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G = M4(2)×Dic7order 448 = 26·7

Direct product of M4(2) and Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — M4(2)×Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4×Dic7 — C2×C4×Dic7 — M4(2)×Dic7
 Lower central C7 — C14 — M4(2)×Dic7
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for M4(2)×Dic7
G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 420 in 142 conjugacy classes, 91 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C4×M4(2), C2×C7⋊C8, C4.Dic7, C4×Dic7, C4×Dic7, C2×C56, C7×M4(2), C22×Dic7, C22×C28, C8×Dic7, C56⋊C4, C2×C4.Dic7, C2×C4×Dic7, C14×M4(2), M4(2)×Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, M4(2), C22×C4, Dic7, D14, C2×C42, C2×M4(2), C4×D7, C2×Dic7, C22×D7, C4×M4(2), C4×Dic7, C2×C4×D7, C22×Dic7, D7×M4(2), C2×C4×Dic7, M4(2)×Dic7

Smallest permutation representation of M4(2)×Dic7
On 224 points
Generators in S224
(1 18 200 139 30 77 107 184)(2 19 201 140 31 78 108 185)(3 20 202 127 32 79 109 186)(4 21 203 128 33 80 110 187)(5 22 204 129 34 81 111 188)(6 23 205 130 35 82 112 189)(7 24 206 131 36 83 99 190)(8 25 207 132 37 84 100 191)(9 26 208 133 38 71 101 192)(10 27 209 134 39 72 102 193)(11 28 210 135 40 73 103 194)(12 15 197 136 41 74 104 195)(13 16 198 137 42 75 105 196)(14 17 199 138 29 76 106 183)(43 92 121 153 162 213 61 171)(44 93 122 154 163 214 62 172)(45 94 123 141 164 215 63 173)(46 95 124 142 165 216 64 174)(47 96 125 143 166 217 65 175)(48 97 126 144 167 218 66 176)(49 98 113 145 168 219 67 177)(50 85 114 146 155 220 68 178)(51 86 115 147 156 221 69 179)(52 87 116 148 157 222 70 180)(53 88 117 149 158 223 57 181)(54 89 118 150 159 224 58 182)(55 90 119 151 160 211 59 169)(56 91 120 152 161 212 60 170)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 213)(86 214)(87 215)(88 216)(89 217)(90 218)(91 219)(92 220)(93 221)(94 222)(95 223)(96 224)(97 211)(98 212)(99 199)(100 200)(101 201)(102 202)(103 203)(104 204)(105 205)(106 206)(107 207)(108 208)(109 209)(110 210)(111 197)(112 198)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(141 180)(142 181)(143 182)(144 169)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)
(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 94 8 87)(2 93 9 86)(3 92 10 85)(4 91 11 98)(5 90 12 97)(6 89 13 96)(7 88 14 95)(15 126 22 119)(16 125 23 118)(17 124 24 117)(18 123 25 116)(19 122 26 115)(20 121 27 114)(21 120 28 113)(29 216 36 223)(30 215 37 222)(31 214 38 221)(32 213 39 220)(33 212 40 219)(34 211 41 218)(35 224 42 217)(43 193 50 186)(44 192 51 185)(45 191 52 184)(46 190 53 183)(47 189 54 196)(48 188 55 195)(49 187 56 194)(57 76 64 83)(58 75 65 82)(59 74 66 81)(60 73 67 80)(61 72 68 79)(62 71 69 78)(63 84 70 77)(99 181 106 174)(100 180 107 173)(101 179 108 172)(102 178 109 171)(103 177 110 170)(104 176 111 169)(105 175 112 182)(127 162 134 155)(128 161 135 168)(129 160 136 167)(130 159 137 166)(131 158 138 165)(132 157 139 164)(133 156 140 163)(141 207 148 200)(142 206 149 199)(143 205 150 198)(144 204 151 197)(145 203 152 210)(146 202 153 209)(147 201 154 208)

