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## G = M4(2)×C33order 432 = 24·33

### Direct product of C33 and M4(2)

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C33
 Chief series C1 — C2 — C4 — C12 — C3×C12 — C32×C12 — C32×C24 — M4(2)×C33
 Lower central C1 — C2 — M4(2)×C33
 Upper central C1 — C32×C12 — M4(2)×C33

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 [×13], C4 [×2], C22, C6 [×13], C6 [×13], C8 [×2], C2×C4, C32 [×13], C12 [×26], C2×C6 [×13], M4(2), C3×C6 [×13], C3×C6 [×13], C24 [×26], C2×C12 [×13], C33, C3×C12 [×26], C62 [×13], C3×M4(2) [×13], C32×C6, C32×C6, C3×C24 [×26], C6×C12 [×13], C32×C12 [×2], C3×C62, C32×M4(2) [×13], C32×C24 [×2], C3×C6×C12, M4(2)×C33
Quotients: C1, C2 [×3], C3 [×13], C4 [×2], C22, C6 [×39], C2×C4, C32 [×13], C12 [×26], C2×C6 [×13], M4(2), C3×C6 [×39], C2×C12 [×13], C33, C3×C12 [×26], C62 [×13], C3×M4(2) [×13], C32×C6 [×3], C6×C12 [×13], C32×C12 [×2], C3×C62, C32×M4(2) [×13], C3×C6×C12, M4(2)×C33

Smallest permutation representation of M4(2)×C33
On 216 points
Generators in S216
(1 106 45)(2 107 46)(3 108 47)(4 109 48)(5 110 41)(6 111 42)(7 112 43)(8 105 44)(9 211 187)(10 212 188)(11 213 189)(12 214 190)(13 215 191)(14 216 192)(15 209 185)(16 210 186)(17 149 130)(18 150 131)(19 151 132)(20 152 133)(21 145 134)(22 146 135)(23 147 136)(24 148 129)(25 53 40)(26 54 33)(27 55 34)(28 56 35)(29 49 36)(30 50 37)(31 51 38)(32 52 39)(57 181 173)(58 182 174)(59 183 175)(60 184 176)(61 177 169)(62 178 170)(63 179 171)(64 180 172)(65 197 140)(66 198 141)(67 199 142)(68 200 143)(69 193 144)(70 194 137)(71 195 138)(72 196 139)(73 205 124)(74 206 125)(75 207 126)(76 208 127)(77 201 128)(78 202 121)(79 203 122)(80 204 123)(81 165 156)(82 166 157)(83 167 158)(84 168 159)(85 161 160)(86 162 153)(87 163 154)(88 164 155)(89 113 97)(90 114 98)(91 115 99)(92 116 100)(93 117 101)(94 118 102)(95 119 103)(96 120 104)
(1 171 119)(2 172 120)(3 173 113)(4 174 114)(5 175 115)(6 176 116)(7 169 117)(8 170 118)(9 22 71)(10 23 72)(11 24 65)(12 17 66)(13 18 67)(14 19 68)(15 20 69)(16 21 70)(25 165 73)(26 166 74)(27 167 75)(28 168 76)(29 161 77)(30 162 78)(31 163 79)(32 164 80)(33 82 125)(34 83 126)(35 84 127)(36 85 128)(37 86 121)(38 87 122)(39 88 123)(40 81 124)(41 183 91)(42 184 92)(43 177 93)(44 178 94)(45 179 95)(46 180 96)(47 181 89)(48 182 90)(49 160 201)(50 153 202)(51 154 203)(52 155 204)(53 156 205)(54 157 206)(55 158 207)(56 159 208)(57 97 108)(58 98 109)(59 99 110)(60 100 111)(61 101 112)(62 102 105)(63 103 106)(64 104 107)(129 140 189)(130 141 190)(131 142 191)(132 143 192)(133 144 185)(134 137 186)(135 138 187)(136 139 188)(145 194 210)(146 195 211)(147 196 212)(148 197 213)(149 198 214)(150 199 215)(151 200 216)(152 193 209)
(1 163 9)(2 164 10)(3 165 11)(4 166 12)(5 167 13)(6 168 14)(7 161 15)(8 162 16)(17 174 74)(18 175 75)(19 176 76)(20 169 77)(21 170 78)(22 171 79)(23 172 80)(24 173 73)(25 65 113)(26 66 114)(27 67 115)(28 68 116)(29 69 117)(30 70 118)(31 71 119)(32 72 120)(33 141 90)(34 142 91)(35 143 92)(36 144 93)(37 137 94)(38 138 95)(39 139 96)(40 140 89)(41 83 191)(42 84 192)(43 85 185)(44 86 186)(45 87 187)(46 88 188)(47 81 189)(48 82 190)(49 193 101)(50 194 102)(51 195 103)(52 196 104)(53 197 97)(54 198 98)(55 199 99)(56 200 100)(57 205 148)(58 206 149)(59 207 150)(60 208 151)(61 201 152)(62 202 145)(63 203 146)(64 204 147)(105 153 210)(106 154 211)(107 155 212)(108 156 213)(109 157 214)(110 158 215)(111 159 216)(112 160 209)(121 134 178)(122 135 179)(123 136 180)(124 129 181)(125 130 182)(126 131 183)(127 132 184)(128 133 177)
