62 59)(13 19 16)(14 20 17)(15 21 18)(22 82 25)(23 83 26)(24 84 27)(28 108 105)(29 106 103)(30 107 104)(31 37 34)(32 38 35)(33 39 36)(40 46 43)(41 47 44)(42 48 45)(49 109 52)(50 110 53)(51 111 54)(55 135 132)(56 133 130)(57 134 131)(64 71 68)(65 72 69)(66 70 67)(73 79 76)(74 80 77)(75 81 78)(85 91 88)(86 92 89)(87 93 90)(94 100 97)(95 101 98)(96 102 99)(112 118 115)(113 119 116)(114 120 117)(121 127 124)(122 128 125)(123 129 126)(136 143 140)(137 144 141)(138 142 139)(145 152 149)(146 153 150)(147 151 148)(154 160 157)(155 161 158)(156 162 159)
(1 59 32)(2 60 33)(3 58 31)(4 39 12)(5 37 10)(6 38 11)(7 61 34)(8 62 35)(9 63 36)(13 67 40)(14 68 41)(15 69 42)(16 70 43)(17 71 44)(18 72 45)(19 66 46)(20 64 47)(21 65 48)(22 73 49)(23 74 50)(24 75 51)(25 76 52)(26 77 53)(27 78 54)(28 162 55)(29 160 56)(30 161 57)(79 109 82)(80 110 83)(81 111 84)(85 139 112)(86 140 113)(87 141 114)(88 142 115)(89 143 116)(90 144 117)(91 138 118)(92 136 119)(93 137 120)(94 148 121)(95 149 122)(96 150 123)(97 151 124)(98 152 125)(99 153 126)(100 147 127)(101 145 128)(102 146 129)(103 154 130)(104 155 131)(105 156 132)(106 157 133)(107 158 134)(108 159 135)
(1 74 14)(2 75 15)(3 73 13)(4 111 65)(5 109 66)(6 110 64)(7 76 16)(8 77 17)(9 78 18)(10 79 19)(11 80 20)(12 81 21)(22 40 31)(23 41 32)(24 42 33)(25 43 34)(26 44 35)(27 45 36)(28 129 120)(29 127 118)(30 128 119)(37 82 46)(38 83 47)(39 84 48)(49 67 58)(50 68 59)(51 69 60)(52 70 61)(53 71 62)(54 72 63)(55 146 137)(56 147 138)(57 145 136)(85 154 94)(86 155 95)(87 156 96)(88 157 97)(89 158 98)(90 159 99)(91 160 100)(92 161 101)(93 162 102)(103 121 112)(104 122 113)(105 123 114)(106 124 115)(107 125 116)(108 126 117)(130 148 139)(131 149 140)(132 150 141)(133 151 142)(134 152 143)(135 153 144)
(2 3)(4 34)(5 36)(6 35)(7 12)(8 11)(9 10)(13 75)(14 74)(15 73)(16 81)(17 80)(18 79)(19 78)(20 77)(21 76)(22 69)(23 68)(24 67)(25 65)(26 64)(27 66)(28 151)(29 153)(30 152)(31 60)(32 59)(33 58)(37 63)(38 62)(39 61)(40 51)(41 50)(42 49)(43 111)(44 110)(45 109)(46 54)(47 53)(48 52)(55 124)(56 126)(57 125)(70 84)(71 83)(72 82)(85 87)(88 93)(89 92)(90 91)(94 156)(95 155)(96 154)(97 162)(98 161)(99 160)(100 159)(101 158)(102 157)(103 150)(104 149)(105 148)(106 146)(107 145)(108 147)(112 141)(113 140)(114 139)(115 137)(116 136)(117 138)(118 144)(119 143)(120 142)(121 132)(122 131)(123 130)(127 135)(128 134)(129 133)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3AN 6A ··· 6AN order 1 2 2 2 3 ··· 3 6 ··· 6 size 1 1 81 81 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 S3 D6 kernel C2×C34⋊C2 C34⋊C2 C33×C6 C32×C6 C33 # reps 1 2 1 40 40

Matrix representation of C2×C34⋊C2 in GL8(ℤ)

 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1
,
 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1
,
 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 1

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1] >;

C2×C34⋊C2 in GAP, Magma, Sage, TeX

C_2\times C_3^4\rtimes C_2
% in TeX

G:=Group("C2xC3^4:C2");
// GroupNames label

G:=SmallGroup(324,175);
// by ID

G=gap.SmallGroup(324,175);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,579,2164,7781]);
// Polycyclic

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

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