direct product, metabelian, nilpotent (class 2), monomial
Aliases: M4(2)×C33, C62.16C12, C12.47C62, C24⋊7(C3×C6), (C3×C24)⋊19C6, C8⋊3(C32×C6), C4.(C32×C12), (C6×C12).50C6, C6.21(C6×C12), (C3×C62).5C4, C4.6(C3×C62), (C3×C12).28C12, C12.10(C3×C12), (C32×C24)⋊15C2, C22.(C32×C12), (C32×C12).16C4, (C32×C12).106C22, C2.3(C3×C6×C12), (C3×C6×C12).20C2, (C2×C6).12(C3×C12), (C3×C6).71(C2×C12), (C2×C12).26(C3×C6), (C2×C4).2(C32×C6), (C3×C12).113(C2×C6), (C32×C6).78(C2×C4), SmallGroup(432,516)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C33
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >
Subgroups: 308 in 280 conjugacy classes, 252 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C3×C6, C24, C2×C12, C33, C3×C12, C62, C3×M4(2), C32×C6, C32×C6, C3×C24, C6×C12, C32×C12, C3×C62, C32×M4(2), C32×C24, C3×C6×C12, M4(2)×C33
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C2×C12, C33, C3×C12, C62, C3×M4(2), C32×C6, C6×C12, C32×C12, C3×C62, C32×M4(2), C3×C6×C12, M4(2)×C33
►On 216 points
Generators in S216
(1 106 33)(2 107 34)(3 108 35)(4 109 36)(5 110 37)(6 111 38)(7 112 39)(8 105 40)(9 211 187)(10 212 188)(11 213 189)(12 214 190)(13 215 191)(14 216 192)(15 209 185)(16 210 186)(17 103 95)(18 104 96)(19 97 89)(20 98 90)(21 99 91)(22 100 92)(23 101 93)(24 102 94)(25 128 160)(26 121 153)(27 122 154)(28 123 155)(29 124 156)(30 125 157)(31 126 158)(32 127 159)(41 114 149)(42 115 150)(43 116 151)(44 117 152)(45 118 145)(46 119 146)(47 120 147)(48 113 148)(49 182 174)(50 183 175)(51 184 176)(52 177 169)(53 178 170)(54 179 171)(55 180 172)(56 181 173)(57 87 163)(58 88 164)(59 81 165)(60 82 166)(61 83 167)(62 84 168)(63 85 161)(64 86 162)(65 197 140)(66 198 141)(67 199 142)(68 200 143)(69 193 144)(70 194 137)(71 195 138)(72 196 139)(73 205 135)(74 206 136)(75 207 129)(76 208 130)(77 201 131)(78 202 132)(79 203 133)(80 204 134)
(1 171 17)(2 172 18)(3 173 19)(4 174 20)(5 175 21)(6 176 22)(7 169 23)(8 170 24)(9 125 71)(10 126 72)(11 127 65)(12 128 66)(13 121 67)(14 122 68)(15 123 69)(16 124 70)(25 141 190)(26 142 191)(27 143 192)(28 144 185)(29 137 186)(30 138 187)(31 139 188)(32 140 189)(33 179 95)(34 180 96)(35 181 89)(36 182 90)(37 183 91)(38 184 92)(39 177 93)(40 178 94)(41 87 133)(42 88 134)(43 81 135)(44 82 136)(45 83 129)(46 84 130)(47 85 131)(48 86 132)(49 98 109)(50 99 110)(51 100 111)(52 101 112)(53 102 105)(54 103 106)(55 104 107)(56 97 108)(57 203 149)(58 204 150)(59 205 151)(60 206 152)(61 207 145)(62 208 146)(63 201 147)(64 202 148)(73 116 165)(74 117 166)(75 118 167)(76 119 168)(77 120 161)(78 113 162)(79 114 163)(80 115 164)(153 199 215)(154 200 216)(155 193 209)(156 194 210)(157 195 211)(158 196 212)(159 197 213)(160 198 214)
