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## G = D7×C2×C42order 448 = 26·7

### Direct product of C2×C42 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C2×C42
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22×C4 — D7×C2×C42
 Lower central C7 — D7×C2×C42
 Upper central C1 — C2×C42

Generators and relations for D7×C2×C42
G = < a,b,c,d,e | a2=b4=c4=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1668 in 498 conjugacy classes, 303 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C42, C42, C22×C4, C22×C4, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C42, C2×C42, C23×C4, C4×D7, C2×Dic7, C2×C28, C22×D7, C22×C14, C22×C42, C4×Dic7, C4×C28, C2×C4×D7, C22×Dic7, C22×C28, C23×D7, D7×C42, C2×C4×Dic7, C2×C4×C28, D7×C22×C4, D7×C2×C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, C22×C4, C24, D14, C2×C42, C23×C4, C4×D7, C22×D7, C22×C42, C2×C4×D7, C23×D7, D7×C42, D7×C22×C4, D7×C2×C42

Smallest permutation representation of D7×C2×C42
On 224 points
Generators in S224
(1 125)(2 126)(3 120)(4 121)(5 122)(6 123)(7 124)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 148)(30 149)(31 150)(32 151)(33 152)(34 153)(35 154)(36 141)(37 142)(38 143)(39 144)(40 145)(41 146)(42 147)(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 168)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 176)(58 177)(59 178)(60 179)(61 180)(62 181)(63 182)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 190)(72 191)(73 192)(74 193)(75 194)(76 195)(77 196)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 204)(86 205)(87 206)(88 207)(89 208)(90 209)(91 210)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 218)(100 219)(101 220)(102 221)(103 222)(104 223)(105 224)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)
(1 104 27 97)(2 105 28 98)(3 99 22 92)(4 100 23 93)(5 101 24 94)(6 102 25 95)(7 103 26 96)(8 106 15 85)(9 107 16 86)(10 108 17 87)(11 109 18 88)(12 110 19 89)(13 111 20 90)(14 112 21 91)(29 57 50 78)(30 58 51 79)(31 59 52 80)(32 60 53 81)(33 61 54 82)(34 62 55 83)(35 63 56 84)(36 64 43 71)(37 65 44 72)(38 66 45 73)(39 67 46 74)(40 68 47 75)(41 69 48 76)(42 70 49 77)(113 211 134 204)(114 212 135 205)(115 213 136 206)(116 214 137 207)(117 215 138 208)(118 216 139 209)(119 217 140 210)(120 218 127 197)(121 219 128 198)(122 220 129 199)(123 221 130 200)(124 222 131 201)(125 223 132 202)(126 224 133 203)(141 169 162 190)(142 170 163 191)(143 171 164 192)(144 172 165 193)(145 173 166 194)(146 174 167 195)(147 175 168 196)(148 176 155 183)(149 177 156 184)(150 178 157 185)(151 179 158 186)(152 180 159 187)(153 181 160 188)(154 182 161 189)
(1 48 20 34)(2 49 21 35)(3 43 15 29)(4 44 16 30)(5 45 17 31)(6 46 18 32)(7 47 19 33)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 99 71 85)(58 100 72 86)(59 101 73 87)(60 102 74 88)(61 103 75 89)(62 104 76 90)(63 105 77 91)(64 106 78 92)(65 107 79 93)(66 108 80 94)(67 109 81 95)(68 110 82 96)(69 111 83 97)(70 112 84 98)(113 155 127 141)(114 156 128 142)(115 157 129 143)(116 158 130 144)(117 159 131 145)(118 160 132 146)(119 161 133 147)(120 162 134 148)(121 163 135 149)(122 164 136 150)(123 165 137 151)(124 166 138 152)(125 167 139 153)(126 168 140 154)(169 211 183 197)(170 212 184 198)(171 213 185 199)(172 214 186 200)(173 215 187 201)(174 216 188 202)(175 217 189 203)(176 218 190 204)(177 219 191 205)(178 220 192 206)(179 221 193 207)(180 222 194 208)(181 223 195 209)(182 224 196 210)
(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 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 17)(9 16)(10 15)(11 21)(12 20)(13 19)(14 18)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)(57 80)(58 79)(59 78)(60 84)(61 83)(62 82)(63 81)(64 73)(65 72)(66 71)(67 77)(68 76)(69 75)(70 74)(85 108)(86 107)(87 106)(88 112)(89 111)(90 110)(91 109)(92 101)(93 100)(94 99)(95 105)(96 104)(97 103)(98 102)(113 136)(114 135)(115 134)(116 140)(117 139)(118 138)(119 137)(120 129)(121 128)(122 127)(123 133)(124 132)(125 131)(126 130)(141 164)(142 163)(143 162)(144 168)(145 167)(146 166)(147 165)(148 157)(149 156)(150 155)(151 161)(152 160)(153 159)(154 158)(169 192)(170 191)(171 190)(172 196)(173 195)(174 194)(175 193)(176 185)(177 184)(178 183)(179 189)(180 188)(181 187)(182 186)(197 220)(198 219)(199 218)(200 224)(201 223)(202 222)(203 221)(204 213)(205 212)(206 211)(207 217)(208 216)(209 215)(210 214)

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

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

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

160 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4X 4Y ··· 4AV 7A 7B 7C 14A ··· 14U 28A ··· 28BT order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 7 ··· 7 1 ··· 1 7 ··· 7 2 2 2 2 ··· 2 2 ··· 2

160 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 D7 D14 D14 C4×D7 kernel D7×C2×C42 D7×C42 C2×C4×Dic7 C2×C4×C28 D7×C22×C4 C2×C4×D7 C2×C42 C42 C22×C4 C2×C4 # reps 1 8 3 1 3 48 3 12 9 72

Matrix representation of D7×C2×C42 in GL4(𝔽29) generated by

 28 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 17 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 1 0 0 0 0 11 1 0 0 27 21
,
 1 0 0 0 0 1 0 0 0 0 21 28 0 0 5 8
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[17,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,11,27,0,0,1,21],[1,0,0,0,0,1,0,0,0,0,21,5,0,0,28,8] >;

D7×C2×C42 in GAP, Magma, Sage, TeX

D_7\times C_2\times C_4^2
% in TeX

G:=Group("D7xC2xC4^2");
// GroupNames label

G:=SmallGroup(448,924);
// by ID

G=gap.SmallGroup(448,924);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,80,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^7=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^-1>;
// generators/relations

׿
×
𝔽