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## G = C22×Dic14order 224 = 25·7

### Direct product of C22 and Dic14

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)]])

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

׿
×
𝔽