C2^4xC14 - GroupNames

G = C₂⁴×C₁₄ order 224 = 2⁵·7

Abelian group of type [2,2,2,2,14]

direct product, abelian, monomial, 2-elementary

Aliases: C₂⁴×C₁₄, SmallGroup(224,197)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴×C₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — C₂³×C₁₄ — C₂⁴×C₁₄

Lower central

C₁ — C₂⁴×C₁₄

Upper central

C₁ — C₂⁴×C₁₄

Generators and relations for C₂⁴×C₁₄
G = < a,b,c,d,e | a²=b²=c²=d²=e¹⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 748, all normal (4 characteristic)
C₁, C₂, C₂², C₇, C₂³, C₁₄, C₂⁴, C₂×C₁₄, C₂⁵, C₂²×C₁₄, C₂³×C₁₄, C₂⁴×C₁₄
Quotients: C₁, C₂, C₂², C₇, C₂³, C₁₄, C₂⁴, C₂×C₁₄, C₂⁵, C₂²×C₁₄, C₂³×C₁₄, C₂⁴×C₁₄

Smallest permutation representation of C₂⁴×C₁₄
►Regular action on 224 points

Generators in S₂₂₄

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

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

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

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

(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)

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

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

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

C₂⁴×C₁₄ is a maximal subgroup of C₂⁵.D₇

224 conjugacy classes

class	1	2A	···	2AE	7A	···	7F	14A	···	14GD
order	1	2	···	2	7	···	7	14	···	14
size	1	1	···	1	1	···	1	1	···	1

224 irreducible representations

dim	1	1	1	1
type	+	+
image	C₁	C₂	C₇	C₁₄
kernel	C₂⁴×C₁₄	C₂³×C₁₄	C₂⁵	C₂⁴
# reps	1	31	6	186

Matrix representation of C₂⁴×C₁₄ ►in GL₅(𝔽₂₉)

28	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	28	0
0	0	0	0	1

28	0	0	0	0
0	28	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	28	0	0
0	0	0	1	0
0	0	0	0	1

5	0	0	0	0
0	5	0	0	0
0	0	7	0	0
0	0	0	7	0
0	0	0	0	28

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1],[28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,5,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,28] >;

C₂⁴×C₁₄ in GAP, Magma, Sage, TeX

C_2^4\times C_{14}

% in TeX

G:=Group("C2^4xC14");

// GroupNames label

G:=SmallGroup(224,197);

// by ID

G=gap.SmallGroup(224,197);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^14=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,d*e=e*d>;

// generators/relations

G = C24×C14 order 224 = 25·7

Abelian group of type [2,2,2,2,14]

G = C₂⁴×C₁₄ order 224 = 2⁵·7