C2^3xDic7 - GroupNames

G = C₂³×Dic₇ order 224 = 2⁵·7

Direct product of C₂³ and Dic₇

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C₂³×Dic₇, C₂⁴.₂D₇, C₁₄.₁₄C₂⁴, C₂³.₃₆D₁₄, C₇⋊₂(C₂³×C₄), (C₂²×C₁₄)⋊₅C₄, C₁₄⋊₂(C₂²×C₄), C₂.₂(C₂³×D₇), (C₂³×C₁₄).₃C₂, (C₂×C₁₄).₆₉C₂³, C₂².₃₃(C₂²×D₇), (C₂²×C₁₄).₄₇C₂², (C₂×C₁₄)⋊₉(C₂×C₄), SmallGroup(224,187)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₇ — C₂³×Dic₇

Chief series

C₁ — C₇ — C₁₄ — Dic₇ — C₂×Dic₇ — C₂²×Dic₇ — C₂³×Dic₇

Lower central

C₇ — C₂³×Dic₇

Upper central

C₁ — C₂⁴

Generators and relations for C₂³×Dic₇
G = < a,b,c,d,e | a²=b²=c²=d¹⁴=1, e²=d⁷, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 542 in 236 conjugacy classes, 185 normal (7 characteristic)
C₁, C₂, C₂, C₄, C₂², C₇, C₂×C₄, C₂³, C₁₄, C₁₄, C₂²×C₄, C₂⁴, Dic₇, C₂×C₁₄, C₂³×C₄, C₂×Dic₇, C₂²×C₁₄, C₂²×Dic₇, C₂³×C₁₄, C₂³×Dic₇
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, D₇, C₂²×C₄, C₂⁴, Dic₇, D₁₄, C₂³×C₄, C₂×Dic₇, C₂²×D₇, C₂²×Dic₇, C₂³×D₇, C₂³×Dic₇

Smallest permutation representation of C₂³×Dic₇
►Regular action on 224 points

Generators in S₂₂₄

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

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

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

(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)

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

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

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

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

C₂³×Dic₇ is a maximal subgroup of
C₂⁴.₄₄D₁₄ C₂³.₄₂D₂₈ C₂⁴.₄₆D₁₄ C₂⁴.₄₇D₁₄ C₂³.₄₅D₂₈ C₂⁴.₁₈D₁₄ C₂⁴.₅₆D₁₄ D₇×C₂³×C₄
C₂³×Dic₇ is a maximal quotient of
C₂⁴.₃₈D₁₄ C₁₄.₄₂2_-¹⁺⁴ C₂₈.₇₆C₂⁴ C₁₄.₁₀₆2_-¹⁺⁴

80 conjugacy classes

class	1	2A	···	2O	4A	···	4P	7A	7B	7C	14A	···	14AS
order	1	2	···	2	4	···	4	7	7	7	14	···	14
size	1	1	···	1	7	···	7	2	2	2	2	···	2

80 irreducible representations

dim	1	1	1	1	2	2	2
type	+	+	+		+	-	+
image	C₁	C₂	C₂	C₄	D₇	Dic₇	D₁₄
kernel	C₂³×Dic₇	C₂²×Dic₇	C₂³×C₁₄	C₂²×C₁₄	C₂⁴	C₂³	C₂³
# reps	1	14	1	16	3	24	21

Matrix representation of C₂³×Dic₇ ►in GL₅(𝔽₂₉)

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

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

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

28	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	7	0
0	0	0	3	25

12	0	0	0	0
0	28	0	0	0
0	0	28	0	0
0	0	0	1	6
0	0	0	0	28

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,3,0,0,0,0,25],[12,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,6,28] >;

C₂³×Dic₇ in GAP, Magma, Sage, TeX

C_2^3\times {\rm Dic}_7

% in TeX

G:=Group("C2^3xDic7");

// GroupNames label

G:=SmallGroup(224,187);

// by ID

G=gap.SmallGroup(224,187);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,6917]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^14=1,e^2=d^7,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^-1=d^-1>;

// generators/relations

G = C23×Dic7 order 224 = 25·7

Direct product of C23 and Dic7

G = C₂³×Dic₇ order 224 = 2⁵·7

Direct product of C₂³ and Dic₇