metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.6D26, Dic13.5D4, (C2×C4).8D26, C22⋊C4⋊5D13, C26.21(C2×D4), C2.10(D4×D13), D26⋊C4⋊6C2, (C2×Dic26)⋊3C2, C13⋊2(C4.4D4), C23.D13⋊5C2, (C4×Dic13)⋊12C2, C26.10(C4○D4), (C2×C52).54C22, (C2×C26).26C23, C2.9(D4⋊2D13), C2.12(D52⋊5C2), (C22×C26).15C22, (C22×D13).4C22, C22.44(C22×D13), (C2×Dic13).30C22, (C13×C22⋊C4)⋊7C2, (C2×C13⋊D4).4C2, SmallGroup(416,106)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.6D26
G = < a,b,c,d,e | a2=b2=c2=1, d26=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd25 >
Subgroups: 536 in 76 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C13, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, D13, C26, C26, C4.4D4, Dic13, Dic13, C52, D26, C2×C26, C2×C26, Dic26, C2×Dic13, C13⋊D4, C2×C52, C22×D13, C22×C26, C4×Dic13, D26⋊C4, C23.D13, C13×C22⋊C4, C2×Dic26, C2×C13⋊D4, C23.6D26
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C4.4D4, D26, C22×D13, D52⋊5C2, D4×D13, D4⋊2D13, C23.6D26
►On 208 points
Generators in S208
(1 116)(2 77)(3 118)(4 79)(5 120)(6 81)(7 122)(8 83)(9 124)(10 85)(11 126)(12 87)(13 128)(14 89)(15 130)(16 91)(17 132)(18 93)(19 134)(20 95)(21 136)(22 97)(23 138)(24 99)(25 140)(26 101)(27 142)(28 103)(29 144)(30 53)(31 146)(32 55)(33 148)(34 57)(35 150)(36 59)(37 152)(38 61)(39 154)(40 63)(41 156)(42 65)(43 106)(44 67)(45 108)(46 69)(47 110)(48 71)(49 112)(50 73)(51 114)(52 75)(54 164)(56 166)(58 168)(60 170)(62 172)(64 174)(66 176)(68 178)(70 180)(72 182)(74 184)(76 186)(78 188)(80 190)(82 192)(84 194)(86 196)(88 198)(90 200)(92 202)(94 204)(96 206)(98 208)(100 158)(102 160)(104 162)(105 175)(107 177)(109 179)(111 181)(113 183)(115 185)(117 187)(119 189)(121 191)(123 193)(125 195)(127 197)(129 199)(131 201)(133 203)(135 205)(137 207)(139 157)(141 159)(143 161)(145 163)(147 165)(149 167)(151 169)(153 171)(155 173)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 147)(122 148)(123 149)(124 150)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)(157 183)(158 184)(159 185)(160 186)(161 187)(162 188)(163 189)(164 190)(165 191)(166 192)(167 193)(168 194)(169 195)(170 196)(171 197)(172 198)(173 199)(174 200)(175 201)(176 202)(177 203)(178 204)(179 205)(180 206)(181 207)(182 208)
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 193)(9 194)(10 195)(11 196)(12 197)(13 198)(14 199)(15 200)(16 201)(17 202)(18 203)(19 204)(20 205)(21 206)(22 207)(23 208)(24 157)(25 158)(26 159)(27 160)(28 161)(29 162)(30 163)(31 164)(32 165)(33 166)(34 167)(35 168)(36 169)(37 170)(38 171)(39 172)(40 173)(41 174)(42 175)(43 176)(44 177)(45 178)(46 179)(47 180)(48 181)(49 182)(50 183)(51 184)(52 185)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)
(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 26 160 185)(2 184 161 25)(3 24 162 183)(4 182 163 23)(5 22 164 181)(6 180 165 21)(7 20 166 179)(8 178 167 19)(9 18 168 177)(10 176 169 17)(11 16 170 175)(12 174 171 15)(13 14 172 173)(27 52 186 159)(28 158 187 51)(29 50 188 157)(30 208 189 49)(31 48 190 207)(32 206 191 47)(33 46 192 205)(34 204 193 45)(35 44 194 203)(36 202 195 43)(37 42 196 201)(38 200 197 41)(39 40 198 199)(53 72 119 138)(54 137 120 71)(55 70 121 136)(56 135 122 69)(57 68 123 134)(58 133 124 67)(59 66 125 132)(60 131 126 65)(61 64 127 130)(62 129 128 63)(73 104 139 118)(74 117 140 103)(75 102 141 116)(76 115 142 101)(77 100 143 114)(78 113 144 99)(79 98 145 112)(80 111 146 97)(81 96 147 110)(82 109 148 95)(83 94 149 108)(84 107 150 93)(85 92 151 106)(86 105 152 91)(87 90 153 156)(88 155 154 89)
