C3^3:D7 - GroupNames

G = C₃³⋊D₇ order 378 = 2·3³·7

3^rd semidirect product of C₃³ and D₇ acting via D₇/C₇=C₂

metabelian, supersoluble, monomial, A-group

Aliases: C₃³⋊₃D₇, C₃²⋊₄D₂₁, C₃⋊(C₃⋊D₂₁), (C₃×C₂₁)⋊₅S₃, C₇⋊(C₃³⋊C₂), C₂₁⋊₁(C₃⋊S₃), (C₃²×C₂₁)⋊₁C₂, SmallGroup(378,59)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₂₁ — C₃³⋊D₇

Chief series

C₁ — C₇ — C₂₁ — C₃×C₂₁ — C₃²×C₂₁ — C₃³⋊D₇

Lower central

C₃²×C₂₁ — C₃³⋊D₇

Upper central

C₁

Generators and relations for C₃³⋊D₇
G = < a,b,c,d,e | a³=b³=c³=d⁷=e²=1, ab=ba, ac=ca, ad=da, eae=a^-1, bc=cb, bd=db, ebe=b^-1, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 1528 in 112 conjugacy classes, 57 normal (5 characteristic)
C₁, C₂, C₃, S₃, C₇, C₃², D₇, C₃⋊S₃, C₂₁, C₃³, D₂₁, C₃³⋊C₂, C₃×C₂₁, C₃⋊D₂₁, C₃²×C₂₁, C₃³⋊D₇
Quotients: C₁, C₂, S₃, D₇, C₃⋊S₃, D₂₁, C₃³⋊C₂, C₃⋊D₂₁, C₃³⋊D₇

Smallest permutation representation of C₃³⋊D₇
►On 189 points

Generators in S₁₈₉

(1 132 69)(2 133 70)(3 127 64)(4 128 65)(5 129 66)(6 130 67)(7 131 68)(8 134 71)(9 135 72)(10 136 73)(11 137 74)(12 138 75)(13 139 76)(14 140 77)(15 141 78)(16 142 79)(17 143 80)(18 144 81)(19 145 82)(20 146 83)(21 147 84)(22 148 85)(23 149 86)(24 150 87)(25 151 88)(26 152 89)(27 153 90)(28 154 91)(29 155 92)(30 156 93)(31 157 94)(32 158 95)(33 159 96)(34 160 97)(35 161 98)(36 162 99)(37 163 100)(38 164 101)(39 165 102)(40 166 103)(41 167 104)(42 168 105)(43 169 106)(44 170 107)(45 171 108)(46 172 109)(47 173 110)(48 174 111)(49 175 112)(50 176 113)(51 177 114)(52 178 115)(53 179 116)(54 180 117)(55 181 118)(56 182 119)(57 183 120)(58 184 121)(59 185 122)(60 186 123)(61 187 124)(62 188 125)(63 189 126)

(1 48 27)(2 49 28)(3 43 22)(4 44 23)(5 45 24)(6 46 25)(7 47 26)(8 50 29)(9 51 30)(10 52 31)(11 53 32)(12 54 33)(13 55 34)(14 56 35)(15 57 36)(16 58 37)(17 59 38)(18 60 39)(19 61 40)(20 62 41)(21 63 42)(64 106 85)(65 107 86)(66 108 87)(67 109 88)(68 110 89)(69 111 90)(70 112 91)(71 113 92)(72 114 93)(73 115 94)(74 116 95)(75 117 96)(76 118 97)(77 119 98)(78 120 99)(79 121 100)(80 122 101)(81 123 102)(82 124 103)(83 125 104)(84 126 105)(127 169 148)(128 170 149)(129 171 150)(130 172 151)(131 173 152)(132 174 153)(133 175 154)(134 176 155)(135 177 156)(136 178 157)(137 179 158)(138 180 159)(139 181 160)(140 182 161)(141 183 162)(142 184 163)(143 185 164)(144 186 165)(145 187 166)(146 188 167)(147 189 168)

