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## G = D7×C3×C9order 378 = 2·33·7

### Direct product of C3×C9 and D7

Aliases: D7×C3×C9, C6319C6, C216C18, C75(C3×C18), (C3×C63)⋊6C2, (C3×C21).10C6, C21.13(C3×C6), C3.1(C32×D7), C32.3(C3×D7), (C32×D7).4C3, (C3×D7).7C32, SmallGroup(378,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C3×C9
 Chief series C1 — C7 — C21 — C63 — C3×C63 — D7×C3×C9
 Lower central C7 — D7×C3×C9
 Upper central C1 — C3×C9

Generators and relations for D7×C3×C9
G = < a,b,c,d | a3=b9=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of D7×C3×C9
On 189 points
Generators in S189
(1 58 131)(2 59 132)(3 60 133)(4 61 134)(5 62 135)(6 63 127)(7 55 128)(8 56 129)(9 57 130)(10 169 103)(11 170 104)(12 171 105)(13 163 106)(14 164 107)(15 165 108)(16 166 100)(17 167 101)(18 168 102)(19 178 112)(20 179 113)(21 180 114)(22 172 115)(23 173 116)(24 174 117)(25 175 109)(26 176 110)(27 177 111)(28 136 79)(29 137 80)(30 138 81)(31 139 73)(32 140 74)(33 141 75)(34 142 76)(35 143 77)(36 144 78)(37 147 90)(38 148 82)(39 149 83)(40 150 84)(41 151 85)(42 152 86)(43 153 87)(44 145 88)(45 146 89)(46 154 97)(47 155 98)(48 156 99)(49 157 91)(50 158 92)(51 159 93)(52 160 94)(53 161 95)(54 162 96)(64 184 118)(65 185 119)(66 186 120)(67 187 121)(68 188 122)(69 189 123)(70 181 124)(71 182 125)(72 183 126)
(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 91 103 76 90 117 69)(2 92 104 77 82 109 70)(3 93 105 78 83 110 71)(4 94 106 79 84 111 72)(5 95 107 80 85 112 64)(6 96 108 81 86 113 65)(7 97 100 73 87 114 66)(8 98 101 74 88 115 67)(9 99 102 75 89 116 68)(10 34 37 24 189 58 49)(11 35 38 25 181 59 50)(12 36 39 26 182 60 51)(13 28 40 27 183 61 52)(14 29 41 19 184 62 53)(15 30 42 20 185 63 54)(16 31 43 21 186 55 46)(17 32 44 22 187 56 47)(18 33 45 23 188 57 48)(118 135 161 164 137 151 178)(119 127 162 165 138 152 179)(120 128 154 166 139 153 180)(121 129 155 167 140 145 172)(122 130 156 168 141 146 173)(123 131 157 169 142 147 174)(124 132 158 170 143 148 175)(125 133 159 171 144 149 176)(126 134 160 163 136 150 177)
(1 69)(2 70)(3 71)(4 72)(5 64)(6 65)(7 66)(8 67)(9 68)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 53)(20 54)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(55 186)(56 187)(57 188)(58 189)(59 181)(60 182)(61 183)(62 184)(63 185)(82 104)(83 105)(84 106)(85 107)(86 108)(87 100)(88 101)(89 102)(90 103)(91 117)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(118 135)(119 127)(120 128)(121 129)(122 130)(123 131)(124 132)(125 133)(126 134)(145 167)(146 168)(147 169)(148 170)(149 171)(150 163)(151 164)(152 165)(153 166)(154 180)(155 172)(156 173)(157 174)(158 175)(159 176)(160 177)(161 178)(162 179)

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

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

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

135 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 7A 7B 7C 9A ··· 9R 18A ··· 18R 21A ··· 21X 63A ··· 63BB order 1 2 3 ··· 3 6 ··· 6 7 7 7 9 ··· 9 18 ··· 18 21 ··· 21 63 ··· 63 size 1 7 1 ··· 1 7 ··· 7 2 2 2 1 ··· 1 7 ··· 7 2 ··· 2 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C3 C6 C6 C9 C18 D7 C3×D7 C3×D7 C9×D7 kernel D7×C3×C9 C3×C63 C9×D7 C32×D7 C63 C3×C21 C3×D7 C21 C3×C9 C9 C32 C3 # reps 1 1 6 2 6 2 18 18 3 18 6 54

Matrix representation of D7×C3×C9 in GL3(𝔽127) generated by

 19 0 0 0 19 0 0 0 19
,
 107 0 0 0 103 0 0 0 103
,
 1 0 0 0 0 1 0 126 24
,
 126 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(127))| [19,0,0,0,19,0,0,0,19],[107,0,0,0,103,0,0,0,103],[1,0,0,0,0,126,0,1,24],[126,0,0,0,0,1,0,1,0] >;

D7×C3×C9 in GAP, Magma, Sage, TeX

D_7\times C_3\times C_9
% in TeX

G:=Group("D7xC3xC9");
// GroupNames label

G:=SmallGroup(378,29);
// by ID

G=gap.SmallGroup(378,29);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,57,8104]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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