(1 163 9)(2 164 10)(3 165 11)(4 166 12)(5 167 13)(6 168 14)(7 161 15)(8 162 16)(17 114 71)(18 115 72)(19 116 65)(20 117 66)(21 118 67)(22 119 68)(23 120 69)(24 113 70)(25 182 136)(26 183 129)(27 184 130)(28 177 131)(29 178 132)(30 179 133)(31 180 134)(32 181 135)(33 87 187)(34 88 188)(35 81 189)(36 82 190)(37 83 191)(38 84 192)(39 85 185)(40 86 186)(41 138 95)(42 139 96)(43 140 89)(44 141 90)(45 142 91)(46 143 92)(47 144 93)(48 137 94)(49 206 160)(50 207 153)(51 208 154)(52 201 155)(53 202 156)(54 203 157)(55 204 158)(56 205 159)(57 211 106)(58 212 107)(59 213 108)(60 214 109)(61 215 110)(62 216 111)(63 209 112)(64 210 105)(73 127 173)(74 128 174)(75 121 175)(76 122 176)(77 123 169)(78 124 170)(79 125 171)(80 126 172)(97 151 197)(98 152 198)(99 145 199)(100 146 200)(101 147 193)(102 148 194)(103 149 195)(104 150 196)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(90 94)(92 96)(98 102)(100 104)(105 109)(107 111)(113 117)(115 119)(122 126)(124 128)(130 134)(132 136)(137 141)(139 143)(146 150)(148 152)(154 158)(156 160)(162 166)(164 168)(170 174)(172 176)(178 182)(180 184)(186 190)(188 192)(194 198)(196 200)(202 206)(204 208)(210 214)(212 216)
G:=sub<Sym(216)| (1,106,33)(2,107,34)(3,108,35)(4,109,36)(5,110,37)(6,111,38)(7,112,39)(8,105,40)(9,211,187)(10,212,188)(11,213,189)(12,214,190)(13,215,191)(14,216,192)(15,209,185)(16,210,186)(17,103,95)(18,104,96)(19,97,89)(20,98,90)(21,99,91)(22,100,92)(23,101,93)(24,102,94)(25,128,160)(26,121,153)(27,122,154)(28,123,155)(29,124,156)(30,125,157)(31,126,158)(32,127,159)(41,114,149)(42,115,150)(43,116,151)(44,117,152)(45,118,145)(46,119,146)(47,120,147)(48,113,148)(49,182,174)(50,183,175)(51,184,176)(52,177,169)(53,178,170)(54,179,171)(55,180,172)(56,181,173)(57,87,163)(58,88,164)(59,81,165)(60,82,166)(61,83,167)(62,84,168)(63,85,161)(64,86,162)(65,197,140)(66,198,141)(67,199,142)(68,200,143)(69,193,144)(70,194,137)(71,195,138)(72,196,139)(73,205,135)(74,206,136)(75,207,129)(76,208,130)(77,201,131)(78,202,132)(79,203,133)(80,204,134), (1,171,17)(2,172,18)(3,173,19)(4,174,20)(5,175,21)(6,176,22)(7,169,23)(8,170,24)(9,125,71)(10,126,72)(11,127,65)(12,128,66)(13,121,67)(14,122,68)(15,123,69)(16,124,70)(25,141,190)(26,142,191)(27,143,192)(28,144,185)(29,137,186)(30,138,187)(31,139,188)(32,140,189)(33,179,95)(34,180,96)(35,181,89)(36,182,90)(37,183,91)(38,184,92)(39,177,93)(40,178,94)(41,87,133)(42,88,134)(43,81,135)(44,82,136)(45,83,129)(46,84,130)(47,85,131)(48,86,132)(49,98,109)(50,99,110)(51,100,111)(52,101,112)(53,102,105)(54,103,106)(55,104,107)(56,97,108)(57,203,149)(58,204,150)(59,205,151)(60,206,152)(61,207,145)(62,208,146)(63,201,147)(64,202,148)(73,116,165)(74,117,166)(75,118,167)(76,119,168)(77,120,161)(78,113,162)(79,114,163)(80,115,164)(153,199,215)(154,200,216)(155,193,209)(156,194,210)(157,195,211)(158,196,212)(159,197,213)(160,198,214), (1,163,9)(2,164,10)(3,165,11)(4,166,12)(5,167,13)(6,168,14)(7,161