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## G = C2×C33⋊5C4order 216 = 23·33

### Direct product of C2 and C33⋊5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×C33⋊5C4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C33⋊5C4 — C2×C33⋊5C4
 Lower central C33 — C2×C33⋊5C4
 Upper central C1 — C22

Generators and relations for C2×C335C4
G = < a,b,c,d,e | a2=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 692 in 224 conjugacy classes, 143 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C6, C2×C4, C32, Dic3, C2×C6, C3×C6, C2×Dic3, C33, C3⋊Dic3, C62, C32×C6, C32×C6, C2×C3⋊Dic3, C335C4, C3×C62, C2×C335C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C3⋊S3, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C33⋊C2, C2×C3⋊Dic3, C335C4, C2×C33⋊C2, C2×C335C4

Smallest permutation representation of C2×C335C4
Regular action on 216 points
Generators in S216
(1 10)(2 11)(3 12)(4 9)(5 83)(6 84)(7 81)(8 82)(13 141)(14 142)(15 143)(16 144)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 106)(26 107)(27 108)(28 105)(29 139)(30 140)(31 137)(32 138)(33 115)(34 116)(35 113)(36 114)(37 119)(38 120)(39 117)(40 118)(41 123)(42 124)(43 121)(44 122)(45 127)(46 128)(47 125)(48 126)(49 131)(50 132)(51 129)(52 130)(53 154)(54 155)(55 156)(56 153)(57 172)(58 169)(59 170)(60 171)(61 176)(62 173)(63 174)(64 175)(65 180)(66 177)(67 178)(68 179)(69 184)(70 181)(71 182)(72 183)(73 188)(74 185)(75 186)(76 187)(77 215)(78 216)(79 213)(80 214)(85 196)(86 193)(87 194)(88 195)(89 200)(90 197)(91 198)(92 199)(93 204)(94 201)(95 202)(96 203)(97 208)(98 205)(99 206)(100 207)(101 212)(102 209)(103 210)(104 211)(109 167)(110 168)(111 165)(112 166)(133 164)(134 161)(135 162)(136 163)(157 192)(158 189)(159 190)(160 191)
(1 162 182)(2 183 163)(3 164 184)(4 181 161)(5 177 132)(6 129 178)(7 179 130)(8 131 180)(9 70 134)(10 135 71)(11 72 136)(12 133 69)(13 74 30)(14 31 75)(15 76 32)(16 29 73)(17 216 87)(18 88 213)(19 214 85)(20 86 215)(21 109 89)(22 90 110)(23 111 91)(24 92 112)(25 45 63)(26 64 46)(27 47 61)(28 62 48)(33 156 96)(34 93 153)(35 154 94)(36 95 155)(37 158 98)(38 99 159)(39 160 100)(40 97 157)(41 170 104)(42 101 171)(43 172 102)(44 103 169)(49 65 82)(50 83 66)(51 67 84)(52 81 68)(53 201 113)(54 114 202)(55 203 115)(56 116 204)(57 209 121)(58 122 210)(59 211 123)(60 124 212)(77 148 193)(78 194 145)(79 146 195)(80 196 147)(105 173 126)(106 127 174)(107 175 128)(108 125 176)(117 191 207)(118 208 192)(119 189 205)(120 206 190)(137 186 142)(138 143 187)(139 188 144)(140 141 185)(149 167 200)(150 197 168)(151 165 198)(152 199 166)
(1 211 112)(2 109 212)(3 209 110)(4 111 210)(5 78 99)(6 100 79)(7 80 97)(8 98 77)(9 165 103)(10 104 166)(11 167 101)(12 102 168)(13 153 128)(14 125 154)(15 155 126)(16 127 156)(17 120 50)(18 51 117)(19 118 52)(20 49 119)(21 124 163)(22 164 121)(23 122 161)(24 162 123)(25 115 188)(26 185 116)(27 113 186)(28 187 114)(29 174 96)(30 93 175)(31 176 94)(32 95 173)(33 73 106)(34 107 74)(35 75 108)(36 105 76)(37 148 131)(38 132 145)(39 146 129)(40 130 147)(41 152 135)(42 136 149)(43 150 133)(44 134 151)(45 55 144)(46 141 56)(47 53 142)(48 143 54)(57 90 184)(58 181 91)(59 92 182)(60 183 89)(61 201 137)(62 138 202)(63 203 139)(64 140 204)(65 189 86)(66 87 190)(67 191 88)(68 85 192)(69 172 197)(70 198 169)(71 170 199)(72 200 171)(81 214 208)(82 205 215)(83 216 206)(84 207 213)(157 179 196)(158 193 180)(159 177 194)(160 195 178)
(1 49 201)(2 202 50)(3 51 203)(4 204 52)(5 72 155)(6 156 69)(7 70 153)(8 154 71)(9 93 130)(10 131 94)(11 95 132)(12 129 96)(13 97 169)(14 170 98)(15 99 171)(16 172 100)(17 109 62)(18 63 110)(19 111 64)(20 61 112)(21 28 87)(22 88 25)(23 26 85)(24 86 27)(29 102 39)(30 40 103)(31 104 37)(32 38 101)(33 133 178)(34 179 134)(35 135 180)(36 177 136)(41 158 75)(42 76 159)(43 160 73)(44 74 157)(45 90 213)(46 214 91)(47 92 215)(48 216 89)(53 182 82)(54 83 183)(55 184 84)(56 81 181)(57 207 144)(58 141 208)(59 205 142)(60 143 206)(65 113 162)(66 163 114)(67 115 164)(68 161 116)(77 125 199)(78 200 126)(79 127 197)(80 198 128)(105 194 149)(106 150 195)(107 196 151)(108 152 193)(117 139 209)(118 210 140)(119 137 211)(120 212 138)(121 191 188)(122 185 192)(123 189 186)(124 187 190)(145 167 173)(146 174 168)(147 165 175)(148 176 166)
(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)

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

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

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

C2×C335C4 is a maximal subgroup of
Dic3×C3⋊Dic3  C62.77D6  C62.78D6  C62.80D6  C62.81D6  C62.82D6  C62.146D6  C62.147D6  C62.148D6  C63.C2  C2×S3×C3⋊Dic3  C62.91D6  C2×Dic3×C3⋊S3  C2×C4×C33⋊C2  C62.100D6
C2×C335C4 is a maximal quotient of
C3318M4(2)  C62.147D6  C63.C2

60 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3M 4A 4B 4C 4D 6A ··· 6AM order 1 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 size 1 1 1 1 2 ··· 2 27 27 27 27 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 S3 Dic3 D6 kernel C2×C33⋊5C4 C33⋊5C4 C3×C62 C32×C6 C62 C3×C6 C3×C6 # reps 1 2 1 4 13 26 13

Matrix representation of C2×C335C4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 9 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 9 0 0 0 0 0 9 3 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 10 2 0 0 0 0 9 3 0 0 0 0 0 0 0 12 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,9,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,9,9,0,0,0,0,0,3,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,10,9,0,0,0,0,2,3,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C2×C335C4 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_5C_4
% in TeX

G:=Group("C2xC3^3:5C4");
// GroupNames label

G:=SmallGroup(216,148);
// by ID

G=gap.SmallGroup(216,148);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,387,1444,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^4=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,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