(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)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(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)(114 118)(116 120)(121 125)(123 127)(130 134)(132 136)(137 141)(139 143)(145 149)(147 151)(153 157)(155 159)(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)| (1,106,45)(2,107,46)(3,108,47)(4,109,48)(5,110,41)(6,111,42)(7,112,43)(8,105,44)(9,211,187)(10,212,188)(11,213,189)(12,214,190)(13,215,191)(14,216,192)(15,209,185)(16,210,186)(17,149,130)(18,150,131)(19,151,132)(20,152,133)(21,145,134)(22,146,135)(23,147,136)(24,148,129)(25,53,40)(26,54,33)(27,55,34)(28,56,35)(29,49,36)(30,50,37)(31,51,38)(32,52,39)(57,181,173)(58,182,174)(59,183,175)(60,184,176)(61,177,169)(62,178,170)(63,179,171)(64,180,172)(65,197,140)(66,198,141)(67,199,142)(68,200,143)(69,193,144)(70,194,137)(71,195,138)(72,196,139)(73,205,124)(74,206,125)(75,207,126)(76,208,127)(77,201,128)(78,202,121)(79,203,122)(80,204,123)(81,165,156)(82,166,157)(83,167,158)(84,168,159)(85,161,160)(86,162,153)(87,163,154)(88,164,155)(89,113,97)(90,114,98)(91,115,99)(92,116,100)(93,117,101)(94,118,102)(95,119,103)(96,120,104), (1,171,119)(2,172,120)(3,173,113)(4,174,114)(5,175,115)(6,176,116)(7,169,117)(8,170,118)(9,22,71)(10,23,72)(11,24,65)(12,17,66)(13,18,67)(14,19,68)(15,20,69)(16,21,70)(25,165,73)(26,166,74)(27,167,75)(28,168,76)(29,161,77)(30,162,78)(31,163,79)(32,164,80)(33,82,125)(34,83,126)(35,84,127)(36,85,128)(37,86,121)(38,87,122)(39,88,123)(40,81,124)(41,183,91)(42,184,92)(43,177,93)(44,178,94)(45,179,95)(46,180,96)(47,181,89)(48,182,90)(49,160,201)(50,153,202)(51,154,203)(52,155,204)(53,156,205)(54,157,206)(55,158,207)(56,159,208)(57,97,108)(58,98,109)(59,99,110)(60,100,111)(61,101,112)(62,102,105)(63,103,106)(64,104,107)(129,140,189)(130,141,190)(131,142,191)(132,143,192)(133,144,185)(134,137,186)(135,138,187)(136,139,188)(145,194,210)(146,195,211)(147,196,212)(148,197,213)(149,198,214)(150,199,215)(151,200,216)(152,193,209), (1,163,9)(2,164,10)(3,165,11)(4,166,12)(5,167,13)(6,168,14)(7,161,15)(8,162,16)(17,174,74)(18,175,75)(19,176,76)(20,169,77)(21,170,78)(22,171,79)(23,172,80)(24,173,73)(25,65,113)(26,66,114)(27,67,115)(28,68,116)(29,69,117)(30,70,118)(31,71,119)(32,72,120)(33,141,90)(34,142,91)(35,143,92)(36,144,93)(37,137,94)(38,138,95)(39,139,96)(40,140,89)(41,83,191)(42,84,192)(43,85,185)(44,86,186)(45,87,187)(46,88,188)(47,81,189)(48,82,190)(49,193,101)(50,194,102)(51,195,103)(52,196,104)(53,197,97)(54,198,98)(55,199,99)(56,200,100)(57,205,148)(58,206,149)(59,207,150)(60,208,151)(61,201,152)(62,202,145)(63,203,146)(64,204,147)(105,153,210)(106,154,211)(107,155,212)(108,156,213)(109,157,214)(110,158,215)(111,159,216)(112,160,209)(121,134,178)(122,135,179)(123,136,180)(124,129,181)(125,130,182)(126,131,183)(