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G = C22×C62order 248 = 23·31

Abelian group of type [2,2,62]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C62, SmallGroup(248,12)

Series: Derived Chief Lower central Upper central

C1 — C22×C62
C1C31C62C2×C62 — C22×C62
C1 — C22×C62
C1 — C22×C62

Generators and relations for C22×C62
 G = < a,b,c | a2=b2=c62=1, ab=ba, ac=ca, bc=cb >


Smallest permutation representation of C22×C62
Regular action on 248 points
Generators in S248
(1 167)(2 168)(3 169)(4 170)(5 171)(6 172)(7 173)(8 174)(9 175)(10 176)(11 177)(12 178)(13 179)(14 180)(15 181)(16 182)(17 183)(18 184)(19 185)(20 186)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(61 165)(62 166)(63 194)(64 195)(65 196)(66 197)(67 198)(68 199)(69 200)(70 201)(71 202)(72 203)(73 204)(74 205)(75 206)(76 207)(77 208)(78 209)(79 210)(80 211)(81 212)(82 213)(83 214)(84 215)(85 216)(86 217)(87 218)(88 219)(89 220)(90 221)(91 222)(92 223)(93 224)(94 225)(95 226)(96 227)(97 228)(98 229)(99 230)(100 231)(101 232)(102 233)(103 234)(104 235)(105 236)(106 237)(107 238)(108 239)(109 240)(110 241)(111 242)(112 243)(113 244)(114 245)(115 246)(116 247)(117 248)(118 187)(119 188)(120 189)(121 190)(122 191)(123 192)(124 193)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(125 204)(126 205)(127 206)(128 207)(129 208)(130 209)(131 210)(132 211)(133 212)(134 213)(135 214)(136 215)(137 216)(138 217)(139 218)(140 219)(141 220)(142 221)(143 222)(144 223)(145 224)(146 225)(147 226)(148 227)(149 228)(150 229)(151 230)(152 231)(153 232)(154 233)(155 234)(156 235)(157 236)(158 237)(159 238)(160 239)(161 240)(162 241)(163 242)(164 243)(165 244)(166 245)(167 246)(168 247)(169 248)(170 187)(171 188)(172 189)(173 190)(174 191)(175 192)(176 193)(177 194)(178 195)(179 196)(180 197)(181 198)(182 199)(183 200)(184 201)(185 202)(186 203)
(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)

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

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

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

C22×C62 is a maximal subgroup of   C23.D31

248 conjugacy classes

class 1 2A···2G31A···31AD62A···62HB
order12···231···3162···62
size11···11···11···1

248 irreducible representations

dim1111
type++
imageC1C2C31C62
kernelC22×C62C2×C62C23C22
# reps1730210

Matrix representation of C22×C62 in GL3(𝔽311) generated by

100
03100
001
,
100
010
00310
,
6100
0830
00225
G:=sub<GL(3,GF(311))| [1,0,0,0,310,0,0,0,1],[1,0,0,0,1,0,0,0,310],[61,0,0,0,83,0,0,0,225] >;

C22×C62 in GAP, Magma, Sage, TeX

C_2^2\times C_{62}
% in TeX

G:=Group("C2^2xC62");
// GroupNames label

G:=SmallGroup(248,12);
// by ID

G=gap.SmallGroup(248,12);
# by ID

G:=PCGroup([4,-2,-2,-2,-31]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^62=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of C22×C62 in TeX

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