direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D28.C4, C28.69C24, C56.46C23, M4(2)⋊25D14, C14⋊2(C8○D4), C4○D28.4C4, C7⋊C8.35C23, D28.28(C2×C4), (C2×D28).16C4, (C2×C8).279D14, (C8×D7)⋊22C22, C4.68(C23×D7), C23.30(C4×D7), C8.43(C22×D7), C8⋊D7⋊18C22, (C2×M4(2))⋊17D7, (C14×M4(2))⋊9C2, C14.32(C23×C4), C28.92(C22×C4), (C4×D7).35C23, (C2×C28).882C23, (C2×C56).238C22, Dic14.29(C2×C4), (C2×Dic14).16C4, C4○D28.49C22, D14.13(C22×C4), (C22×C4).374D14, (C7×M4(2))⋊25C22, Dic7.13(C22×C4), (C22×C28).264C22, C7⋊2(C2×C8○D4), (D7×C2×C8)⋊29C2, C4.123(C2×C4×D7), C22.8(C2×C4×D7), (C22×C7⋊C8)⋊10C2, (C2×C7⋊C8)⋊33C22, (C2×C4).87(C4×D7), C7⋊D4.3(C2×C4), (C2×C8⋊D7)⋊27C2, C2.33(D7×C22×C4), (C4×D7).24(C2×C4), (C2×C7⋊D4).14C4, (C2×C28).131(C2×C4), (C2×C4○D28).21C2, (C2×C4×D7).303C22, (C2×C14).25(C22×C4), (C22×C14).78(C2×C4), (C2×Dic7).72(C2×C4), (C22×D7).46(C2×C4), (C2×C4).825(C22×D7), SmallGroup(448,1197)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 932 in 266 conjugacy classes, 151 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C7, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D7 [×4], C14, C14 [×2], C14 [×2], C2×C8 [×2], C2×C8 [×14], M4(2) [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic7 [×4], C28 [×2], C28 [×2], D14 [×4], D14 [×4], C2×C14, C2×C14 [×2], C2×C14 [×2], C22×C8 [×3], C2×M4(2), C2×M4(2) [×2], C8○D4 [×8], C2×C4○D4, C7⋊C8 [×4], C56 [×4], Dic14 [×4], C4×D7 [×8], D28 [×4], C2×Dic7 [×2], C7⋊D4 [×8], C2×C28 [×2], C2×C28 [×4], C22×D7 [×2], C22×C14, C2×C8○D4, C8×D7 [×8], C8⋊D7 [×8], C2×C7⋊C8 [×2], C2×C7⋊C8 [×4], C2×C56 [×2], C7×M4(2) [×4], C2×Dic14, C2×C4×D7 [×2], C2×D28, C4○D28 [×8], C2×C7⋊D4 [×2], C22×C28, D7×C2×C8 [×2], C2×C8⋊D7 [×2], D28.C4 [×8], C22×C7⋊C8, C14×M4(2), C2×C4○D28, C2×D28.C4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C24, D14 [×7], C8○D4 [×2], C23×C4, C4×D7 [×4], C22×D7 [×7], C2×C8○D4, C2×C4×D7 [×6], C23×D7, D28.C4 [×2], D7×C22×C4, C2×D28.C4
Generators and relations
G = < a,b,c,d | a2=b28=c2=1, d4=b14, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b15, dcd-1=b14c >
►On 224 points
Generators in S224
(1 165)(2 166)(3 167)(4 168)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 161)(26 162)(27 163)(28 164)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)(169 209)(170 210)(171 211)(172 212)(173 213)(174 214)(175 215)(176 216)(177 217)(178 218)(179 219)(180 220)(181 221)(182 222)(183 223)(184 224)(185 197)(186 198)(187 199)(188 200)(189 201)(190 202)(191 203)(192 204)(193 205)(194 206)(195 207)(196 208)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(46 56)(47 55)(48 54)(49 53)(50 52)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(74 84)(75 83)(76 82)(77 81)(78 80)(85 87)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(113 115)(116 140)(117 139)(118 138)(119 137)(120 136)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(141 167)(142 166)(143 165)(144 164)(145 163)(146 162)(147 161)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)(169 187)(170 186)(171 185)(172 184)(173 183)(174 182)(175 181)(176 180)(177 179)(188 196)(189 195)(190 194)(191 193)(197 211)(198 210)(199 209)(200 208)(201 207)(202 206)(203 205)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)
