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## G = C7×Dic9order 252 = 22·32·7

### Direct product of C7 and Dic9

Aliases: C7×Dic9, C9⋊C28, C632C4, C18.C14, C42.5S3, C14.2D9, C126.2C2, C21.2Dic3, C2.(C7×D9), C6.1(S3×C7), C3.(C7×Dic3), SmallGroup(252,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C7×Dic9
 Chief series C1 — C3 — C9 — C18 — C126 — C7×Dic9
 Lower central C9 — C7×Dic9
 Upper central C1 — C14

Generators and relations for C7×Dic9
G = < a,b,c | a7=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×Dic9
Regular action on 252 points
Generators in S252
(1 116 94 83 69 40 33)(2 117 95 84 70 41 34)(3 118 96 85 71 42 35)(4 119 97 86 72 43 36)(5 120 98 87 55 44 19)(6 121 99 88 56 45 20)(7 122 100 89 57 46 21)(8 123 101 90 58 47 22)(9 124 102 73 59 48 23)(10 125 103 74 60 49 24)(11 126 104 75 61 50 25)(12 109 105 76 62 51 26)(13 110 106 77 63 52 27)(14 111 107 78 64 53 28)(15 112 108 79 65 54 29)(16 113 91 80 66 37 30)(17 114 92 81 67 38 31)(18 115 93 82 68 39 32)(127 235 221 199 190 171 154)(128 236 222 200 191 172 155)(129 237 223 201 192 173 156)(130 238 224 202 193 174 157)(131 239 225 203 194 175 158)(132 240 226 204 195 176 159)(133 241 227 205 196 177 160)(134 242 228 206 197 178 161)(135 243 229 207 198 179 162)(136 244 230 208 181 180 145)(137 245 231 209 182 163 146)(138 246 232 210 183 164 147)(139 247 233 211 184 165 148)(140 248 234 212 185 166 149)(141 249 217 213 186 167 150)(142 250 218 214 187 168 151)(143 251 219 215 188 169 152)(144 252 220 216 189 170 153)
(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)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)(235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)
(1 144 10 135)(2 143 11 134)(3 142 12 133)(4 141 13 132)(5 140 14 131)(6 139 15 130)(7 138 16 129)(8 137 17 128)(9 136 18 127)(19 149 28 158)(20 148 29 157)(21 147 30 156)(22 146 31 155)(23 145 32 154)(24 162 33 153)(25 161 34 152)(26 160 35 151)(27 159 36 150)(37 173 46 164)(38 172 47 163)(39 171 48 180)(40 170 49 179)(41 169 50 178)(42 168 51 177)(43 167 52 176)(44 166 53 175)(45 165 54 174)(55 185 64 194)(56 184 65 193)(57 183 66 192)(58 182 67 191)(59 181 68 190)(60 198 69 189)(61 197 70 188)(62 196 71 187)(63 195 72 186)(73 208 82 199)(74 207 83 216)(75 206 84 215)(76 205 85 214)(77 204 86 213)(78 203 87 212)(79 202 88 211)(80 201 89 210)(81 200 90 209)(91 223 100 232)(92 222 101 231)(93 221 102 230)(94 220 103 229)(95 219 104 228)(96 218 105 227)(97 217 106 226)(98 234 107 225)(99 233 108 224)(109 241 118 250)(110 240 119 249)(111 239 120 248)(112 238 121 247)(113 237 122 246)(114 236 123 245)(115 235 124 244)(116 252 125 243)(117 251 126 242)