(1 20 13)(2 21 14)(3 15 8)(4 16 9)(5 17 10)(6 18 11)(7 19 12)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)(64 78 71)(65 79 72)(66 80 73)(67 81 74)(68 82 75)(69 83 76)(70 84 77)(85 99 92)(86 100 93)(87 101 94)(88 102 95)(89 103 96)(90 104 97)(91 105 98)(106 120 113)(107 121 114)(108 122 115)(109 123 116)(110 124 117)(111 125 118)(112 126 119)(127 141 134)(128 142 135)(129 143 136)(130 144 137)(131 145 138)(132 146 139)(133 147 140)(148 162 155)(149 163 156)(150 164 157)(151 165 158)(152 166 159)(153 167 160)(154 168 161)(169 183 176)(170 184 177)(171 185 178)(172 186 179)(173 187 180)(174 188 181)(175 189 182)

(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)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)

(1 7)(2 6)(3 5)(8 17)(9 16)(10 15)(11 21)(12 20)(13 19)(14 18)(22 45)(23 44)(24 43)(25 49)(26 48)(27 47)(28 46)(29 59)(30 58)(31 57)(32 63)(33 62)(34 61)(35 60)(36 52)(37 51)(38 50)(39 56)(40 55)(41 54)(42 53)(64 129)(65 128)(66 127)(67 133)(68 132)(69 131)(70 130)(71 143)(72 142)(73 141)(74 147)(75 146)(76 145)(77 144)(78 136)(79 135)(80 134)(81 140)(82 139)(83 138)(84 137)(85 171)(86 170)(87 169)(88 175)(89 174)(90 173)(91 172)(92 185)(93 184)(94 183)(95 189)(96 188)(97 187)(98 186)(99 178)(100 177)(101 176)(102 182)(103 181)(104 180)(105 179)(106 150)(107 149)(108 148)(109 154)(110 153)(111 152)(112 151)(113 164)(114 163)(115 162)(116 168)(117 167)(118 166)(119 165)(120 157)(121 156)(122 155)(123 161)(124 160)(125 159)(126 158)

G:=sub<Sym(189)| (1,132,69)(2,133,70)(3,127,64)(4,128,65)(5,129,66)(6,130,67)(7,131,68)(8,134,71)(9,135,72)(10,136,73)(11,137,74)(12,138,75)(13,139,76)(14,140,77)(15,141,78)(16,142,79)(17,143,80)(18,144,81)(19,145,82)(20,146,83)(21,147,84)(22,148,85)(23,149,86)(24,150,87)(25,151,88)(26,152,89)(27,153,90)(28,154,91)(29,155,92)(30,156,93)(31,157,94)(32,158,95)(33,159,96)(34,160,97)(35,161,98)(36,162,99)(37,163,100)(38,164,101)(39,165,102)(40,166,103)(41,167,104)(42,168,105)(43,169,106)(44,170,107)(45,171,108)(46,172,109)(47,173,110)(48,174,111)(49,175,112)(50,176,113)(51,177,114)(52,178,115)(53,179,116)(54,180,117)(55,181,118)(56,182,119)(57,183,120)(58,184,121)(59,185,122)(60,186,123)(61,187,124)(62,188,125)(63,189,126), (1,48,27)(2,49,28)(3,43,22)(4,44,23)(5,45,24)(6,46,25)(7,47,26)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42)(64,106,85)(65,107,86)(66,108,87)(67,109,88)(68,110,89)(69,111,90)(70,112,91)(71,113,92)(72,114,93)(73,115,94)(74,116,95)(75,117,96)(76,118,97)(77,119,98)(78,120,99)(79,121,100)(80,122,101)(81,123,102)(82,124,103)(83,125,104)(84,126,105)(127,169,148)(128,170,149)(129,171,150)(130,172,151)(131,173,152)(132,174,153)(133,175,154)(134,176,155)(135,177,156)(136,178,157)(137,179,158)(138,180,159)(139,181,160)(140,182,161)(141,183,162)(142,184,163)(143,185,164)(144,186,165)(145,187,166)(146,188,167)(147,189,168), (1,20,13)(2,21,14)(3,15,8)(4,16,9)(5,17,10)(6,18,11)(7,19,12)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56)(64,78,71)(65,79,72)(66,80,73)(67,81,74)(68,82,75)(69,83,76)(70,84,77)(85,99,92)(86,100,93)(87,101,94)(88,102,95)(89,103,96)(90,104,97)(91,105,98)(106,120,113)(107,121,114)(108,122,115)(109,123,116)(110,124,117)(111,125,118)(112,126,119)(127,141,134)(128,142,135)(129,143,136)(130,144,137)(131,145,138)(132,146,139)(133,147,140)(148,162,155)(149,163,156)(150,164,157)(151,165,158)(152,166,159)(153,167,160)(154,168,161)(169,183,176)(170,184,177)(171,185,178)(172,186,179)(173,187,180)(174,188,181)(175,189,182), (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)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189), (1,7)(2,6)(3,5)(8,17)(9,16)(10,15)(11,21)(12,20)(13,19)(14,18)(22,45)(23,44)(24,43)(25,49)(26,48)(27,47)(28,46)(29,59)(30,58)(31,57)(32,63)(33,62)(34,61)(35,60)(36,52)(37,51)(38,50)(39,56)(40,55)(41,54)(42,53)(64,129)(65,128)(66,127)(67,133)(68,132)(69,131)(70,130)(71,143)(72,142)(73,141)(74,147)(75,146)(76,145)(77,144)(78,136)(79,135)(80,134)(81,140)(82,139)(83,138)(84,137)(85,171)(86,170)(87,169)(88,175)(89,174)(90,173)(91,172)(92,185)(93,184)(94,183)(95,189)(96,188)(97,187)(98,186)(99,178)(100,177)(101,176)(102,182)(103,181)(104,180)(105,179)(106,150)(107,149)(108,148)(109,154)(110,153)(111,152)(112,151)(113,164)(114,163)(115,162)(116,168)(117,167)(118,166)(119,165)(120,157)(121,156)(122,155)(123,161)(124,160)(125,159)(126,158)>;

