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G = C6×D41order 492 = 22·3·41

Direct product of C6 and D41

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

Aliases: C6×D41, C82⋊C6, C2462C2, C1233C22, C41⋊(C2×C6), SmallGroup(492,9)

Series: Derived Chief Lower central Upper central

C1C41 — C6×D41
C1C41C123C3×D41 — C6×D41
C41 — C6×D41
C1C6

Generators and relations for C6×D41
 G = < a,b,c | a6=b41=c2=1, ab=ba, ac=ca, cbc=b-1 >

41C2
41C2
41C22
41C6
41C6
41C2×C6

Smallest permutation representation of C6×D41
On 246 points
Generators in S246
(1 179 86 130 72 207)(2 180 87 131 73 208)(3 181 88 132 74 209)(4 182 89 133 75 210)(5 183 90 134 76 211)(6 184 91 135 77 212)(7 185 92 136 78 213)(8 186 93 137 79 214)(9 187 94 138 80 215)(10 188 95 139 81 216)(11 189 96 140 82 217)(12 190 97 141 42 218)(13 191 98 142 43 219)(14 192 99 143 44 220)(15 193 100 144 45 221)(16 194 101 145 46 222)(17 195 102 146 47 223)(18 196 103 147 48 224)(19 197 104 148 49 225)(20 198 105 149 50 226)(21 199 106 150 51 227)(22 200 107 151 52 228)(23 201 108 152 53 229)(24 202 109 153 54 230)(25 203 110 154 55 231)(26 204 111 155 56 232)(27 205 112 156 57 233)(28 165 113 157 58 234)(29 166 114 158 59 235)(30 167 115 159 60 236)(31 168 116 160 61 237)(32 169 117 161 62 238)(33 170 118 162 63 239)(34 171 119 163 64 240)(35 172 120 164 65 241)(36 173 121 124 66 242)(37 174 122 125 67 243)(38 175 123 126 68 244)(39 176 83 127 69 245)(40 177 84 128 70 246)(41 178 85 129 71 206)
(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)
(1 129)(2 128)(3 127)(4 126)(5 125)(6 124)(7 164)(8 163)(9 162)(10 161)(11 160)(12 159)(13 158)(14 157)(15 156)(16 155)(17 154)(18 153)(19 152)(20 151)(21 150)(22 149)(23 148)(24 147)(25 146)(26 145)(27 144)(28 143)(29 142)(30 141)(31 140)(32 139)(33 138)(34 137)(35 136)(36 135)(37 134)(38 133)(39 132)(40 131)(41 130)(42 167)(43 166)(44 165)(45 205)(46 204)(47 203)(48 202)(49 201)(50 200)(51 199)(52 198)(53 197)(54 196)(55 195)(56 194)(57 193)(58 192)(59 191)(60 190)(61 189)(62 188)(63 187)(64 186)(65 185)(66 184)(67 183)(68 182)(69 181)(70 180)(71 179)(72 178)(73 177)(74 176)(75 175)(76 174)(77 173)(78 172)(79 171)(80 170)(81 169)(82 168)(83 209)(84 208)(85 207)(86 206)(87 246)(88 245)(89 244)(90 243)(91 242)(92 241)(93 240)(94 239)(95 238)(96 237)(97 236)(98 235)(99 234)(100 233)(101 232)(102 231)(103 230)(104 229)(105 228)(106 227)(107 226)(108 225)(109 224)(110 223)(111 222)(112 221)(113 220)(114 219)(115 218)(116 217)(117 216)(118 215)(119 214)(120 213)(121 212)(122 211)(123 210)

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

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

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

132 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F41A···41T82A···82T123A···123AN246A···246AN
order12223366666641···4182···82123···123246···246
size1141411111414141412···22···22···22···2

132 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D41D82C3×D41C6×D41
kernelC6×D41C3×D41C246D82D41C82C6C3C2C1
# reps12124220204040

Matrix representation of C6×D41 in GL2(𝔽739) generated by

4190
0419
,
1661
720267
,
668445
54571
G:=sub<GL(2,GF(739))| [419,0,0,419],[166,720,1,267],[668,545,445,71] >;

C6×D41 in GAP, Magma, Sage, TeX

C_6\times D_{41}
% in TeX

G:=Group("C6xD41");
// GroupNames label

G:=SmallGroup(492,9);
// by ID

G=gap.SmallGroup(492,9);
# by ID

G:=PCGroup([4,-2,-2,-3,-41,7683]);
// Polycyclic

G:=Group<a,b,c|a^6=b^41=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D41 in TeX

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