G:=sub<Sym(252)| (1,116,94,83,69,40,33)(2,117,95,84,70,41,34)(3,118,96,85,71,42,35)(4,119,97,86,72,43,36)(5,120,98,87,55,44,19)(6,121,99,88,56,45,20)(7,122,100,89,57,46,21)(8,123,101,90,58,47,22)(9,124,102,73,59,48,23)(10,125,103,74,60,49,24)(11,126,104,75,61,50,25)(12,109,105,76,62,51,26)(13,110,106,77,63,52,27)(14,111,107,78,64,53,28)(15,112,108,79,65,54,29)(16,113,91,80,66,37,30)(17,114,92,81,67,38,31)(18,115,93,82,68,39,32)(127,235,221,199,190,171,154)(128,236,222,200,191,172,155)(129,237,223,201,192,173,156)(130,238,224,202,193,174,157)(131,239,225,203,194,175,158)(132,240,226,204,195,176,159)(133,241,227,205,196,177,160)(134,242,228,206,197,178,161)(135,243,229,207,198,179,162)(136,244,230,208,181,180,145)(137,245,231,209,182,163,146)(138,246,232,210,183,164,147)(139,247,233,211,184,165,148)(140,248,234,212,185,166,149)(141,249,217,213,186,167,150)(142,250,218,214,187,168,151)(143,251,219,215,188,169,152)(144,252,220,216,189,170,153), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252), (1,144,10,135)(2,143,11,134)(3,142,12,133)(4,141,13,132)(5,140,14,131)(6,139,15,130)(7,138,16,129)(8,137,17,128)(9,136,18,127)(19,149,28,158)(20,148,29,157)(21,147,30,156)(22,146,31,155)(23,145,32,154)(24,162,33,153)(25,161,34,152)(26,160,35,151)(27,159,36,150)(37,173,46,164)(38,172,47,163)(39,171,48,180)(40,170,49,179)(41,169,50,178)(42,168,51,177)(43,167,52,176)(44,166,53,175)(45,165,54,174)(55,185,64,194)(56,184,65,193)(57,183,66,192)(58,182,67,191)(59,181,68,190)(60,198,69,189)(61,197,70,188)(62,196,71,187)(63,195,72,186)(73,208,82,199)(74,207,83,216)(75,206,84,215)(76,205,85,214)(77,204,86,213)(78,203,87,212)(79,202,88,211)(80,201,89,210)(81,200,90,209)(91,223,100,232)(92,222,101,231)(93,221,102,230)(94,220,103,229)(95,219,104,228)(96,218,105,227)(97,217,106,226)(98,234,107,225)(99,233,108,224)(109,241,118,250)(110,240,119,249)(111,239,120,248)(112,238,121,247)(113,237,122,246)(114,236,123,245)(115,235,124,244)(116,252,125,243)(117,251,126,242)>;

G:=Group( (1,116,94,83,69,40,33)(2,117,95,84,70,41,34)(3,118,96,85,71,42,35)(4,119,97,86,72,43,36)(5,120,98,87,55,44,19)(6,121,99,88,56,45,20)(7,122,100,89,57,46,21)(8,123,101,90,58,47,22)(9,124,102,73,59,48,23)(10,125,103,74,60,49,24)(11,126,104,75,61,50,25)(12,109,105,76,62,51,26)(13,110,106,77,63,52,27)(14,111,107,78,64,53,28)(15,112,108,79,65,54,29)(16,113,91,80,66,37,30)(17,114,92,81,67,38,31)(18,115,93,82,68,39,32)(127,235,221,199,190,171,154)(128,236,222,200,191,172,155)(129,237,223,201,192,173,156)(130,238,224,202,193,174,157)(131,239,225,203,194,175,158)(132,240,226,204,195,176,159)(133,241,227,205,196,177,160)(134,242,228,206,197,178,161)(135,243,229,207,198,179,162)(136,244,230,208,181,180,145)(137,245,231,209,182,163,146)(138,246,232,210,183,164,147)(139,247,233,211,184,165,148)(140,248,234,212,185,166,149)(141,249,217,213,186,167,150)(142,250,218,214,187,168,151)(143,251,219,215,188,169,152)(144,252,220,216,189,170,153), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252), (1,144,10,135)(2,143,11,134)(3,142,12,133)(4,141,13,132)(5,140,14,131)(6,139,15,130)(7,138,16,129)(8,137,17,128)(9,136,18,127)(19,149,28,158)(20,148,29,157)(21,147,30,156)(22,146,31,155)(23,145,32,154)(24,162,33,153)(25,161,34,152)(26,160,35,151)(27,159,36,150)(37,173,46,164)(38,172,47,163)(39,171,48,180)(40,170,49,179)(41,169,50,178)(42,168,51,177)(43,167,52,176)(44,166,53,175)(45,165,54,174)(55,185,64,194)(56,184,65,193)(57,183,66,192)(58,182,67,191)(59,181,68,190)(60,198,69,189)(61,197,70,188)(62,196,71,187)(63,195,72,186)(73,208,82,199)(74,207,83,216)(75,206,84,215)(76,205,85,214)(77,204,86,213)(78,203,87,212)(79,202,88,211)(80,201,89,210)(81,200,90,209)(91,223,100,232)(92,222,101,231)(93,221,102,230)(94,220,103,229)(95,219,104,228)(96,218,105,227)(97,217,106,226)(98,234,107,225)(99,233,108,224)(109,241,118,250)(110,240,119,249)(111,239,120,248)(112,238,121,247)(113,237,122,246)(114,236,123,245)(115,235,124,244)(116,252,125,243)(117,251,126,242) );