G:=Group( (1,132,69)(2,133,70)(3,127,64)(4,128,65)(5,129,66)(6,130,67)(7,131,68)(8,134,71)(9,135,72)(10,136,73)(11,137,74)(12,138,75)(13,139,76)(14,140,77)(15,141,78)(16,142,79)(17,143,80)(18,144,81)(19,145,82)(20,146,83)(21,147,84)(22,148,85)(23,149,86)(24,150,87)(25,151,88)(26,152,89)(27,153,90)(28,154,91)(29,155,92)(30,156,93)(31,157,94)(32,158,95)(33,159,96)(34,160,97)(35,161,98)(36,162,99)(37,163,100)(38,164,101)(39,165,102)(40,166,103)(41,167,104)(42,168,105)(43,169,106)(44,170,107)(45,171,108)(46,172,109)(47,173,110)(48,174,111)(49,175,112)(50,176,113)(51,177,114)(52,178,115)(53,179,116)(54,180,117)(55,181,118)(56,182,119)(57,183,120)(58,184,121)(59,185,122)(60,186,123)(61,187,124)(62,188,125)(63,189,126), (1,48,27)(2,49,28)(3,43,22)(4,44,23)(5,45,24)(6,46,25)(7,47,26)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42)(64,106,85)(65,107,86)(66,108,87)(67,109,88)(68,110,89)(69,111,90)(70,112,91)(71,113,92)(72,114,93)(73,115,94)(74,116,95)(75,117,96)(76,118,97)(77,119,98)(78,120,99)(79,121,100)(80,122,101)(81,123,102)(82,124,103)(83,125,104)(84,126,105)(127,169,148)(128,170,149)(129,171,150)(130,172,151)(131,173,152)(132,174,153)(133,175,154)(134,176,155)(135,177,156)(136,178,157)(137,179,158)(138,180,159)(139,181,160)(140,182,161)(141,183,162)(142,184,163)(143,185,164)(144,186,165)(145,187,166)(146,188,167)(147,189,168), (1,20,13)(2,21,14)(3,15,8)(4,16,9)(5,17,10)(6,18,11)(7,19,12)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56)(64,78,71)(65,79,72)(66,80,73)(67,81,74)(68,82,75)(69,83,76)(70,84,77)(85,99,92)(86,100,93)(87,101,94)(88,102,95)(89,103,96)(90,104,97)(91,105,98)(106,120,113)(107,121,114)(108,122,115)(109,123,116)(110,124,117)(111,125,118)(112,126,119)(127,141,134)(128,142,135)(129,143,136)(130,144,137)(131,145,138)(132,146,139)(133,147,140)(148,162,155)(149,163,156)(150,164,157)(151,165,158)(152,166,159)(153,167,160)(154,168,161)(169,183,176)(170,184,177)(171,185,178)(172,186,179)(173,187,180)(174,188,181)(175,189,182), (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)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189), (1,7)(2,6)(3,5)(8,17)(9,16)(10,15)(11,21)(12,20)(13,19)(14,18)(22,45)(23,44)(24,43)(25,49)(26,48)(27,47)(28,46)(29,59)(30,58)(31,57)(32,63)(33,62)(34,61)(35,60)(36,52)(37,51)(38,50)(39,56)(40,55)(41,54)(42,53)(64,129)(65,128)(66,127)(67,133)(68,132)(69,131)(70,130)(71,143)(72,142)(73,141)(74,147)(75,146)(76,145)(77,144)(78,136)(79,135)(80,134)(81,140)(82,139)(83,138)(84,137)(85,171)(86,170)(87,169)(88,175)(89,174)(90,173)(91,172)(92,185)(93,184)(94,183)(95,189)(96,188)(97,187)(98,186)(99,178)(100,177)(101,176)(102,182)(103,181)(104,180)(105,179)(106,150)(107,149)(108,148)(109,154)(110,153)(111,152)(112,151)(113,164)(114,163)(115,162)(116,168)(117,167)(118,166)(119,165)(120,157)(121,156)(122,155)(123,161)(124,160)(125,159)(126,158) );