,15)(8,162,16)(17,114,71)(18,115,72)(19,116,65)(20,117,66)(21,118,67)(22,119,68)(23,120,69)(24,113,70)(25,182,136)(26,183,129)(27,184,130)(28,177,131)(29,178,132)(30,179,133)(31,180,134)(32,181,135)(33,87,187)(34,88,188)(35,81,189)(36,82,190)(37,83,191)(38,84,192)(39,85,185)(40,86,186)(41,138,95)(42,139,96)(43,140,89)(44,141,90)(45,142,91)(46,143,92)(47,144,93)(48,137,94)(49,206,160)(50,207,153)(51,208,154)(52,201,155)(53,202,156)(54,203,157)(55,204,158)(56,205,159)(57,211,106)(58,212,107)(59,213,108)(60,214,109)(61,215,110)(62,216,111)(63,209,112)(64,210,105)(73,127,173)(74,128,174)(75,121,175)(76,122,176)(77,123,169)(78,124,170)(79,125,171)(80,126,172)(97,151,197)(98,152,198)(99,145,199)(100,146,200)(101,147,193)(102,148,194)(103,149,195)(104,150,196), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(90,94)(92,96)(98,102)(100,104)(105,109)(107,111)(113,117)(115,119)(122,126)(124,128)(130,134)(132,136)(137,141)(139,143)(146,150)(148,152)(154,158)(156,160)(162,166)(164,168)(170,174)(172,176)(178,182)(180,184)(186,190)(188,192)(194,198)(196,200)(202,206)(204,208)(210,214)(212,216)>;
G:=Group( (1,106,33)(2,107,34)(3,108,35)(4,109,36)(5,110,37)(6,111,38)(7,112,39)(8,105,40)(9,211,187)(10,212,188)(11,213,189)(12,214,190)(13,215,191)(14,216,192)(15,209,185)(16,210,186)(17,103,95)(18,104,96)(19,97,89)(20,98,90)(21,99,91)(22,100,92)(23,101,93)(24,102,94)(25,128,160)(26,121,153)(27,122,154)(28,123,155)(29,124,156)(30,125,157)(31,126,158)(32,127,159)(41,114,149)(42,115,150)(43,116,151)(44,117,152)(45,118,145)(46,119,146)(47,120,147)(48,113,148)(49,182,174)(50,183,175)(51,184,176)(52,177,169)(53,178,170)(54,179,171)(55,180,172)(56,181,173)(57,87,163)(58,88,164)(59,81,165)(60,82,166)(61,83,167)(62,84,168)(63,85,161)(64,86,162)(65,197,140)(66,198,141)(67,199,142)(68,200,143)(69,193,144)(70,194,137)(71,195,138)(72,196,139)(73,205,135)(74,206,136)(75,207,129)(76,208,130)(77,201,131)(78,202,132)(79,203,133)(80,204,134), (1,171,17)(2,172,18)(3,173,19)(4,174,20)(5,175,21)(6,176,22)(7,169,23)(8,170,24)(9,125,71)(10,126,72)(11,127,65)(12,128,66)(13,121,67)(14,122,68)(15,123,69)(16,124,70)(25,141,190)(26,142,191)(27,143,192)(28,144,185)(29,137,186)(30,138,187)(31,139,188)(32,140,189)(33,179,95)(34,180,96)(35,181,89)(36,182,90)(37,183,91)(38,184,92)(39,177,93)(40,178,94)(41,87,133)(42,88,134)(43,81,135)(44,82,136)(45,83,129)(46,84,130)(47,85,131)(48,86,132)(49,98,109)(50,99,110)(51,100,111)(52,101,112)(53,102,105)(54,103,106)(55,104,107)(56,97,108)(57,203,149)(58,204,150)(59,205,151)(60,206,152)(61,207,145)(62,208,146)(63,201,147)(64,202,148)(73,116,165)(74,117,166)(75,118,167)(76,119,168)(77,120,161)(78,113,162)(79,114,163)(80,115,164)(153,199,215)(154,200,216)(155,193,209)(156,194,210)(157,195,211)(158,196,212)(159,197,213)(160,198,214), (1,163,9)(2,164,10