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G = C22×C66order 264 = 23·3·11

Abelian group of type [2,2,66]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C66, SmallGroup(264,39)

Series: Derived Chief Lower central Upper central

C1 — C22×C66
C1C11C33C66C2×C66 — C22×C66
C1 — C22×C66
C1 — C22×C66

Generators and relations for C22×C66
 G = < a,b,c | a2=b2=c66=1, ab=ba, ac=ca, bc=cb >

Subgroups: 64, all normal (8 characteristic)
C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C11, C2×C6 [×7], C22 [×7], C22×C6, C33, C2×C22 [×7], C66 [×7], C22×C22, C2×C66 [×7], C22×C66
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C11, C2×C6 [×7], C22 [×7], C22×C6, C33, C2×C22 [×7], C66 [×7], C22×C22, C2×C66 [×7], C22×C66

Smallest permutation representation of C22×C66
Regular action on 264 points
Generators in S264
(1 187)(2 188)(3 189)(4 190)(5 191)(6 192)(7 193)(8 194)(9 195)(10 196)(11 197)(12 198)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 171)(52 172)(53 173)(54 174)(55 175)(56 176)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 214)(68 215)(69 216)(70 217)(71 218)(72 219)(73 220)(74 221)(75 222)(76 223)(77 224)(78 225)(79 226)(80 227)(81 228)(82 229)(83 230)(84 231)(85 232)(86 233)(87 234)(88 235)(89 236)(90 237)(91 238)(92 239)(93 240)(94 241)(95 242)(96 243)(97 244)(98 245)(99 246)(100 247)(101 248)(102 249)(103 250)(104 251)(105 252)(106 253)(107 254)(108 255)(109 256)(110 257)(111 258)(112 259)(113 260)(114 261)(115 262)(116 263)(117 264)(118 199)(119 200)(120 201)(121 202)(122 203)(123 204)(124 205)(125 206)(126 207)(127 208)(128 209)(129 210)(130 211)(131 212)(132 213)
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 120)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 128)(42 129)(43 130)(44 131)(45 132)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(133 247)(134 248)(135 249)(136 250)(137 251)(138 252)(139 253)(140 254)(141 255)(142 256)(143 257)(144 258)(145 259)(146 260)(147 261)(148 262)(149 263)(150 264)(151 199)(152 200)(153 201)(154 202)(155 203)(156 204)(157 205)(158 206)(159 207)(160 208)(161 209)(162 210)(163 211)(164 212)(165 213)(166 214)(167 215)(168 216)(169 217)(170 218)(171 219)(172 220)(173 221)(174 222)(175 223)(176 224)(177 225)(178 226)(179 227)(180 228)(181 229)(182 230)(183 231)(184 232)(185 233)(186 234)(187 235)(188 236)(189 237)(190 238)(191 239)(192 240)(193 241)(194 242)(195 243)(196 244)(197 245)(198 246)
(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)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264)

G:=sub<Sym(264)| (1,187)(2,188)(3,189)(4,190)(5,191)(6,192)(7,193)(8,194)(9,195)(10,196)(11,197)(12,198)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,214)(68,215)(69,216)(70,217)(71,218)(72,219)(73,220)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,228)(82,229)(83,230)(84,231)(85,232)(86,233)(87,234)(88,235)(89,236)(90,237)(91,238)(92,239)(93,240)(94,241)(95,242)(96,243)(97,244)(98,245)(99,246)(100,247)(101,248)(102,249)(103,250)(104,251)(105,252)(106,253)(107,254)(108,255)(109,256)(110,257)(111,258)(112,259)(113,260)(114,261)(115,262)(116,263)(117,264)(118,199)(119,200)(120,201)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,128)(42,129)(43,130)(44,131)(45,132)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(133,247)(134,248)(135,249)(136,250)(137,251)(138,252)(139,253)(140,254)(141,255)(142,256)(143,257)(144,258)(145,259)(146,260)(147,261)(148,262)(149,263)(150,264)(151,199)(152,200)(153,201)(154,202)(155,203)(156,204)(157,205)(158,206)(159,207)(160,208)(161,209)(162,210)(163,211)(164,212)(165,213)(166,214)(167,215)(168,216)(169,217)(170,218)(171,219)(172,220)(173,221)(174,222)(175,223)(176,224)(177,225)(178,226)(179,227)(180,228)(181,229)(182,230)(183,231)(184,232)(185,233)(186,234)(187,235)(188,236)(189,237)(190,238)(191,239)(192,240)(193,241)(194,242)(195,243)(196,244)(197,245)(198,246), (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264)>;

