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G = C22×D55order 440 = 23·5·11

Direct product of C22 and D55

Aliases: C22×D55, C102D22, C222D10, C552C23, C1102C22, (C2×C22)⋊3D5, (C2×C110)⋊3C2, (C2×C10)⋊3D11, C112(C22×D5), C52(C22×D11), SmallGroup(440,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C55 — C22×D55
 Chief series C1 — C11 — C55 — D55 — D110 — C22×D55
 Lower central C55 — C22×D55
 Upper central C1 — C22

Generators and relations for C22×D55
G = < a,b,c,d | a2=b2=c55=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 812 in 64 conjugacy classes, 31 normal (9 characteristic)
C1, C2, C2, C22, C22, C5, C23, D5, C10, C11, D10, C2×C10, D11, C22, C22×D5, D22, C2×C22, C55, C22×D11, D55, C110, D110, C2×C110, C22×D55
Quotients: C1, C2, C22, C23, D5, D10, D11, C22×D5, D22, C22×D11, D55, D110, C22×D55

Smallest permutation representation of C22×D55
On 220 points
Generators in S220
(1 117)(2 118)(3 119)(4 120)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 135)(20 136)(21 137)(22 138)(23 139)(24 140)(25 141)(26 142)(27 143)(28 144)(29 145)(30 146)(31 147)(32 148)(33 149)(34 150)(35 151)(36 152)(37 153)(38 154)(39 155)(40 156)(41 157)(42 158)(43 159)(44 160)(45 161)(46 162)(47 163)(48 164)(49 165)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 201)(57 202)(58 203)(59 204)(60 205)(61 206)(62 207)(63 208)(64 209)(65 210)(66 211)(67 212)(68 213)(69 214)(70 215)(71 216)(72 217)(73 218)(74 219)(75 220)(76 166)(77 167)(78 168)(79 169)(80 170)(81 171)(82 172)(83 173)(84 174)(85 175)(86 176)(87 177)(88 178)(89 179)(90 180)(91 181)(92 182)(93 183)(94 184)(95 185)(96 186)(97 187)(98 188)(99 189)(100 190)(101 191)(102 192)(103 193)(104 194)(105 195)(106 196)(107 197)(108 198)(109 199)(110 200)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 109)(44 110)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(111 206)(112 207)(113 208)(114 209)(115 210)(116 211)(117 212)(118 213)(119 214)(120 215)(121 216)(122 217)(123 218)(124 219)(125 220)(126 166)(127 167)(128 168)(129 169)(130 170)(131 171)(132 172)(133 173)(134 174)(135 175)(136 176)(137 177)(138 178)(139 179)(140 180)(141 181)(142 182)(143 183)(144 184)(145 185)(146 186)(147 187)(148 188)(149 189)(150 190)(151 191)(152 192)(153 193)(154 194)(155 195)(156 196)(157 197)(158 198)(159 199)(160 200)(161 201)(162 202)(163 203)(164 204)(165 205)
(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)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(56 77)(57 76)(58 75)(59 74)(60 73)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(111 122)(112 121)(113 120)(114 119)(115 118)(116 117)(123 165)(124 164)(125 163)(126 162)(127 161)(128 160)(129 159)(130 158)(131 157)(132 156)(133 155)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(166 202)(167 201)(168 200)(169 199)(170 198)(171 197)(172 196)(173 195)(174 194)(175 193)(176 192)(177 191)(178 190)(179 189)(180 188)(181 187)(182 186)(183 185)(203 220)(204 219)(205 218)(206 217)(207 216)(208 215)(209 214)(210 213)(211 212)

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

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

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

116 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 10A ··· 10F 11A ··· 11E 22A ··· 22O 55A ··· 55T 110A ··· 110BH order 1 2 2 2 2 2 2 2 5 5 10 ··· 10 11 ··· 11 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 1 1 55 55 55 55 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

116 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 D5 D10 D11 D22 D55 D110 kernel C22×D55 D110 C2×C110 C2×C22 C22 C2×C10 C10 C22 C2 # reps 1 6 1 2 6 5 15 20 60

Matrix representation of C22×D55 in GL4(𝔽331) generated by

 330 0 0 0 0 330 0 0 0 0 330 0 0 0 0 330
,
 1 0 0 0 0 1 0 0 0 0 330 0 0 0 0 330
,
 124 172 0 0 159 330 0 0 0 0 34 122 0 0 209 234
,
 124 172 0 0 28 207 0 0 0 0 207 159 0 0 303 124
G:=sub<GL(4,GF(331))| [330,0,0,0,0,330,0,0,0,0,330,0,0,0,0,330],[1,0,0,0,0,1,0,0,0,0,330,0,0,0,0,330],[124,159,0,0,172,330,0,0,0,0,34,209,0,0,122,234],[124,28,0,0,172,207,0,0,0,0,207,303,0,0,159,124] >;

C22×D55 in GAP, Magma, Sage, TeX

C_2^2\times D_{55}
% in TeX

G:=Group("C2^2xD55");
// GroupNames label

G:=SmallGroup(440,50);
// by ID

G=gap.SmallGroup(440,50);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,643,10004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^55=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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