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## G = M4(2)×C2×C14order 448 = 26·7

### Direct product of C2×C14 and M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C2×C14
 Chief series C1 — C2 — C4 — C28 — C56 — C7×M4(2) — C14×M4(2) — M4(2)×C2×C14
 Lower central C1 — C2 — M4(2)×C2×C14
 Upper central C1 — C22×C28 — M4(2)×C2×C14

Generators and relations for M4(2)×C2×C14
G = < a,b,c,d | a2=b14=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 338 in 298 conjugacy classes, 258 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C23, C23, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C22×C4, C24, C28, C28, C2×C14, C2×C14, C22×C8, C2×M4(2), C23×C4, C56, C2×C28, C22×C14, C22×C14, C22×C14, C22×M4(2), C2×C56, C7×M4(2), C22×C28, C22×C28, C23×C14, C22×C56, C14×M4(2), C23×C28, M4(2)×C2×C14
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, M4(2), C22×C4, C24, C28, C2×C14, C2×M4(2), C23×C4, C2×C28, C22×C14, C22×M4(2), C7×M4(2), C22×C28, C23×C14, C14×M4(2), C23×C28, M4(2)×C2×C14

Smallest permutation representation of M4(2)×C2×C14
On 224 points
Generators in S224
(1 39)(2 40)(3 41)(4 42)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 124)(16 125)(17 126)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(43 184)(44 185)(45 186)(46 187)(47 188)(48 189)(49 190)(50 191)(51 192)(52 193)(53 194)(54 195)(55 196)(56 183)(57 207)(58 208)(59 209)(60 210)(61 197)(62 198)(63 199)(64 200)(65 201)(66 202)(67 203)(68 204)(69 205)(70 206)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 99)(97 100)(98 101)(141 177)(142 178)(143 179)(144 180)(145 181)(146 182)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)(155 213)(156 214)(157 215)(158 216)(159 217)(160 218)(161 219)(162 220)(163 221)(164 222)(165 223)(166 224)(167 211)(168 212)
(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)
(1 223 58 172 116 104 139 192)(2 224 59 173 117 105 140 193)(3 211 60 174 118 106 127 194)(4 212 61 175 119 107 128 195)(5 213 62 176 120 108 129 196)(6 214 63 177 121 109 130 183)(7 215 64 178 122 110 131 184)(8 216 65 179 123 111 132 185)(9 217 66 180 124 112 133 186)(10 218 67 181 125 99 134 187)(11 219 68 182 126 100 135 188)(12 220 69 169 113 101 136 189)(13 221 70 170 114 102 137 190)(14 222 57 171 115 103 138 191)(15 95 71 45 33 159 202 144)(16 96 72 46 34 160 203 145)(17 97 73 47 35 161 204 146)(18 98 74 48 36 162 205 147)(19 85 75 49 37 163 206 148)(20 86 76 50 38 164 207 149)(21 87 77 51 39 165 208 150)(22 88 78 52 40 166 209 151)(23 89 79 53 41 167 210 152)(24 90 80 54 42 168 197 153)(25 91 81 55 29 155 198 154)(26 92 82 56 30 156 199 141)(27 93 83 43 31 157 200 142)(28 94 84 44 32 158 201 143)
(1 28)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 191)(44 192)(45 193)(46 194)(47 195)(48 196)(49 183)(50 184)(51 185)(52 186)(53 187)(54 188)(55 189)(56 190)(57 83)(58 84)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)(85 109)(86 110)(87 111)(88 112)(89 99)(90 100)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)(127 203)(128 204)(129 205)(130 206)(131 207)(132 208)(133 209)(134 210)(135 197)(136 198)(137 199)(138 200)(139 201)(140 202)(141 170)(142 171)(143 172)(144 173)(145 174)(146 175)(147 176)(148 177)(149 178)(150 179)(151 180)(152 181)(153 182)(154 169)(155 220)(156 221)(157 222)(158 223)(159 224)(160 211)(161 212)(162 213)(163 214)(164 215)(165 216)(166 217)(167 218)(168 219)

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

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

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

280 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 7A ··· 7F 8A ··· 8P 14A ··· 14AP 14AQ ··· 14BN 28A ··· 28AV 28AW ··· 28BT 56A ··· 56CR order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

280 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 M4(2) C7×M4(2) kernel M4(2)×C2×C14 C22×C56 C14×M4(2) C23×C28 C22×C28 C23×C14 C22×M4(2) C22×C8 C2×M4(2) C23×C4 C22×C4 C24 C2×C14 C22 # reps 1 2 12 1 14 2 6 12 72 6 84 12 8 48

Matrix representation of M4(2)×C2×C14 in GL4(𝔽113) generated by

 112 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 112 0 0 0 0 7 0 0 0 0 7
,
 1 0 0 0 0 112 0 0 0 0 72 111 0 0 57 41
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 72 112
G:=sub<GL(4,GF(113))| [112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,112,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,112,0,0,0,0,72,57,0,0,111,41],[1,0,0,0,0,1,0,0,0,0,1,72,0,0,0,112] >;

M4(2)×C2×C14 in GAP, Magma, Sage, TeX

M_4(2)\times C_2\times C_{14}
% in TeX

G:=Group("M4(2)xC2xC14");
// GroupNames label

G:=SmallGroup(448,1349);
// by ID

G=gap.SmallGroup(448,1349);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,3165,124]);
// Polycyclic

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

׿
×
𝔽