G:=sub<Sym(224)| (1,18,200,139,30,77,107,184)(2,19,201,140,31,78,108,185)(3,20,202,127,32,79,109,186)(4,21,203,128,33,80,110,187)(5,22,204,129,34,81,111,188)(6,23,205,130,35,82,112,189)(7,24,206,131,36,83,99,190)(8,25,207,132,37,84,100,191)(9,26,208,133,38,71,101,192)(10,27,209,134,39,72,102,193)(11,28,210,135,40,73,103,194)(12,15,197,136,41,74,104,195)(13,16,198,137,42,75,105,196)(14,17,199,138,29,76,106,183)(43,92,121,153,162,213,61,171)(44,93,122,154,163,214,62,172)(45,94,123,141,164,215,63,173)(46,95,124,142,165,216,64,174)(47,96,125,143,166,217,65,175)(48,97,126,144,167,218,66,176)(49,98,113,145,168,219,67,177)(50,85,114,146,155,220,68,178)(51,86,115,147,156,221,69,179)(52,87,116,148,157,222,70,180)(53,88,117,149,158,223,57,181)(54,89,118,150,159,224,58,182)(55,90,119,151,160,211,59,169)(56,91,120,152,161,212,60,170), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,213)(86,214)(87,215)(88,216)(89,217)(90,218)(91,219)(92,220)(93,221)(94,222)(95,223)(96,224)(97,211)(98,212)(99,199)(100,200)(101,201)(102,202)(103,203)(104,204)(105,205)(106,206)(107,207)(108,208)(109,209)(110,210)(111,197)(112,198)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,180)(142,181)(143,182)(144,169)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,94,8,87)(2,93,9,86)(3,92,10,85)(4,91,11,98)(5,90,12,97)(6,89,13,96)(7,88,14,95)(15,126,22,119)(16,125,23,118)(17,124,24,117)(18,123,25,116)(19,122,26,115)(20,121,27,114)(21,120,28,113)(29,216,36,223)(30,215,37,222)(31,214,38,221)(32,213,39,220)(33,212,40,219)(34,211,41,218)(35,224,42,217)(43,193,50,186)(44,192,51,185)(45,191,52,184)(46,190,53,183)(47,189,54,196)(48,188,55,195)(49,187,56,194)(57,76,64,83)(58,75,65,82)(59,74,66,81)(60,73,67,80)(61,72,68,79)(62,71,69,78)(63,84,70,77)(99,181,106,174)(100,180,107,173)(101,179,108,172)(102,178,109,171)(103,177,110,170)(104,176,111,169)(105,175,112,182)(127,162,134,155)(128,161,135,168)(129,160,136,167)(130,159,137,166)(131,158,138,165)(132,157,139,164)(133,156,140,163)(141,207,148,200)(142,206,149,199)(143,205,150,198)(144,204,151,197)(145,203,152,210)(146,202,153,209)(147,201,154,208)>;

G:=Group( (1,18,200,139,30,77,107,184)(2,19,201,140,31,78,108,185)(3,20,202,127,32,79,109,186)(4,21,203,128,33,80,110,187)(5,22,204,129,34,81,111,188)(6,23,205,130,35,82,112,189)(7,24,206,131,36,83,99,190)(8,25,207,132,37,84,100,191)(9,26,208,133,38,71,101,192)(10,27,209,134,39,72,102,193)(11,28,210,135,40,73,103,194)(12,15,197,136,41,74,104,195)(13,16,198,137,42,75,105,196)(14,17,199,138,29,76,106,183)(43,92,121,153,162,213,61,171)(44,93,122,154,163,214,62,172)(45,94,123,141,164,215,63,173)(46,95,124,142,165,216,64,174)(47,96,125,143,166,217,65,175)(48,97,126,144,167,218,66,176)(49,98,113,145,168,219,67,177)(50,85,114,146,155,220,68,178)(51,86,115,147,156,221,69,179)(52,87,116,148,157,222,70,180)(53,88,117,149,158,223,57,181)(54,89,118,150,159,224,58,182)(55,90,119,151,160,211,59,169)(56,91,120,152,161,212,60,170), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,213)(86,214)(87,215)(88,216)(89,217)(90,218)(91,219)(92,220)(93,221)(94,222)(95,223)(96,224)(97,211)(98,212)(99,199)(100,200)(101,201)(102,202)(103,203)(104,204)(105,205)(106,206)(107,207)(108,208)(109,209)(110,210)(111,197)(112,198)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,180)(142,181)(143,182)(144,169)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,94,8,87)(2,93,9,86)(3,92,10,85)(4,91,11,98)(5,90,12,97)(6,89,13,96)(7,88,14,95)(15,126,22,119)(16,125,23,118)(17,124,24,117)(18,123,25,116)(19,122,26,115)(20,121,27,114)(21,120,28,113)(29,216,36,223)(30,215,37,222)(31,214,38,221)(32,213,39,220)(33,212,40,219)(34,211,41,218)(35,224,42,217)(43,193,50,186)(44,192,51,185)(45,191,52,184)(46,190,53,183)(47,189,54,196)(48,188,55,195)(49,187,56,194)(57,76,64,83)(58,75,65,82)(59,74,66,81)(60,73,67,80)(61,72,68,79)(62,71,69,78)(63,84,70,77)(99,181,106,174)(100,180,107,173)(101,179,108,172)(102,178,109,171)(103,177,110,170)(104,176,111,169)(105,175,112,182)(127,162,134,155)(128,161,135,168)(129,160,136,167)(130,159,137,166)(131,158,138,165)(132,157,139,164)(133,156,140,163)(141,207,148,200)(142,206,149,199)(143,205,150,198)(144,204,151,197)(145,203,152,210)(146,202,153,209)(147,201,154,208) );

