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G = D4×C61order 488 = 23·61

Direct product of C61 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C61, C4⋊C122, C2443C2, C22⋊C122, C122.6C22, (C2×C122)⋊1C2, C2.1(C2×C122), SmallGroup(488,10)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C61
C1C2C122C2×C122 — D4×C61
C1C2 — D4×C61
C1C122 — D4×C61

Generators and relations for D4×C61
 G = < a,b,c | a61=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C122
2C122

Smallest permutation representation of D4×C61
On 244 points
Generators in S244
(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)
(1 228 104 132)(2 229 105 133)(3 230 106 134)(4 231 107 135)(5 232 108 136)(6 233 109 137)(7 234 110 138)(8 235 111 139)(9 236 112 140)(10 237 113 141)(11 238 114 142)(12 239 115 143)(13 240 116 144)(14 241 117 145)(15 242 118 146)(16 243 119 147)(17 244 120 148)(18 184 121 149)(19 185 122 150)(20 186 62 151)(21 187 63 152)(22 188 64 153)(23 189 65 154)(24 190 66 155)(25 191 67 156)(26 192 68 157)(27 193 69 158)(28 194 70 159)(29 195 71 160)(30 196 72 161)(31 197 73 162)(32 198 74 163)(33 199 75 164)(34 200 76 165)(35 201 77 166)(36 202 78 167)(37 203 79 168)(38 204 80 169)(39 205 81 170)(40 206 82 171)(41 207 83 172)(42 208 84 173)(43 209 85 174)(44 210 86 175)(45 211 87 176)(46 212 88 177)(47 213 89 178)(48 214 90 179)(49 215 91 180)(50 216 92 181)(51 217 93 182)(52 218 94 183)(53 219 95 123)(54 220 96 124)(55 221 97 125)(56 222 98 126)(57 223 99 127)(58 224 100 128)(59 225 101 129)(60 226 102 130)(61 227 103 131)
(123 219)(124 220)(125 221)(126 222)(127 223)(128 224)(129 225)(130 226)(131 227)(132 228)(133 229)(134 230)(135 231)(136 232)(137 233)(138 234)(139 235)(140 236)(141 237)(142 238)(143 239)(144 240)(145 241)(146 242)(147 243)(148 244)(149 184)(150 185)(151 186)(152 187)(153 188)(154 189)(155 190)(156 191)(157 192)(158 193)(159 194)(160 195)(161 196)(162 197)(163 198)(164 199)(165 200)(166 201)(167 202)(168 203)(169 204)(170 205)(171 206)(172 207)(173 208)(174 209)(175 210)(176 211)(177 212)(178 213)(179 214)(180 215)(181 216)(182 217)(183 218)

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

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

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

305 conjugacy classes

class 1 2A2B2C 4 61A···61BH122A···122BH122BI···122FX244A···244BH
order1222461···61122···122122···122244···244
size112221···11···12···22···2

305 irreducible representations

dim11111122
type++++
imageC1C2C2C61C122C122D4D4×C61
kernelD4×C61C244C2×C122D4C4C22C61C1
# reps1126060120160

Matrix representation of D4×C61 in GL2(𝔽733) generated by

6120
0612
,
0732
10
,
10
0732
G:=sub<GL(2,GF(733))| [612,0,0,612],[0,1,732,0],[1,0,0,732] >;

D4×C61 in GAP, Magma, Sage, TeX

D_4\times C_{61}
% in TeX

G:=Group("D4xC61");
// GroupNames label

G:=SmallGroup(488,10);
// by ID

G=gap.SmallGroup(488,10);
# by ID

G:=PCGroup([4,-2,-2,-61,-2,1969]);
// Polycyclic

G:=Group<a,b,c|a^61=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D4×C61 in TeX

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