C2^2xDic14 - GroupNames

G = C₂²×Dic₁₄ order 224 = 2⁵·7

Direct product of C₂² and Dic₁₄

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

Aliases: C₂²×Dic₁₄, C₁₄.₁C₂⁴, C₂₈.₃₄C₂³, C₂³.₃₃D₁₄, Dic₇.₁C₂³, (C₂×C₁₄)⋊₄Q₈, C₁₄⋊₁(C₂×Q₈), C₇⋊₁(C₂²×Q₈), (C₂×C₄).₈₆D₁₄, C₂.₃(C₂³×D₇), (C₂²×C₄).₈D₇, (C₂²×C₂₈).₈C₂, C₄.₃₂(C₂²×D₇), (C₂×C₂₈).₉₅C₂², (C₂×C₁₄).₆₂C₂³, (C₂²×Dic₇).₆C₂, C₂².₂₈(C₂²×D₇), (C₂²×C₁₄).₄₃C₂², (C₂×Dic₇).₄₂C₂², SmallGroup(224,174)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₂²×Dic₁₄

Chief series

C₁ — C₇ — C₁₄ — Dic₇ — C₂×Dic₇ — C₂²×Dic₇ — C₂²×Dic₁₄

Lower central

C₇ — C₁₄ — C₂²×Dic₁₄

Upper central

C₁ — C₂³ — C₂²×C₄

Generators and relations for C₂²×Dic₁₄
G = < a,b,c,d | a²=b²=c²⁸=1, d²=c¹⁴, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 462 in 156 conjugacy classes, 105 normal (9 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₇, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₄, C₁₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, Dic₇, C₂₈, C₂×C₁₄, C₂²×Q₈, Dic₁₄, C₂×Dic₇, C₂×C₂₈, C₂²×C₁₄, C₂×Dic₁₄, C₂²×Dic₇, C₂²×C₂₈, C₂²×Dic₁₄
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₇, C₂×Q₈, C₂⁴, D₁₄, C₂²×Q₈, Dic₁₄, C₂²×D₇, C₂×Dic₁₄, C₂³×D₇, C₂²×Dic₁₄

Smallest permutation representation of C₂²×Dic₁₄
►Regular action on 224 points

Generators in S₂₂₄

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

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

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

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

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

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

C₂²×Dic₁₄ is a maximal subgroup of
(C₂×C₂₈)⋊Q₈ (C₂×Dic₇)⋊Q₈ (C₂×C₄).₂₀D₂₈ Dic₁₄⋊₁₄D₄ C₂²⋊Dic₂₈ (C₂×C₂₈)⋊₁₀Q₈ C₂³⋊Dic₁₄ (C₂×Dic₇)⋊₆Q₈ (C₂×C₄)⋊Dic₁₄ (C₂×C₄).₄₇D₂₈ Dic₁₄⋊₁₇D₄ Dic₁₄.₃₇D₄ C₂³.₄₆D₂₈ C₄².₈₇D₁₄ C₄².₉₂D₁₄ Dic₁₄⋊₂₃D₄ Dic₁₄⋊₁₉D₄ Dic₁₄⋊₂₁D₄ C₁₄.₇₉2_-¹⁺⁴ C₁₄.₁₀₅2_-¹⁺⁴ C₂²×Q₈×D₇
C₂²×Dic₁₄ is a maximal quotient of
C₄².₂₇₄D₁₄ C₂³⋊₂Dic₁₄ C₁₄.₇2₊¹⁺⁴ C₄².₈₈D₁₄ C₄².₉₀D₁₄ D₄⋊₅Dic₁₄ D₄⋊₆Dic₁₄ Q₈⋊₅Dic₁₄ Q₈⋊₆Dic₁₄

68 conjugacy classes

class	1	2A	···	2G	4A	4B	4C	4D	4E	···	4L	7A	7B	7C	14A	···	14U	28A	···	28X
order	1	2	···	2	4	4	4	4	4	···	4	7	7	7	14	···	14	28	···	28
size	1	1	···	1	2	2	2	2	14	···	14	2	2	2	2	···	2	2	···	2

68 irreducible representations

dim	1	1	1	1	2	2	2	2	2
type	+	+	+	+	-	+	+	+	-
image	C₁	C₂	C₂	C₂	Q₈	D₇	D₁₄	D₁₄	Dic₁₄
kernel	C₂²×Dic₁₄	C₂×Dic₁₄	C₂²×Dic₇	C₂²×C₂₈	C₂×C₁₄	C₂²×C₄	C₂×C₄	C₂³	C₂²
# reps	1	12	2	1	4	3	18	3	24

Matrix representation of C₂²×Dic₁₄ ►in GL₆(𝔽₂₉)

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

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

0	1	0	0	0	0
28	0	0	0	0	0
0	0	12	0	0	0
0	0	3	17	0	0
0	0	0	0	8	16
0	0	0	0	28	9

14	21	0	0	0	0
21	15	0	0	0	0
0	0	6	10	0	0
0	0	5	23	0	0
0	0	0	0	26	19
0	0	0	0	1	3

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,12,3,0,0,0,0,0,17,0,0,0,0,0,0,8,28,0,0,0,0,16,9],[14,21,0,0,0,0,21,15,0,0,0,0,0,0,6,5,0,0,0,0,10,23,0,0,0,0,0,0,26,1,0,0,0,0,19,3] >;

C₂²×Dic₁₄ in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{14}

% in TeX

G:=Group("C2^2xDic14");

// GroupNames label

G:=SmallGroup(224,174);

// by ID

G=gap.SmallGroup(224,174);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,579,69,6917]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^28=1,d^2=c^14,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

G = C22×Dic14 order 224 = 25·7

Direct product of C22 and Dic14

G = C₂²×Dic₁₄ order 224 = 2⁵·7

Direct product of C₂² and Dic₁₄