G=PermutationGroup([[(1,18,200,139,30,77,107,184),(2,19,201,140,31,78,108,185),(3,20,202,127,32,79,109,186),(4,21,203,128,33,80,110,187),(5,22,204,129,34,81,111,188),(6,23,205,130,35,82,112,189),(7,24,206,131,36,83,99,190),(8,25,207,132,37,84,100,191),(9,26,208,133,38,71,101,192),(10,27,209,134,39,72,102,193),(11,28,210,135,40,73,103,194),(12,15,197,136,41,74,104,195),(13,16,198,137,42,75,105,196),(14,17,199,138,29,76,106,183),(43,92,121,153,162,213,61,171),(44,93,122,154,163,214,62,172),(45,94,123,141,164,215,63,173),(46,95,124,142,165,216,64,174),(47,96,125,143,166,217,65,175),(48,97,126,144,167,218,66,176),(49,98,113,145,168,219,67,177),(50,85,114,146,155,220,68,178),(51,86,115,147,156,221,69,179),(52,87,116,148,157,222,70,180),(53,88,117,149,158,223,57,181),(54,89,118,150,159,224,58,182),(55,90,119,151,160,211,59,169),(56,91,120,152,161,212,60,170)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,213),(86,214),(87,215),(88,216),(89,217),(90,218),(91,219),(92,220),(93,221),(94,222),(95,223),(96,224),(97,211),(98,212),(99,199),(100,200),(101,201),(102,202),(103,203),(104,204),(105,205),(106,206),(107,207),(108,208),(109,209),(110,210),(111,197),(112,198),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(133,140),(141,180),(142,181),(143,182),(144,169),(145,170),(146,171),(147,172),(148,173),(149,174),(150,175),(151,176),(152,177),(153,178),(154,179),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(161,168),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196)], [(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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,94,8,87),(2,93,9,86),(3,92,10,85),(4,91,11,98),(5,90,12,97),(6,89,13,96),(7,88,14,95),(15,126,22,119),(16,125,23,118),(17,124,24,117),(18,123,25,116),(19,122,26,115),(20,121,27,114),(21,120,28,113),(29,216,36,223),(30,215,37,222),(31,214,38,221),(32,213,39,220),(33,212,40,219),(34,211,41,218),(35,224,42,217),(43,193,50,186),(44,192,51,185),(45,191,52,184),(46,190,53,183),(47,189,54,196),(48,188,55,195),(49,187,56,194),(57,76,64,83),(58,75,65,82),(59,74,66,81),(60,73,67,80),(61,72,68,79),(62,71,69,78),(63,84,70,77),(99,181,106,174),(100,180,107,173),(101,179,108,172),(102,178,109,171),(103,177,110,170),(104,176,111,169),(105,175,112,182),(127,162,134,155),(128,161,135,168),(129,160,136,167),(130,159,137,166),(131,158,138,165),(132,157,139,164),(133,156,140,163),(141,207,148,200),(142,206,149,199),(143,205,150,198),(144,204,151,197),(145,203,152,210),(146,202,153,209),(147,201,154,208)]])

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 4Q 4R 7A 7B 7C 8A ··· 8H 8I ··· 8P 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 4 4 4 7 7 7 8 ··· 8 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 1 1 1 1 2 2 7 ··· 7 14 14 14 14 2 2 2 2 ··· 2 14 ··· 14 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D7 M4(2) D14 Dic7 D14 C4×D7 C4×D7 D7×M4(2) kernel M4(2)×Dic7 C8×Dic7 C56⋊C4 C2×C4.Dic7 C2×C4×Dic7 C14×M4(2) C4.Dic7 C4×Dic7 C7×M4(2) C22×Dic7 C2×M4(2) Dic7 C2×C8 M4(2) C22×C4 C2×C4 C23 C2 # reps 1 2 2 1 1 1 8 4 8 4 3 8 6 12 3 18 6 12

Matrix representation of M4(2)×Dic7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 0 1 0 0 15 0
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 112
,
 1 112 0 0 26 88 0 0 0 0 112 0 0 0 0 112
,
 98 15 0 0 0 15 0 0 0 0 98 0 0 0 0 98
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,0,15,0,0,1,0],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112],[1,26,0,0,112,88,0,0,0,0,112,0,0,0,0,112],[98,0,0,0,15,15,0,0,0,0,98,0,0,0,0,98] >;

M4(2)×Dic7 in GAP, Magma, Sage, TeX

M_4(2)\times {\rm Dic}_7
% in TeX

G:=Group("M4(2)xDic7");
// GroupNames label

G:=SmallGroup(448,651);
// by ID

G=gap.SmallGroup(448,651);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,387,100,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=c^7,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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