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## G = D7×C33order 378 = 2·33·7

### Direct product of C33 and D7

Aliases: D7×C33, C216(C3×C6), C73(C32×C6), (C3×C21)⋊17C6, (C32×C21)⋊4C2, SmallGroup(378,53)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C33
 Chief series C1 — C7 — C21 — C3×C21 — C32×C21 — D7×C33
 Lower central C7 — D7×C33
 Upper central C1 — C33

Generators and relations for D7×C33
G = < a,b,c,d,e | a3=b3=c3=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 280 in 112 conjugacy classes, 84 normal (6 characteristic)
C1, C2, C3, C6, C7, C32, D7, C3×C6, C21, C33, C3×D7, C32×C6, C3×C21, C32×D7, C32×C21, D7×C33
Quotients: C1, C2, C3, C6, C32, D7, C3×C6, C33, C3×D7, C32×C6, C32×D7, D7×C33

Smallest permutation representation of D7×C33
On 189 points
Generators in S189
(1 188 97)(2 189 98)(3 183 92)(4 184 93)(5 185 94)(6 186 95)(7 187 96)(8 169 99)(9 170 100)(10 171 101)(11 172 102)(12 173 103)(13 174 104)(14 175 105)(15 176 85)(16 177 86)(17 178 87)(18 179 88)(19 180 89)(20 181 90)(21 182 91)(22 141 113)(23 142 114)(24 143 115)(25 144 116)(26 145 117)(27 146 118)(28 147 119)(29 127 120)(30 128 121)(31 129 122)(32 130 123)(33 131 124)(34 132 125)(35 133 126)(36 134 106)(37 135 107)(38 136 108)(39 137 109)(40 138 110)(41 139 111)(42 140 112)(43 162 71)(44 163 72)(45 164 73)(46 165 74)(47 166 75)(48 167 76)(49 168 77)(50 148 78)(51 149 79)(52 150 80)(53 151 81)(54 152 82)(55 153 83)(56 154 84)(57 155 64)(58 156 65)(59 157 66)(60 158 67)(61 159 68)(62 160 69)(63 161 70)
(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 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 87)(88 91)(89 90)(92 94)(95 98)(96 97)(99 101)(102 105)(103 104)(106 108)(109 112)(110 111)(113 115)(116 119)(117 118)(120 122)(123 126)(124 125)(127 129)(130 133)(131 132)(134 136)(137 140)(138 139)(141 143)(144 147)(145 146)(148 150)(151 154)(152 153)(155 157)(158 161)(159 160)(162 164)(165 168)(166 167)(169 171)(172 175)(173 174)(176 178)(179 182)(180 181)(183 185)(186 189)(187 188)

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

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

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

135 conjugacy classes

 class 1 2 3A ··· 3Z 6A ··· 6Z 7A 7B 7C 21A ··· 21BZ order 1 2 3 ··· 3 6 ··· 6 7 7 7 21 ··· 21 size 1 7 1 ··· 1 7 ··· 7 2 2 2 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 D7 C3×D7 kernel D7×C33 C32×C21 C32×D7 C3×C21 C33 C32 # reps 1 1 26 26 3 78

Matrix representation of D7×C33 in GL4(𝔽43) generated by

 36 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 36 0 0 0 0 6 0 0 0 0 36 0 0 0 0 36
,
 6 0 0 0 0 6 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 42 19
,
 42 0 0 0 0 42 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(43))| [36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,6,0,0,0,0,36,0,0,0,0,36],[6,0,0,0,0,6,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,42,0,0,1,19],[42,0,0,0,0,42,0,0,0,0,0,1,0,0,1,0] >;

D7×C33 in GAP, Magma, Sage, TeX

D_7\times C_3^3
% in TeX

G:=Group("D7xC3^3");
// GroupNames label

G:=SmallGroup(378,53);
// by ID

G=gap.SmallGroup(378,53);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,8104]);
// 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,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