G:=sub<Sym(208)| (1,116)(2,77)(3,118)(4,79)(5,120)(6,81)(7,122)(8,83)(9,124)(10,85)(11,126)(12,87)(13,128)(14,89)(15,130)(16,91)(17,132)(18,93)(19,134)(20,95)(21,136)(22,97)(23,138)(24,99)(25,140)(26,101)(27,142)(28,103)(29,144)(30,53)(31,146)(32,55)(33,148)(34,57)(35,150)(36,59)(37,152)(38,61)(39,154)(40,63)(41,156)(42,65)(43,106)(44,67)(45,108)(46,69)(47,110)(48,71)(49,112)(50,73)(51,114)(52,75)(54,164)(56,166)(58,168)(60,170)(62,172)(64,174)(66,176)(68,178)(70,180)(72,182)(74,184)(76,186)(78,188)(80,190)(82,192)(84,194)(86,196)(88,198)(90,200)(92,202)(94,204)(96,206)(98,208)(100,158)(102,160)(104,162)(105,175)(107,177)(109,179)(111,181)(113,183)(115,185)(117,187)(119,189)(121,191)(123,193)(125,195)(127,197)(129,199)(131,201)(133,203)(135,205)(137,207)(139,157)(141,159)(143,161)(145,163)(147,165)(149,167)(151,169)(153,171)(155,173), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(157,183)(158,184)(159,185)(160,186)(161,187)(162,188)(163,189)(164,190)(165,191)(166,192)(167,193)(168,194)(169,195)(170,196)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208), (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,201)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,157)(25,158)(26,159)(27,160)(28,161)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,169)(37,170)(38,171)(39,172)(40,173)(41,174)(42,175)(43,176)(44,177)(45,178)(46,179)(47,180)(48,181)(49,182)(50,183)(51,184)(52,185)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144), (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), (1,26,160,185)(2,184,161,25)(3,24,162,183)(4,182,163,23)(5,22,164,181)(6,180,165,21)(7,20,166,179)(8,178,167,19)(9,18,168,177)(10,176,169,17)(11,16,170,175)(12,174,171,15)(13,14,172,173)(27,52,186,159)(28,158,187,51)(29,50,188,157)(30,208,189,49)(31,48,190,207)(32,206,191,47)(33,46,192,205)(34,204,193,45)(35,44,194,203)(36,202,195,43)(37,42,196,201)(38,200,197,41)(39,40,198,199)(53,72,119,138)(54,137,120,71)(55,70,121,136)(56,135,122,69)(57,68,123,134)(58,133,124,67)(59,66,125,132)(60,131,126,65)(61,64,127,130)(62,129,128,63)(73,104,139,118)(74,117,140,103)(75,102,141,116)(76,115,142,101)(77,100,143,114)(78,113,144,99)(79,98,145,112)(80,111,146,97)(81,96,147,110)(82,109,148,95)(83,94,149,108)(84,107,150,93)(85,92,151,106)(86,105,152,91)(87,90,153,156)(88,155,154,89)>;
G:=Group( (1,116)(2,77)(3,118)(4,79)(5,120)(6,81)(7,122)(8,83)(9,124)(10,85)(11,126)(12,87)(13,128)(14,89)(15,130)(16,91)(17,132)(18,93)(19,134)(20,95)(21,136)(22,97)(23,138)(24,99)(25,140)(26,101)(27,142)(28,103)(29,144)(30,53)(31,146)(32,55)(33,148)(34,57)(35,150)(36,59)(37,152)(38,61)(39,154)(40,63)(41,156)(42,65)(43,106)(44,67)(45,108)(46,69)(47,110)(48,71)(49,112)(50,73)(51,114)(52,75)(54,164)(56,166)(58,168)(60,170)(62,172)(64,174)(66,176)(68,178)(70,180)(72,182)(74,184)(76,186)(78,188)(80,190)(82,192)(84,194)(86,196)(88,198)(90,200)(