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G = C2×C34⋊C2  order 324 = 22·34

Direct product of C2 and C34⋊C2

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: C2×C34⋊C2, C33⋊21D6, C34⋊10C22, C6⋊(C33⋊C2), (C33×C6)⋊4C2, (C32×C6)⋊9S3, (C3×C6)⋊4(C3⋊S3), C32⋊9(C2×C3⋊S3), C3⋊2(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, C3, C22, S3, C6, C32, D6, C3⋊S3, C3×C6, C33, C2×C3⋊S3, C33⋊C2, C32×C6, C34, C2×C33⋊C2, C34⋊C2, C33×C6, C2×C34⋊C2
Quotients: C1, C2, C22, S3, D6, C3⋊S3, C2×C3⋊S3, C33⋊C2, C2×C33⋊C2, 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 2A2B2C3A···3AN6A···6AN
order12223···36···6
size1181812···22···2

84 irreducible representations

dim11122
type+++++
imageC1C2C2S3D6
kernelC2×C34⋊C2C34⋊C2C33×C6C32×C6C33
# reps1214040

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

-10000000
0-1000000
00100000
00010000
00001000
00000100
00000010
00000001
,
-1-1000000
10000000
00100000
00010000
00000100
0000-1-100
00000001
000000-1-1
,
-1-1000000
10000000
00-1-10000
00100000
00001000
00000100
00000001
000000-1-1
,
10000000
01000000
00100000
00010000
0000-1-100
00001000
00000010
00000001
,
01000000
-1-1000000
00010000
00-1-10000
00000100
0000-1-100
00000001
000000-1-1
,
10000000
-1-1000000
00110000
000-10000
00000-100
0000-1000
000000-10
00000011

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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