)(3,165,11)(4,166,12)(5,167,13)(6,168,14)(7,161,15)(8,162,16)(17,114,71)(18,115,72)(19,116,65)(20,117,66)(21,118,67)(22,119,68)(23,120,69)(24,113,70)(25,182,136)(26,183,129)(27,184,130)(28,177,131)(29,178,132)(30,179,133)(31,180,134)(32,181,135)(33,87,187)(34,88,188)(35,81,189)(36,82,190)(37,83,191)(38,84,192)(39,85,185)(40,86,186)(41,138,95)(42,139,96)(43,140,89)(44,141,90)(45,142,91)(46,143,92)(47,144,93)(48,137,94)(49,206,160)(50,207,153)(51,208,154)(52,201,155)(53,202,156)(54,203,157)(55,204,158)(56,205,159)(57,211,106)(58,212,107)(59,213,108)(60,214,109)(61,215,110)(62,216,111)(63,209,112)(64,210,105)(73,127,173)(74,128,174)(75,121,175)(76,122,176)(77,123,169)(78,124,170)(79,125,171)(80,126,172)(97,151,197)(98,152,198)(99,145,199)(100,146,200)(101,147,193)(102,148,194)(103,149,195)(104,150,196), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(90,94)(92,96)(98,102)(100,104)(105,109)(107,111)(113,117)(115,119)(122,126)(124,128)(130,134)(132,136)(137,141)(139,143)(146,150)(148,152)(154,158)(156,160)(162,166)(164,168)(170,174)(172,176)(178,182)(180,184)(186,190)(188,192)(194,198)(196,200)(202,206)(204,208)(210,214)(212,216) );
G=PermutationGroup([[(1,106,33),(2,107,34),(3,108,35),(4,109,36),(5,110,37),(6,111,38),(7,112,39),(8,105,40),(9,211,187),(10,212,188),(11,213,189),(12,214,190),(13,215,191),(14,216,192),(15,209,185),(16,210,186),(17,103,95),(18,104,96),(19,97,89),(20,98,90),(21,99,91),(22,100,92),(23,101,93),(24,102,94),(25,128,160),(26,121,153),(27,122,154),(28,123,155),(29,124,156),(30,125,157),(31,126,158),(32,127,159),(41,114,149),(42,115,150),(43,116,151),(44,117,152),(45,118,145),(46,119,146),(47,120,147),(48,113,148),(49,182,174),(50,183,175),(51,184,176),(52,177,169),(53,178,170),(54,179,171),(55,180,172),(56,181,173),(57,87,163),(58,88,164),(59,81,165),(60,82,166),(61,83,167),(62,84,168),(63,85,161),(64,86,162),(65,197,140),(66,198,141),(67,199,142),(68,200,143),(69,193,144),(70,194,137),(71,195,138),(72,196,139),(73,205,135),(74,206,136),(75,207,129),(76,208,130),(77,201,131),(78,202,132),(79,203,133),(80,204,134)], [(1,171,17),(2,172,18),(3,173,19),(4,174,20),(5,175,21),(6,176,22),(7,169,23),(8,170,24),(9,125,71),(10,126,72),(11,127,65),(12,128,66),(13,121,67),(14,122,68),(15,123,69),(16,124,70),(25,141,190),(26,142,191),(27,143,192),(28,144,185),(29,137,186),(30,138,187),(31,139,188),(32,140,189),(33,179,95),(34,180,96),(35,181,89),(36,182,90),(37,183,91),(38,184,92),(39,177,93),(40,178,94),(41,87,133),(42,88,134),(43,81,135),(44,82,136),(45,83,129),(46,84,130),(47,85,131),(48,86,132),(49,98,109),(50,99,110),(51,100,111),(52,101,112),(53,102,105),(54,103,106),(55,104,107),(56,97,108),(57,203,149),(58,204,150),(59,205,151),(60,206,152),(61,207,145),(62,208,146),(63,201,147),(64,202,148),(73,116,165),(74,117,166),(75,118