G:=Group( (1,187)(2,188)(3,189)(4,190)(5,191)(6,192)(7,193)(8,194)(9,195)(10,196)(11,197)(12,198)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,214)(68,215)(69,216)(70,217)(71,218)(72,219)(73,220)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,228)(82,229)(83,230)(84,231)(85,232)(86,233)(87,234)(88,235)(89,236)(90,237)(91,238)(92,239)(93,240)(94,241)(95,242)(96,243)(97,244)(98,245)(99,246)(100,247)(101,248)(102,249)(103,250)(104,251)(105,252)(106,253)(107,254)(108,255)(109,256)(110,257)(111,258)(112,259)(113,260)(114,261)(115,262)(116,263)(117,264)(118,199)(119,200)(120,201)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,128)(42,129)(43,130)(44,131)(45,132)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(133,247)(134,248)(135,249)(136,250)(137,251)(138,252)(139,253)(140,254)(141,255)(142,256)(143,257)(144,258)(145,259)(146,260)(147,261)(148,262)(149,263)(150,264)(151,199)(152,200)(153,201)(154,202)(155,203)(156,204)(157,205)(158,206)(159,207)(160,208)(161,209)(162,210)(163,211)(164,212)(165,213)(166,214)(167,215)(168,216)(169,217)(170,218)(171,219)(172,220)(173,221)(174,222)(175,223)(176,224)(177,225)(178,226)(179,227)(180,228)(181,229)(182,230)(183,231)(184,232)(185,233)(186,234)(187,235)(188,236)(189,237)(190,238)(191,239)(192,240)(193,241)(194,242)(195,243)(196,244)(197,245)(198,246), (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264) );

G=PermutationGroup([(1,187),(2,188),(3,189),(4,190),(5,191),(6,192),(7,193),(8,194),(9,195),(10,196),(11,197),(12,198),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,141),(22,142),(23,143),(24,144),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(36,156),(37,157),(38,158),(39,159),(40,160),(41,161),(42,162),(43,163),(44,164),(45,165),(46,166),(47,167),(48,168),(49,169),(50,170),(51,171),(52,172),(53,173),(54,174),(55,175),(56,176),(57,177),(58,178),(59,179),(60,180),(61,181),(62,182),(63,183),(64,184),(65,185),(66,186),(67,214),(68,215),(69,216),(70,217),(71,218),(72,219),(73,220),(74,221),(75,222),(76,223),(77,224),(78,225),(79,226),(80,227),(81,228),(82,229),(83,230),(84,231),(85,232),(86,233),(87,234),(88,235),(89,236),(90,237),(91,238),(92,239),(93,240),(94,241),(95,242),(96,243),(97,244),(98,245),(99,246),(100,247),(101,248),(102,249),(103,250),(104,251),(105,252),(106,253),(107,254),(108,255),(109,256),(110,257),(111,258),(112,259),(113,260),(114,261),(115,262),(116,263),(117,264),(118,199),(119,200),(120,201),(121,202),(122,203),(123,204),(124,205),(125,206),(126,207),(127,208),(128,209),(129,210),(130,211),(131,212),(132,213)], [(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,112),(26,113),(27,114),(28,115),(29,116),(30,117),(31,118),(32,119),(33,120),(34,121),(35,122),(36,123),(37,124),(38,125),(39,126),(40,127),(41,128),(42,129),(43,130),(44,131),(45,132),(46,67),(47,68),(48,69),(49,70),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(133,247),(134,248),(135,249),(136,250),(137,251),(138,252),(139,253),(140,254),(141,255),(142,256),(143,257),(144,258),(145,259),(146,260),(147,261),(148,262),(149,263),(150,264),(151,199),(152,200),(153,201),(154,202),(155,203),(156,204),(157,205),(158,206),(159,207),(160,208),(161,209),(162,210),(163,211),(164,212),(165,213),(166,214),(167,215),(168,216),(169,217),(170,218),(171,219),(172,220),(173,221),(174,222),(175,223),(176,224),(177,225),(178,226),(179,227),(180,228),(181,229),(182,230),(183,231),(184,232),(185,233),(186,234),(187,235),(188,236),(189,237),(190,238),(191,239),(192,240),(193,241),(194,242),(195,243),(196,244),(197,245),(198,246)], [(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),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264)])

264 conjugacy classes

class 1 2A···2G3A3B6A···6N11A···11J22A···22BR33A···33T66A···66EJ
order12···2336···611···1122···2233···3366···66
size11···1111···11···11···11···11···1

264 irreducible representations

dim11111111
type++
imageC1C2C3C6C11C22C33C66
kernelC22×C66C2×C66C22×C22C2×C22C22×C6C2×C6C23C22
# reps17214107020140

Matrix representation of C22×C66 in GL3(𝔽67) generated by

6600
0660
0066
,
6600
010
0066
,
4000
0380
0051
G:=sub<GL(3,GF(67))| [66,0,0,0,66,0,0,0,66],[66,0,0,0,1,0,0,0,66],[40,0,0,0,38,0,0,0,51] >;

C22×C66 in GAP, Magma, Sage, TeX

C_2^2\times C_{66}
% in TeX

G:=Group("C2^2xC66");
// GroupNames label

G:=SmallGroup(264,39);
// by ID

G=gap.SmallGroup(264,39);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-11]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^66=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