127,132,184)(128,133,177), (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)(17,21)(19,23)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(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)(114,118)(116,120)(121,125)(123,127)(130,134)(132,136)(137,141)(139,143)(145,149)(147,151)(153,157)(155,159)(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( (1,106,45)(2,107,46)(3,108,47)(4,109,48)(5,110,41)(6,111,42)(7,112,43)(8,105,44)(9,211,187)(10,212,188)(11,213,189)(12,214,190)(13,215,191)(14,216,192)(15,209,185)(16,210,186)(17,149,130)(18,150,131)(19,151,132)(20,152,133)(21,145,134)(22,146,135)(23,147,136)(24,148,129)(25,53,40)(26,54,33)(27,55,34)(28,56,35)(29,49,36)(30,50,37)(31,51,38)(32,52,39)(57,181,173)(58,182,174)(59,183,175)(60,184,176)(61,177,169)(62,178,170)(63,179,171)(64,180,172)(65,197,140)(66,198,141)(67,199,142)(68,200,143)(69,193,144)(70,194,137)(71,195,138)(72,196,139)(73,205,124)(74,206,125)(75,207,126)(76,208,127)(77,201,128)(78,202,121)(79,203,122)(80,204,123)(81,165,156)(82,166,157)(83,167,158)(84,168,159)(85,161,160)(86,162,153)(87,163,154)(88,164,155)(89,113,97)(90,114,98)(91,115,99)(92,116,100)(93,117,101)(94,118,102)(95,119,103)(96,120,104), (1,171,119)(2,172,120)(3,173,113)(4,174,114)(5,175,115)(6,176,116)(7,169,117)(8,170,118)(9,22,71)(10,23,72)(11,24,65)(12,17,66)(13,18,67)(14,19,68)(15,20,69)(16,21,70)(25,165,73)(26,166,74)(27,167,75)(28,168,76)(29,161,77)(30,162,78)(31,163,79)(32,164,80)(33,82,125)(34,83,126)(35,84,127)(36,85,128)(37,86,121)(38,87,122)(39,88,123)(40,81,124)(41,183,91)(42,184,92)(43,177,93)(44,178,94)(45,179,95)(46,180,96)(47,181,89)(48,182,90)(49,160,201)(50,153,202)(51,154,203)(52,155,204)(53,156,205)(54,157,206)(55,158,207)(56,159,208)(57,97,108)(58,98,109)(59,99,110)(60,100,111)(61,101,112)(62,102,105)(63,103,106)(64,104,107)(129,140,189)(130,141,190)(131,142,191)(132,143,192)(133,144,185)(134,137,186)(135,138,187)(136,139,188)(145,194,210)(146,195,211)(147,196,212)(148,197,213)(149,198,214)(150,199,215)(151,200,216)(152,193,209), (1,163,9)(2,164,10)(3,165,11)(4,166,12)(5,167,13)(6,168,14)(7,161,15)(8,162,16)(17,174,74)(18,175,75)(19,176,76)(20,169,77)(21,170,78)(22,171,79)(23,172,80)(24,173,73)(25,65,113)(26,66,114)(27,67,115)(28,68,116)(29,69,117)(30,70,118)(31,71,119)(32,72,120)(33,141,90)(34,142,91)(35,143,92)(36,144,93)(37,137,94)(38,138,95)(39,139,96)(40,140,89)(41,83,191)(42,84,192)(43,85,185)(44,86,186)(45,87,187)(46,88,188)(47,81,189)(48,82,190)(49,193,101)(50,194,102)(51,195,103)(52,196,104)(53,197,97)(54,198,98)(55,199,99)(56,200,100)(57,205,148)(58,206,149)(59,207,150)(60,208,151)(61,201,152)(62,202,145)(63,203,146)(64,204,147)(105,153,210)(106,154,211)(107,155,212)(108,156,213)(109,157,214)(110,158,215)(111,159,216)(112,160,209)(121,134,178)(122,135,179)(123,136,180)(124,129,181)(125,130,182)(126,131,183)(127,132,184)(128,133,177), (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)(17,21)(19,23)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(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)(114,118)(116,120)(121,125)(123,127)(130,134)(132,136)(137,141)(139,143)(145,149)(147,151)(153,157)(155,159)(