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G = D7×C34order 476 = 22·7·17

Direct product of C34 and D7

Aliases: D7×C34, C14⋊C34, C2383C2, C1194C22, C7⋊(C2×C34), SmallGroup(476,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C34
 Chief series C1 — C7 — C119 — D7×C17 — D7×C34
 Lower central C7 — D7×C34
 Upper central C1 — C34

Generators and relations for D7×C34
G = < a,b,c | a34=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D7×C34
On 238 points
Generators in S238
(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)
(1 193 124 86 170 233 42)(2 194 125 87 137 234 43)(3 195 126 88 138 235 44)(4 196 127 89 139 236 45)(5 197 128 90 140 237 46)(6 198 129 91 141 238 47)(7 199 130 92 142 205 48)(8 200 131 93 143 206 49)(9 201 132 94 144 207 50)(10 202 133 95 145 208 51)(11 203 134 96 146 209 52)(12 204 135 97 147 210 53)(13 171 136 98 148 211 54)(14 172 103 99 149 212 55)(15 173 104 100 150 213 56)(16 174 105 101 151 214 57)(17 175 106 102 152 215 58)(18 176 107 69 153 216 59)(19 177 108 70 154 217 60)(20 178 109 71 155 218 61)(21 179 110 72 156 219 62)(22 180 111 73 157 220 63)(23 181 112 74 158 221 64)(24 182 113 75 159 222 65)(25 183 114 76 160 223 66)(26 184 115 77 161 224 67)(27 185 116 78 162 225 68)(28 186 117 79 163 226 35)(29 187 118 80 164 227 36)(30 188 119 81 165 228 37)(31 189 120 82 166 229 38)(32 190 121 83 167 230 39)(33 191 122 84 168 231 40)(34 192 123 85 169 232 41)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)(103 166)(104 167)(105 168)(106 169)(107 170)(108 137)(109 138)(110 139)(111 140)(112 141)(113 142)(114 143)(115 144)(116 145)(117 146)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 161)(133 162)(134 163)(135 164)(136 165)(171 228)(172 229)(173 230)(174 231)(175 232)(176 233)(177 234)(178 235)(179 236)(180 237)(181 238)(182 205)(183 206)(184 207)(185 208)(186 209)(187 210)(188 211)(189 212)(190 213)(191 214)(192 215)(193 216)(194 217)(195 218)(196 219)(197 220)(198 221)(199 222)(200 223)(201 224)(202 225)(203 226)(204 227)

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

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

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

170 conjugacy classes

 class 1 2A 2B 2C 7A 7B 7C 14A 14B 14C 17A ··· 17P 34A ··· 34P 34Q ··· 34AV 119A ··· 119AV 238A ··· 238AV order 1 2 2 2 7 7 7 14 14 14 17 ··· 17 34 ··· 34 34 ··· 34 119 ··· 119 238 ··· 238 size 1 1 7 7 2 2 2 2 2 2 1 ··· 1 1 ··· 1 7 ··· 7 2 ··· 2 2 ··· 2

170 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C17 C34 C34 D7 D14 D7×C17 D7×C34 kernel D7×C34 D7×C17 C238 D14 D7 C14 C34 C17 C2 C1 # reps 1 2 1 16 32 16 3 3 48 48

Matrix representation of D7×C34 in GL3(𝔽239) generated by

 238 0 0 0 163 0 0 0 163
,
 1 0 0 0 0 1 0 238 198
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(239))| [238,0,0,0,163,0,0,0,163],[1,0,0,0,0,238,0,1,198],[1,0,0,0,0,1,0,1,0] >;

D7×C34 in GAP, Magma, Sage, TeX

D_7\times C_{34}
% in TeX

G:=Group("D7xC34");
// GroupNames label

G:=SmallGroup(476,9);
// by ID

G=gap.SmallGroup(476,9);
# by ID

G:=PCGroup([4,-2,-2,-17,-7,6531]);
// Polycyclic

G:=Group<a,b,c|a^34=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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