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## G = C7×C9⋊S3order 378 = 2·33·7

### Direct product of C7 and C9⋊S3

Aliases: C7×C9⋊S3, C632S3, C213D9, C9⋊(S3×C7), C3⋊(C7×D9), (C3×C63)⋊5C2, (C3×C9)⋊3C14, (C3×C21).6S3, C21.3(C3⋊S3), C32.3(S3×C7), C3.(C7×C3⋊S3), SmallGroup(378,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C7×C9⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C63 — C7×C9⋊S3
 Lower central C3×C9 — C7×C9⋊S3
 Upper central C1 — C7

Generators and relations for C7×C9⋊S3
G = < a,b,c,d | a7=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Smallest permutation representation of C7×C9⋊S3
On 189 points
Generators in S189
(1 181 154 127 100 73 46)(2 182 155 128 101 74 47)(3 183 156 129 102 75 48)(4 184 157 130 103 76 49)(5 185 158 131 104 77 50)(6 186 159 132 105 78 51)(7 187 160 133 106 79 52)(8 188 161 134 107 80 53)(9 189 162 135 108 81 54)(10 170 143 116 89 62 35)(11 171 144 117 90 63 36)(12 163 136 109 82 55 28)(13 164 137 110 83 56 29)(14 165 138 111 84 57 30)(15 166 139 112 85 58 31)(16 167 140 113 86 59 32)(17 168 141 114 87 60 33)(18 169 142 115 88 61 34)(19 176 149 122 95 68 41)(20 177 150 123 96 69 42)(21 178 151 124 97 70 43)(22 179 152 125 98 71 44)(23 180 153 126 99 72 45)(24 172 145 118 91 64 37)(25 173 146 119 92 65 38)(26 174 147 120 93 66 39)(27 175 148 121 94 67 40)
(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 33 42)(2 34 43)(3 35 44)(4 36 45)(5 28 37)(6 29 38)(7 30 39)(8 31 40)(9 32 41)(10 22 183)(11 23 184)(12 24 185)(13 25 186)(14 26 187)(15 27 188)(16 19 189)(17 20 181)(18 21 182)(46 60 69)(47 61 70)(48 62 71)(49 63 72)(50 55 64)(51 56 65)(52 57 66)(53 58 67)(54 59 68)(73 87 96)(74 88 97)(75 89 98)(76 90 99)(77 82 91)(78 83 92)(79 84 93)(80 85 94)(81 86 95)(100 114 123)(101 115 124)(102 116 125)(103 117 126)(104 109 118)(105 110 119)(106 111 120)(107 112 121)(108 113 122)(127 141 150)(128 142 151)(129 143 152)(130 144 153)(131 136 145)(132 137 146)(133 138 147)(134 139 148)(135 140 149)(154 168 177)(155 169 178)(156 170 179)(157 171 180)(158 163 172)(159 164 173)(160 165 174)(161 166 175)(162 167 176)
(2 9)(3 8)(4 7)(5 6)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(28 38)(29 37)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(47 54)(48 53)(49 52)(50 51)(55 65)(56 64)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(74 81)(75 80)(76 79)(77 78)(82 92)(83 91)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(101 108)(102 107)(103 106)(104 105)(109 119)(110 118)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 120)(128 135)(129 134)(130 133)(131 132)(136 146)(137 145)(138 153)(139 152)(140 151)(141 150)(142 149)(143 148)(144 147)(155 162)(156 161)(157 160)(158 159)(163 173)(164 172)(165 180)(166 179)(167 178)(168 177)(169 176)(170 175)(171 174)(182 189)(183 188)(184 187)(185 186)

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

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

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

105 conjugacy classes

 class 1 2 3A 3B 3C 3D 7A ··· 7F 9A ··· 9I 14A ··· 14F 21A ··· 21X 63A ··· 63BB order 1 2 3 3 3 3 7 ··· 7 9 ··· 9 14 ··· 14 21 ··· 21 63 ··· 63 size 1 27 2 2 2 2 1 ··· 1 2 ··· 2 27 ··· 27 2 ··· 2 2 ··· 2

105 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C7 C14 S3 S3 D9 S3×C7 S3×C7 C7×D9 kernel C7×C9⋊S3 C3×C63 C9⋊S3 C3×C9 C63 C3×C21 C21 C9 C32 C3 # reps 1 1 6 6 3 1 9 18 6 54

Matrix representation of C7×C9⋊S3 in GL4(𝔽127) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 126 126 0 0 1 0 0 0 0 0 9 22 0 0 105 31
,
 126 126 0 0 1 0 0 0 0 0 0 126 0 0 1 126
,
 0 1 0 0 1 0 0 0 0 0 9 22 0 0 31 118
G:=sub<GL(4,GF(127))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[126,1,0,0,126,0,0,0,0,0,9,105,0,0,22,31],[126,1,0,0,126,0,0,0,0,0,0,1,0,0,126,126],[0,1,0,0,1,0,0,0,0,0,9,31,0,0,22,118] >;

C7×C9⋊S3 in GAP, Magma, Sage, TeX

C_7\times C_9\rtimes S_3
% in TeX

G:=Group("C7xC9:S3");
// GroupNames label

G:=SmallGroup(378,40);
// by ID

G=gap.SmallGroup(378,40);
# by ID

G:=PCGroup([5,-2,-7,-3,-3,-3,2312,282,1683,6304]);
// Polycyclic

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

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