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([(1,106,45),(2,107,46),(3,108,47),(4,109,48),(5,110,41),(6,111,42),(7,112,43),(8,105,44),(9,211,187),(10,212,188),(11,213,189),(12,214,190),(13,215,191),(14,216,192),(15,209,185),(16,210,186),(17,149,130),(18,150,131),(19,151,132),(20,152,133),(21,145,134),(22,146,135),(23,147,136),(24,148,129),(25,53,40),(26,54,33),(27,55,34),(28,56,35),(29,49,36),(30,50,37),(31,51,38),(32,52,39),(57,181,173),(58,182,174),(59,183,175),(60,184,176),(61,177,169),(62,178,170),(63,179,171),(64,180,172),(65,197,140),(66,198,141),(67,199,142),(68,200,143),(69,193,144),(70,194,137),(71,195,138),(72,196,139),(73,205,124),(74,206,125),(75,207,126),(76,208,127),(77,201,128),(78,202,121),(79,203,122),(80,204,123),(81,165,156),(82,166,157),(83,167,158),(84,168,159),(85,161,160),(86,162,153),(87,163,154),(88,164,155),(89,113,97),(90,114,98),(91,115,99),(92,116,100),(93,117,101),(94,118,102),(95,119,103),(96,120,104)], [(1,171,119),(2,172,120),(3,173,113),(4,174,114),(5,175,115),(6,176,116),(7,169,117),(8,170,118),(9,22,71),(10,23,72),(11,24,65),(12,17,66),(13,18,67),(14,19,68),(15,20,69),(16,21,70),(25,165,73),(26,166,74),(27,167,75),(28,168,76),(29,161,77),(30,162,78),(31,163,79),(32,164,80),(33,82,125),(34,83,126),(35,84,127),(36,85,128),(37,86,121),(38,87,122),(39,88,123),(40,81,124),(41,183,91),(42,184,92),(43,177,93),(44,178,94),(45,179,95),(46,180,96),(47,181,89),(48,182,90),(49,160,201),(50,153,202),(51,154,203),(52,155,204),(53,156,205),(54,157,206),(55,158,207),(56,159,208),(57,97,108),(58,98,109),(59,99,110),(60,100,111),(61,101,112),(62,102,105),(63,103,106),(64,104,107),(129,140,189),(130,141,190),(131,142,191),(132,143,192),(133,144,185),(134,137,186),(135,138,187),(136,139,188),(145,194,210),(146,195,211),(147,196,212),(148,197,213),(149,198,214),(150,199,215),(151,200,216),(152,193,209)], [(1,163,9),(2,164,10),(3,165,11),(4,166,12),(5,167,13),(6,168,14),(7,161,15),(8,162,16),(17,174,74),(18,175,75),(19,176,76),(20,169,77),(21,170,78),(22,171,79),(23,172,80),(24,173,73),(25,65,113),(26,66,114),(27,67,115),(28,68,116),(29,69,117),(30,70,118),(31,71,119),(32,72,120),(33,141,90),(34,142,91),(35,143,92),(36,144,93),(37,137,94),(38,138,95),(39,139,96),(40,140,89),(41,83,191),(42,84,192),(43,85,185),(44,86,186),(45,87,187),(46,88,188),(47,81,189),(48,82,190),(49,193,101),(50,194,102),(51,195,103),(52,196,104),(53,197,97),(54,198,98),(55,199,99),(56,200,100),(57,205,148),(58,206,149),(59,207,150),(60,208,151),(61,201,152),(62,202,145),(63,203,146),(64,204,147),(105,153,210),(106,154,211),(107,155,212),(108,156,213),(109,157,214),(110,158,215),(111,159,216),(112,160,209),(121,134,178),(122,135,179),(123,136,180),(124,129,181),(125,130,182),(126,131,183),(127,132,184),(128,133,177)], [(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),(17,21),(19,23),(26,30),(28,32),(33,37),(35,39),(42,46),(44,48),(50,54),(52,56),(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),(114,118),(116,120),(121,125),(123,127),(130,134),(132,136),(137,141),(139,143),(145,149),(147,151),(153,157),(155,159),(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

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