Copied to
clipboard

## G = C9×Dic7order 252 = 22·32·7

### Direct product of C9 and Dic7

Aliases: C9×Dic7, C633C4, C73C36, C18.2D7, C14.3C18, C126.3C2, C21.3C12, C42.10C6, C2.(C9×D7), C6.2(C3×D7), C3.(C3×Dic7), (C3×Dic7).2C3, SmallGroup(252,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C9×Dic7
 Chief series C1 — C7 — C21 — C42 — C126 — C9×Dic7
 Lower central C7 — C9×Dic7
 Upper central C1 — C18

Generators and relations for C9×Dic7
G = < a,b,c | a9=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C9×Dic7
Regular action on 252 points
Generators in S252
(1 209 127 67 241 103 193 223 51)(2 210 128 68 242 104 194 224 52)(3 197 129 69 243 105 195 211 53)(4 198 130 70 244 106 196 212 54)(5 199 131 57 245 107 183 213 55)(6 200 132 58 246 108 184 214 56)(7 201 133 59 247 109 185 215 43)(8 202 134 60 248 110 186 216 44)(9 203 135 61 249 111 187 217 45)(10 204 136 62 250 112 188 218 46)(11 205 137 63 251 99 189 219 47)(12 206 138 64 252 100 190 220 48)(13 207 139 65 239 101 191 221 49)(14 208 140 66 240 102 192 222 50)(15 32 113 162 150 89 226 174 79)(16 33 114 163 151 90 227 175 80)(17 34 115 164 152 91 228 176 81)(18 35 116 165 153 92 229 177 82)(19 36 117 166 154 93 230 178 83)(20 37 118 167 141 94 231 179 84)(21 38 119 168 142 95 232 180 71)(22 39 120 155 143 96 233 181 72)(23 40 121 156 144 97 234 182 73)(24 41 122 157 145 98 235 169 74)(25 42 123 158 146 85 236 170 75)(26 29 124 159 147 86 237 171 76)(27 30 125 160 148 87 238 172 77)(28 31 126 161 149 88 225 173 78)
(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 80 8 73)(2 79 9 72)(3 78 10 71)(4 77 11 84)(5 76 12 83)(6 75 13 82)(7 74 14 81)(15 203 22 210)(16 202 23 209)(17 201 24 208)(18 200 25 207)(19 199 26 206)(20 198 27 205)(21 197 28 204)(29 138 36 131)(30 137 37 130)(31 136 38 129)(32 135 39 128)(33 134 40 127)(34 133 41 140)(35 132 42 139)(43 169 50 176)(44 182 51 175)(45 181 52 174)(46 180 53 173)(47 179 54 172)(48 178 55 171)(49 177 56 170)(57 124 64 117)(58 123 65 116)(59 122 66 115)(60 121 67 114)(61 120 68 113)(62 119 69 126)(63 118 70 125)(85 191 92 184)(86 190 93 183)(87 189 94 196)(88 188 95 195)(89 187 96 194)(90 186 97 193)(91 185 98 192)(99 141 106 148)(100 154 107 147)(101 153 108 146)(102 152 109 145)(103 151 110 144)(104 150 111 143)(105 149 112 142)(155 242 162 249)(156 241 163 248)(157 240 164 247)(158 239 165 246)(159 252 166 245)(160 251 167 244)(161 250 168 243)(211 225 218 232)(212 238 219 231)(213 237 220 230)(214 236 221 229)(215 235 222 228)(216 234 223 227)(217 233 224 226)

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

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

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

90 conjugacy classes

 class 1 2 3A 3B 4A 4B 6A 6B 7A 7B 7C 9A ··· 9F 12A 12B 12C 12D 14A 14B 14C 18A ··· 18F 21A ··· 21F 36A ··· 36L 42A ··· 42F 63A ··· 63R 126A ··· 126R order 1 2 3 3 4 4 6 6 7 7 7 9 ··· 9 12 12 12 12 14 14 14 18 ··· 18 21 ··· 21 36 ··· 36 42 ··· 42 63 ··· 63 126 ··· 126 size 1 1 1 1 7 7 1 1 2 2 2 1 ··· 1 7 7 7 7 2 2 2 1 ··· 1 2 ··· 2 7 ··· 7 2 ··· 2 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C9 C12 C18 C36 D7 Dic7 C3×D7 C3×Dic7 C9×D7 C9×Dic7 kernel C9×Dic7 C126 C3×Dic7 C63 C42 Dic7 C21 C14 C7 C18 C9 C6 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 3 3 6 6 18 18

Matrix representation of C9×Dic7 in GL4(𝔽757) generated by

 673 0 0 0 0 27 0 0 0 0 729 0 0 0 0 729
,
 1 0 0 0 0 756 0 0 0 0 0 1 0 0 756 136
,
 756 0 0 0 0 670 0 0 0 0 17 748 0 0 32 740
G:=sub<GL(4,GF(757))| [673,0,0,0,0,27,0,0,0,0,729,0,0,0,0,729],[1,0,0,0,0,756,0,0,0,0,0,756,0,0,1,136],[756,0,0,0,0,670,0,0,0,0,17,32,0,0,748,740] >;

C9×Dic7 in GAP, Magma, Sage, TeX

C_9\times {\rm Dic}_7
% in TeX

G:=Group("C9xDic7");
// GroupNames label

G:=SmallGroup(252,4);
// by ID

G=gap.SmallGroup(252,4);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-7,30,66,5404]);
// Polycyclic

G:=Group<a,b,c|a^9=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

׿
×
𝔽