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## G = C22×C4×D13order 416 = 25·13

### Direct product of C22×C4 and D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C22×C4×D13
 Chief series C1 — C13 — C26 — D26 — C22×D13 — C23×D13 — C22×C4×D13
 Lower central C13 — C22×C4×D13
 Upper central C1 — C22×C4

Generators and relations for C22×C4×D13
G = < a,b,c,d,e | a2=b2=c4=d13=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1328 in 236 conjugacy classes, 145 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C23, C13, C22×C4, C22×C4, C24, D13, C26, C26, C23×C4, Dic13, C52, D26, C2×C26, C4×D13, C2×Dic13, C2×C52, C22×D13, C22×C26, C2×C4×D13, C22×Dic13, C22×C52, C23×D13, C22×C4×D13
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, D13, C23×C4, D26, C4×D13, C22×D13, C2×C4×D13, C23×D13, C22×C4×D13

Smallest permutation representation of C22×C4×D13
On 208 points
Generators in S208
(1 128)(2 129)(3 130)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 105)(26 106)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 144)(38 145)(39 146)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 131)(49 132)(50 133)(51 134)(52 135)(53 180)(54 181)(55 182)(56 170)(57 171)(58 172)(59 173)(60 174)(61 175)(62 176)(63 177)(64 178)(65 179)(66 160)(67 161)(68 162)(69 163)(70 164)(71 165)(72 166)(73 167)(74 168)(75 169)(76 157)(77 158)(78 159)(79 196)(80 197)(81 198)(82 199)(83 200)(84 201)(85 202)(86 203)(87 204)(88 205)(89 206)(90 207)(91 208)(92 183)(93 184)(94 185)(95 186)(96 187)(97 188)(98 189)(99 190)(100 191)(101 192)(102 193)(103 194)(104 195)
(1 78)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 53)(26 54)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 90)(41 91)(42 79)(43 80)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 89)(105 180)(106 181)(107 182)(108 170)(109 171)(110 172)(111 173)(112 174)(113 175)(114 176)(115 177)(116 178)(117 179)(118 162)(119 163)(120 164)(121 165)(122 166)(123 167)(124 168)(125 169)(126 157)(127 158)(128 159)(129 160)(130 161)(131 202)(132 203)(133 204)(134 205)(135 206)(136 207)(137 208)(138 196)(139 197)(140 198)(141 199)(142 200)(143 201)(144 186)(145 187)(146 188)(147 189)(148 190)(149 191)(150 192)(151 193)(152 194)(153 195)(154 183)(155 184)(156 185)
(1 29 24 41)(2 30 25 42)(3 31 26 43)(4 32 14 44)(5 33 15 45)(6 34 16 46)(7 35 17 47)(8 36 18 48)(9 37 19 49)(10 38 20 50)(11 39 21 51)(12 27 22 52)(13 28 23 40)(53 79 66 101)(54 80 67 102)(55 81 68 103)(56 82 69 104)(57 83 70 92)(58 84 71 93)(59 85 72 94)(60 86 73 95)(61 87 74 96)(62 88 75 97)(63 89 76 98)(64 90 77 99)(65 91 78 100)(105 138 129 150)(106 139 130 151)(107 140 118 152)(108 141 119 153)(109 142 120 154)(110 143 121 155)(111 131 122 156)(112 132 123 144)(113 133 124 145)(114 134 125 146)(115 135 126 147)(116 136 127 148)(117 137 128 149)(157 189 177 206)(158 190 178 207)(159 191 179 208)(160 192 180 196)(161 193 181 197)(162 194 182 198)(163 195 170 199)(164 183 171 200)(165 184 172 201)(166 185 173 202)(167 186 174 203)(168 187 175 204)(169 188 176 205)
(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)
(1 158)(2 157)(3 169)(4 168)(5 167)(6 166)(7 165)(8 164)(9 163)(10 162)(11 161)(12 160)(13 159)(14 175)(15 174)(16 173)(17 172)(18 171)(19 170)(20 182)(21 181)(22 180)(23 179)(24 178)(25 177)(26 176)(27 192)(28 191)(29 190)(30 189)(31 188)(32 187)(33 186)(34 185)(35 184)(36 183)(37 195)(38 194)(39 193)(40 208)(41 207)(42 206)(43 205)(44 204)(45 203)(46 202)(47 201)(48 200)(49 199)(50 198)(51 197)(52 196)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 108)(61 107)(62 106)(63 105)(64 117)(65 116)(66 126)(67 125)(68 124)(69 123)(70 122)(71 121)(72 120)(73 119)(74 118)(75 130)(76 129)(77 128)(78 127)(79 135)(80 134)(81 133)(82 132)(83 131)(84 143)(85 142)(86 141)(87 140)(88 139)(89 138)(90 137)(91 136)(92 156)(93 155)(94 154)(95 153)(96 152)(97 151)(98 150)(99 149)(100 148)(101 147)(102 146)(103 145)(104 144)

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

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

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

128 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4H 4I ··· 4P 13A ··· 13F 26A ··· 26AP 52A ··· 52AV order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 13 ··· 13 1 ··· 1 13 ··· 13 2 ··· 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 D13 D26 D26 C4×D13 kernel C22×C4×D13 C2×C4×D13 C22×Dic13 C22×C52 C23×D13 C22×D13 C22×C4 C2×C4 C23 C22 # reps 1 12 1 1 1 16 6 36 6 48

Matrix representation of C22×C4×D13 in GL5(𝔽53)

 52 0 0 0 0 0 52 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 52 0 0 0 0 0 1 0 0 0 0 0 1
,
 52 0 0 0 0 0 23 0 0 0 0 0 1 0 0 0 0 0 52 0 0 0 0 0 52
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 52 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(53))| [52,0,0,0,0,0,52,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,52,0,0,0,0,0,1,0,0,0,0,0,1],[52,0,0,0,0,0,23,0,0,0,0,0,1,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,52,0,0,0,1,9],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C22×C4×D13 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times D_{13}
% in TeX

G:=Group("C2^2xC4xD13");
// GroupNames label

G:=SmallGroup(416,213);
// by ID

G=gap.SmallGroup(416,213);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,69,13829]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^13=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