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G = C7×Dic9order 252 = 22·32·7

Direct product of C7 and Dic9

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

C1C9 — C7×Dic9
C1C3C9C18C126 — C7×Dic9
C9 — C7×Dic9
C1C14

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

9C4
3Dic3
9C28
3C7×Dic3

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

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

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

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

84 conjugacy classes

class 1  2  3 4A4B 6 7A···7F9A9B9C14A···14F18A18B18C21A···21F28A···28L42A···42F63A···63R126A···126R
order1234467···799914···1418181821···2128···2842···4263···63126···126
size1129921···12221···12222···29···92···22···22···2

84 irreducible representations

dim11111122222222
type+++-+-
imageC1C2C4C7C14C28S3Dic3D9Dic9S3×C7C7×Dic3C7×D9C7×Dic9
kernelC7×Dic9C126C63Dic9C18C9C42C21C14C7C6C3C2C1
# reps11266121133661818

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

100
02320
00232
,
75600
019158
0699133
,
8700
048933
0301268
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

Export

Subgroup lattice of C7×Dic9 in TeX

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