G=PermutationGroup([[(1,116,94,83,69,40,33),(2,117,95,84,70,41,34),(3,118,96,85,71,42,35),(4,119,97,86,72,43,36),(5,120,98,87,55,44,19),(6,121,99,88,56,45,20),(7,122,100,89,57,46,21),(8,123,101,90,58,47,22),(9,124,102,73,59,48,23),(10,125,103,74,60,49,24),(11,126,104,75,61,50,25),(12,109,105,76,62,51,26),(13,110,106,77,63,52,27),(14,111,107,78,64,53,28),(15,112,108,79,65,54,29),(16,113,91,80,66,37,30),(17,114,92,81,67,38,31),(18,115,93,82,68,39,32),(127,235,221,199,190,171,154),(128,236,222,200,191,172,155),(129,237,223,201,192,173,156),(130,238,224,202,193,174,157),(131,239,225,203,194,175,158),(132,240,226,204,195,176,159),(133,241,227,205,196,177,160),(134,242,228,206,197,178,161),(135,243,229,207,198,179,162),(136,244,230,208,181,180,145),(137,245,231,209,182,163,146),(138,246,232,210,183,164,147),(139,247,233,211,184,165,148),(140,248,234,212,185,166,149),(141,249,217,213,186,167,150),(142,250,218,214,187,168,151),(143,251,219,215,188,169,152),(144,252,220,216,189,170,153)], [(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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234),(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)], [(1,144,10,135),(2,143,11,134),(3,142,12,133),(4,141,13,132),(5,140,14,131),(6,139,15,130),(7,138,16,129),(8,137,17,128),(9,136,18,127),(19,149,28,158),(20,148,29,157),(21,147,30,156),(22,146,31,155),(23,145,32,154),(24,162,33,153),(25,161,34,152),(26,160,35,151),(27,159,36,150),(37,173,46,164),(38,172,47,163),(39,171,48,180),(40,170,49,179),(41,169,50,178),(42,168,51,177),(43,167,52,176),(44,166,53,175),(45,165,54,174),(55,185,64,194),(56,184,65,193),(57,183,66,192),(58,182,67,191),(59,181,68,190),(60,198,69,189),(61,197,70,188),(62,196,71,187),(63,195,72,186),(73,208,82,199),(74,207,83,216),(75,206,84,215),(76,205,85,214),(77,204,86,213),(78,203,87,212),(79,202,88,211),(80,201,89,210),(81,200,90,209),(91,223,100,232),(92,222,101,231),(93,221,102,230),(94,220,103,229),(95,219,104,228),(96,218,105,227),(97,217,106,226),(98,234,107,225),(99,233,108,224),(109,241,118,250),(110,240,119,249),(111,239,120,248),(112,238,121,247),(113,237,122,246),(114,236,123,245),(115,235,124,244),(116,252,125,243),(117,251,126,242)]])

84 conjugacy classes

 class 1 2 3 4A 4B 6 7A ··· 7F 9A 9B 9C 14A ··· 14F 18A 18B 18C 21A ··· 21F 28A ··· 28L 42A ··· 42F 63A ··· 63R 126A ··· 126R order 1 2 3 4 4 6 7 ··· 7 9 9 9 14 ··· 14 18 18 18 21 ··· 21 28 ··· 28 42 ··· 42 63 ··· 63 126 ··· 126 size 1 1 2 9 9 2 1 ··· 1 2 2 2 1 ··· 1 2 2 2 2 ··· 2 9 ··· 9 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - + - image C1 C2 C4 C7 C14 C28 S3 Dic3 D9 Dic9 S3×C7 C7×Dic3 C7×D9 C7×Dic9 kernel C7×Dic9 C126 C63 Dic9 C18 C9 C42 C21 C14 C7 C6 C3 C2 C1 # reps 1 1 2 6 6 12 1 1 3 3 6 6 18 18

Matrix representation of C7×Dic9 in GL3(𝔽757) generated by

 1 0 0 0 232 0 0 0 232
,
 756 0 0 0 191 58 0 699 133
,
 87 0 0 0 489 33 0 301 268
G:=sub<GL(3,GF(757))| [1,0,0,0,232,0,0,0,232],[756,0,0,0,191,699,0,58,133],[87,0,0,0,489,301,0,33,268] >;

C7×Dic9 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_9
% in TeX

G:=Group("C7xDic9");
// GroupNames label

G:=SmallGroup(252,3);
// by ID

G=gap.SmallGroup(252,3);
# by ID

G:=PCGroup([5,-2,-7,-2,-3,-3,70,2803,138,4204]);
// Polycyclic

G:=Group<a,b,c|a^7=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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