G=PermutationGroup([[(1,132,69),(2,133,70),(3,127,64),(4,128,65),(5,129,66),(6,130,67),(7,131,68),(8,134,71),(9,135,72),(10,136,73),(11,137,74),(12,138,75),(13,139,76),(14,140,77),(15,141,78),(16,142,79),(17,143,80),(18,144,81),(19,145,82),(20,146,83),(21,147,84),(22,148,85),(23,149,86),(24,150,87),(25,151,88),(26,152,89),(27,153,90),(28,154,91),(29,155,92),(30,156,93),(31,157,94),(32,158,95),(33,159,96),(34,160,97),(35,161,98),(36,162,99),(37,163,100),(38,164,101),(39,165,102),(40,166,103),(41,167,104),(42,168,105),(43,169,106),(44,170,107),(45,171,108),(46,172,109),(47,173,110),(48,174,111),(49,175,112),(50,176,113),(51,177,114),(52,178,115),(53,179,116),(54,180,117),(55,181,118),(56,182,119),(57,183,120),(58,184,121),(59,185,122),(60,186,123),(61,187,124),(62,188,125),(63,189,126)], [(1,48,27),(2,49,28),(3,43,22),(4,44,23),(5,45,24),(6,46,25),(7,47,26),(8,50,29),(9,51,30),(10,52,31),(11,53,32),(12,54,33),(13,55,34),(14,56,35),(15,57,36),(16,58,37),(17,59,38),(18,60,39),(19,61,40),(20,62,41),(21,63,42),(64,106,85),(65,107,86),(66,108,87),(67,109,88),(68,110,89),(69,111,90),(70,112,91),(71,113,92),(72,114,93),(73,115,94),(74,116,95),(75,117,96),(76,118,97),(77,119,98),(78,120,99),(79,121,100),(80,122,101),(81,123,102),(82,124,103),(83,125,104),(84,126,105),(127,169,148),(128,170,149),(129,171,150),(130,172,151),(131,173,152),(132,174,153),(133,175,154),(134,176,155),(135,177,156),(136,178,157),(137,179,158),(138,180,159),(139,181,160),(140,182,161),(141,183,162),(142,184,163),(143,185,164),(144,186,165),(145,187,166),(146,188,167),(147,189,168)], [(1,20,13),(2,21,14),(3,15,8),(4,16,9),(5,17,10),(6,18,11),(7,19,12),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56),(64,78,71),(65,79,72),(66,80,73),(67,81,74),(68,82,75),(69,83,76),(70,84,77),(85,99,92),(86,100,93),(87,101,94),(88,102,95),(89,103,96),(90,104,97),(91,105,98),(106,120,113),(107,121,114),(108,122,115),(109,123,116),(110,124,117),(111,125,118),(112,126,119),(127,141,134),(128,142,135),(129,143,136),(130,144,137),(131,145,138),(132,146,139),(133,147,140),(148,162,155),(149,163,156),(150,164,157),(151,165,158),(152,166,159),(153,167,160),(154,168,161),(169,183,176),(170,184,177),(171,185,178),(172,186,179),(173,187,180),(174,188,181),(175,189,182)], [(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),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189)], [(1,7),(2,6),(3,5),(8,17),(9,16),(10,15),(11,21),(12,20),(13,19),(14,18),(22,45),(23,44),(24,43),(25,49),(26,48),(27,47),(28,46),(29,59),(30,58),(31,57),(32,63),(33,62),(34,61),(35,60),(36,52),(37,51),(38,50),(39,56),(40,55),(41,54),(42,53),(64,129),(65,128),(66,127),(67,133),(68,132),(69,131),(70,130),(71,143),(72,142),(73,141),(74,147),(75,146),(76,145),(77,144),(78,136),(79,135),(80,134),(81,140),(82,139),(83,138),(84,137),(85,171),(86,170),(87,169),(88,175),(89,174),(90,173),(91,172),(92,185),(93,184),(94,183),(95,189),(96,188),(97,187),(98,186),(99,178),(100,177),(101,176),(102,182),(103,181),(104,180),(105,179),(106,150),(107,149),(108,148),(109,154),(110,153),(111,152),(112,151),(113,164),(114,163),(115,162),(116,168),(117,167),(118,166),(119,165),(120,157),(121,156),(122,155),(123,161),(124,160),(125,159),(126,158)]])