,167),(76,119,168),(77,120,161),(78,113,162),(79,114,163),(80,115,164),(153,199,215),(154,200,216),(155,193,209),(156,194,210),(157,195,211),(158,196,212),(159,197,213),(160,198,214)], [(1,163,9),(2,164,10),(3,165,11),(4,166,12),(5,167,13),(6,168,14),(7,161,15),(8,162,16),(17,114,71),(18,115,72),(19,116,65),(20,117,66),(21,118,67),(22,119,68),(23,120,69),(24,113,70),(25,182,136),(26,183,129),(27,184,130),(28,177,131),(29,178,132),(30,179,133),(31,180,134),(32,181,135),(33,87,187),(34,88,188),(35,81,189),(36,82,190),(37,83,191),(38,84,192),(39,85,185),(40,86,186),(41,138,95),(42,139,96),(43,140,89),(44,141,90),(45,142,91),(46,143,92),(47,144,93),(48,137,94),(49,206,160),(50,207,153),(51,208,154),(52,201,155),(53,202,156),(54,203,157),(55,204,158),(56,205,159),(57,211,106),(58,212,107),(59,213,108),(60,214,109),(61,215,110),(62,216,111),(63,209,112),(64,210,105),(73,127,173),(74,128,174),(75,121,175),(76,122,176),(77,123,169),(78,124,170),(79,125,171),(80,126,172),(97,151,197),(98,152,198),(99,145,199),(100,146,200),(101,147,193),(102,148,194),(103,149,195),(104,150,196)], [(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)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(49,53),(51,55),(58,62),(60,64),(66,70),(68,72),(74,78),(76,80),(82,86),(84,88),(90,94),(92,96),(98,102),(100,104),(105,109),(107,111),(113,117),(115,119),(122,126),(124,128),(130,134),(132,136),(137,141),(139,143),(146,150),(148,152),(154,158),(156,160),(162,166),(164,168),(170,174),(172,176),(178,182),(180,184),(186,190),(188,192),(194,198),(196,200),(202,206),(204,208),(210,214),(212,216)]])
270 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3Z | 4A | 4B | 4C | 6A | ··· | 6Z | 6AA | ··· | 6AZ | 8A | 8B | 8C | 8D | 12A | ··· | 12AZ | 12BA | ··· | 12BZ | 24A | ··· | 24CZ |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | ··· | 1 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
270 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) |
| kernel | M4(2)×C33 | C32×C24 | C3×C6×C12 | C32×M4(2) | C32×C12 | C3×C62 | C3×C24 | C6×C12 | C3×C12 | C62 | C33 | C32 |
| # reps | 1 | 2 | 1 | 26 | 2 | 2 | 52 | 26 | 52 | 52 | 2 | 52 |
Matrix representation of M4(2)×C33 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 71 |
| 0 | 0 | 4 | 45 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[64,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,28,4,0,0,71,45],[72,0,0,0,0,72,0,0,0,0,1,28,0,0,0,72] >;
M4(2)×C33 in GAP, Magma, Sage, TeX
M_4(2)\times C_3^3
% in TeX
G:=Group("M4(2)xC3^3"); // GroupNames label
G:=SmallGroup(432,516);
// by ID
G=gap.SmallGroup(432,516);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,-2,756,3053,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=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^5>;
// generators/relations