92,202)(94,204)(96,206)(98,208)(100,158)(102,160)(104,162)(105,175)(107,177)(109,179)(111,181)(113,183)(115,185)(117,187)(119,189)(121,191)(123,193)(125,195)(127,197)(129,199)(131,201)(133,203)(135,205)(137,207)(139,157)(141,159)(143,161)(145,163)(147,165)(149,167)(151,169)(153,171)(155,173), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(157,183)(158,184)(159,185)(160,186)(161,187)(162,188)(163,189)(164,190)(165,191)(166,192)(167,193)(168,194)(169,195)(170,196)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208), (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,201)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,157)(25,158)(26,159)(27,160)(28,161)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,169)(37,170)(38,171)(39,172)(40,173)(41,174)(42,175)(43,176)(44,177)(45,178)(46,179)(47,180)(48,181)(49,182)(50,183)(51,184)(52,185)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144), (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), (1,26,160,185)(2,184,161,25)(3,24,162,183)(4,182,163,23)(5,22,164,181)(6,180,165,21)(7,20,166,179)(8,178,167,19)(9,18,168,177)(10,176,169,17)(11,16,170,175)(12,174,171,15)(13,14,172,173)(27,52,186,159)(28,158,187,51)(29,50,188,157)(30,208,189,49)(31,48,190,207)(32,206,191,47)(33,46,192,205)(34,204,193,45)(35,44,194,203)(36,202,195,43)(37,42,196,201)(38,200,197,41)(39,40,198,199)(53,72,119,138)(54,137,120,71)(55,70,121,136)(56,135,122,69)(57,68,123,134)(58,133,124,67)(59,66,125,132)(60,131,126,65)(61,64,127,130)(62,129,128,63)(73,104,139,118)(74,117,140,103)(75,102,141,116)(76,115,142,101)(77,100,143,114)(78,113,144,99)(79,98,145,112)(80,111,146,97)(81,96,147,110)(82,109,148,95)(83,94,149,108)(84,107,150,93)(85,92,151,106)(86,105,152,91)(87,90,153,156)(88,155,154,89) );
G=PermutationGroup([[(1,116),(2,77),(3,118),(4,79),(5,120),(6,81),(7,122),(8,83),(9,124),(10,85),(11,126),(12,87),(13,128),(14,89),(15,130),(16,91),(17,132),(18,93),(19,134),(20,95),(21,136),(22,97),(23,138),(24,99),(25,140),(26,101),(27,142),(28,103),(29,144),(30,53),(31,146),(32,55),(33,148),(34,57),(35,150),(36,59),(37,152),(38,61),(39,154),(40,63),(41,156),(42,65),(43,106),(44,67),(45,108),(46,69),(47,110),(48,71),(49,112),(50,73),(51,114),(52,75),(54,164),(56,166),(58,168),(60,170),(62,172),(64,174),(66,176),(68,178),(70,180),(72,182),(74,184),(76,186),(78,188),(80,190),(82,192),(84,194),(86,196),(88,198),(90,200),(92,202),(94,204),(96,206),(98,208),(100,158),(102,160),(104,162),(105,175),(107,177),(109,179),(111,181),(113,183),(115,185),(117,187),(119,189),(121,191),(123,193),(125,195),(127,197),(129,199),(131,201),(133,203),(135,205),(137,207),(139,157),(141,159),(143,161),(145,163),(147,165),(149,167),(151,169),(153,171),(155,173)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(53,79),(54,80),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138),(113,139),(114,140),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146),(121,147),(122,148),(123,149),(124,150),(125,151),(126,152),(127,153),(128,154),(129,155),(130,156),(157,183),(158,184),(159,185),(160,186),(161,187),(162,188),(163,189),(164,190),(165,191),(166,192),(167,193),(168,194),(169,195),(170,196),(171,197),(172,198),(173,199