96 conjugacy classes

class	1	2	3A	···	3M	7A	7B	7C	21A	···	21BZ
order	1	2	3	···	3	7	7	7	21	···	21
size	1	189	2	···	2	2	2	2	2	···	2

96 irreducible representations

dim	1	1	2	2	2
type	+	+	+	+	+
image	C₁	C₂	S₃	D₇	D₂₁
kernel	C₃³⋊D₇	C₃²×C₂₁	C₃×C₂₁	C₃³	C₃²
# reps	1	1	13	3	78

Matrix representation of C₃³⋊D₇ ►in GL₆(𝔽₄₃)

16	35	0	0	0	0
18	26	0	0	0	0
0	0	16	12	0	0
0	0	31	26	0	0
0	0	0	0	1	0
0	0	0	0	0	1

26	8	0	0	0	0
25	16	0	0	0	0
0	0	16	12	0	0
0	0	31	26	0	0
0	0	0	0	16	12
0	0	0	0	31	26

26	8	0	0	0	0
25	16	0	0	0	0
0	0	26	31	0	0
0	0	12	16	0	0
0	0	0	0	16	12
0	0	0	0	31	26

35	1	0	0	0	0
30	23	0	0	0	0
0	0	0	1	0	0
0	0	42	8	0	0
0	0	0	0	0	1
0	0	0	0	42	8

23	42	0	0	0	0
12	20	0	0	0	0
0	0	0	42	0	0
0	0	42	0	0	0
0	0	0	0	31	27
0	0	0	0	17	12

G:=sub<GL(6,GF(43))| [16,18,0,0,0,0,35,26,0,0,0,0,0,0,16,31,0,0,0,0,12,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,25,0,0,0,0,8,16,0,0,0,0,0,0,16,31,0,0,0,0,12,26,0,0,0,0,0,0,16,31,0,0,0,0,12,26],[26,25,0,0,0,0,8,16,0,0,0,0,0,0,26,12,0,0,0,0,31,16,0,0,0,0,0,0,16,31,0,0,0,0,12,26],[35,30,0,0,0,0,1,23,0,0,0,0,0,0,0,42,0,0,0,0,1,8,0,0,0,0,0,0,0,42,0,0,0,0,1,8],[23,12,0,0,0,0,42,20,0,0,0,0,0,0,0,42,0,0,0,0,42,0,0,0,0,0,0,0,31,17,0,0,0,0,27,12] >;

C₃³⋊D₇ in GAP, Magma, Sage, TeX

C_3^3\rtimes D_7

% in TeX

G:=Group("C3^3:D7");

// GroupNames label

G:=SmallGroup(378,59);

// by ID

G=gap.SmallGroup(378,59);

# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,41,182,723,8104]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = C33⋊D7 order 378 = 2·33·7

3rd semidirect product of C33 and D7 acting via D7/C7=C2

G = C₃³⋊D₇ order 378 = 2·3³·7

3^rd semidirect product of C₃³ and D₇ acting via D₇/C₇=C₂