),(174,200),(175,201),(176,202),(177,203),(178,204),(179,205),(180,206),(181,207),(182,208)], [(1,186),(2,187),(3,188),(4,189),(5,190),(6,191),(7,192),(8,193),(9,194),(10,195),(11,196),(12,197),(13,198),(14,199),(15,200),(16,201),(17,202),(18,203),(19,204),(20,205),(21,206),(22,207),(23,208),(24,157),(25,158),(26,159),(27,160),(28,161),(29,162),(30,163),(31,164),(32,165),(33,166),(34,167),(35,168),(36,169),(37,170),(38,171),(39,172),(40,173),(41,174),(42,175),(43,176),(44,177),(45,178),(46,179),(47,180),(48,181),(49,182),(50,183),(51,184),(52,185),(53,145),(54,146),(55,147),(56,148),(57,149),(58,150),(59,151),(60,152),(61,153),(62,154),(63,155),(64,156),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(73,113),(74,114),(75,115),(76,116),(77,117),(78,118),(79,119),(80,120),(81,121),(82,122),(83,123),(84,124),(85,125),(86,126),(87,127),(88,128),(89,129),(90,130),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144)], [(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)], [(1,26,160,185),(2,184,161,25),(3,24,162,183),(4,182,163,23),(5,22,164,181),(6,180,165,21),(7,20,166,179),(8,178,167,19),(9,18,168,177),(10,176,169,17),(11,16,170,175),(12,174,171,15),(13,14,172,173),(27,52,186,159),(28,158,187,51),(29,50,188,157),(30,208,189,49),(31,48,190,207),(32,206,191,47),(33,46,192,205),(34,204,193,45),(35,44,194,203),(36,202,195,43),(37,42,196,201),(38,200,197,41),(39,40,198,199),(53,72,119,138),(54,137,120,71),(55,70,121,136),(56,135,122,69),(57,68,123,134),(58,133,124,67),(59,66,125,132),(60,131,126,65),(61,64,127,130),(62,129,128,63),(73,104,139,118),(74,117,140,103),(75,102,141,116),(76,115,142,101),(77,100,143,114),(78,113,144,99),(79,98,145,112),(80,111,146,97),(81,96,147,110),(82,109,148,95),(83,94,149,108),(84,107,150,93),(85,92,151,106),(86,105,152,91),(87,90,153,156),(88,155,154,89)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AD | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 4 | 52 | 2 | 2 | 4 | 26 | 26 | 26 | 26 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | D13 | D26 | D26 | D52⋊5C2 | D4×D13 | D4⋊2D13 |
| kernel | C23.6D26 | C4×Dic13 | D26⋊C4 | C23.D13 | C13×C22⋊C4 | C2×Dic26 | C2×C13⋊D4 | Dic13 | C26 | C22⋊C4 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 6 | 12 | 6 | 24 | 6 | 6 |
Matrix representation of C23.6D26 ►in GL4(𝔽53) generated by
| 14 | 47 | 0 | 0 |
| 6 | 39 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 23 | 52 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 43 | 22 | 0 | 0 |
| 31 | 22 | 0 | 0 |
| 0 | 0 | 47 | 42 |
| 0 | 0 | 13 | 6 |
| 22 | 43 | 0 | 0 |
| 22 | 31 | 0 | 0 |
| 0 | 0 | 6 | 11 |
| 0 | 0 | 45 | 47 |
G:=sub<GL(4,GF(53))| [14,6,0,0,47,39,0,0,0,0,1,23,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[43,31,0,0,22,22,0,0,0,0,47,13,0,0,42,6],[22,22,0,0,43,31,0,0,0,0,6,45,0,0,11,47] >;
C23.6D26 in GAP, Magma, Sage, TeX
C_2^3._6D_{26} % in TeX
G:=Group("C2^3.6D26"); // GroupNames label
G:=SmallGroup(416,106);
// by ID
G=gap.SmallGroup(416,106);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,55,506,188,13829]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^26=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^25>;
// generators/relations