(1 69 201 90 15 83 215 104)(2 84 202 105 16 70 216 91)(3 71 203 92 17 57 217 106)(4 58 204 107 18 72 218 93)(5 73 205 94 19 59 219 108)(6 60 206 109 20 74 220 95)(7 75 207 96 21 61 221 110)(8 62 208 111 22 76 222 97)(9 77 209 98 23 63 223 112)(10 64 210 85 24 78 224 99)(11 79 211 100 25 65 197 86)(12 66 212 87 26 80 198 101)(13 81 213 102 27 67 199 88)(14 68 214 89 28 82 200 103)(29 191 120 153 43 177 134 167)(30 178 121 168 44 192 135 154)(31 193 122 155 45 179 136 141)(32 180 123 142 46 194 137 156)(33 195 124 157 47 181 138 143)(34 182 125 144 48 196 139 158)(35 169 126 159 49 183 140 145)(36 184 127 146 50 170 113 160)(37 171 128 161 51 185 114 147)(38 186 129 148 52 172 115 162)(39 173 130 163 53 187 116 149)(40 188 131 150 54 174 117 164)(41 175 132 165 55 189 118 151)(42 190 133 152 56 176 119 166)
G:=sub<Sym(224)| (1,165)(2,166)(3,167)(4,168)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,161)(26,162)(27,163)(28,164)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)(177,217)(178,218)(179,219)(180,220)(181,221)(182,222)(183,223)(184,224)(185,197)(186,198)(187,199)(188,200)(189,201)(190,202)(191,203)(192,204)(193,205)(194,206)(195,207)(196,208), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(74,84)(75,83)(76,82)(77,81)(78,80)(85,87)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(113,115)(116,140)(117,139)(118,138)(119,137)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155)(169,187)(170,186)(171,185)(172,184)(173,183)(174,182)(175,181)(176,180)(177,179)(188,196)(189,195)(190,194)(191,193)(197,211)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,69,201,90,15,83,215,104)(2,84,202,105,16,70,216,91)(3,71,203,92,17,57,217,106)(4,58,204,107,18,72,218,93)(5,73,205,94,19,59,219,108)(6,60,206,109,20,74,220,95)(7,75,207,96,21,61,221,110)(8,62,208,111,22,76,222,97)(9,77,209,98,23,63,223,112)(10,64,210,85,24,78,224,99)(11,79,211,100,25,65,197,86)(12,66,212,87,26,80,198,101)(13,81,213,102,27,67,199,88)(14,68,214,89,28,82,200,103)(29,191,120,153,43,177,134,167)(30,178,121,168,44,192,135,154)(31,193,122,155,45,179,136,141)(32,180,123,142,46,194,137,156)(33,195,124,157,47,181,138,143)(34,182,125,144,48,196,139,158)(35,169,126,159,49,183,140,145)(36,184,127,146,50,170,113,160)(37,171,128,161,51,185,114,147)(38,186,129,148,52,172,115,162)(39,173,130,163,53,187,116,149)(40,188,131,150,54,174,117,164)(41,175,132,165,55,189,118,151)(42,190,133,152,56,176,119,166)>;
G:=Group( (1,165)(2,166)(3,167)(4,168)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,161)(26,162)(27,163)(28,164)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)(177,217)(178,218)(179,219)(180,220)(181,221)(182,222)(183,223)(184,224)(185,197)(186,198)(187,199)(188,200)(189,201)(190,202)(191,203)(192,204)(193,205)(194,206)(195,207)(196,208), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(74,84)(75,83)(76,82)(77,81)(78,80)(85,87)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(113,115)(116,140)(117,139)(118,138)(119,137)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155)(169,187)(170,186)(171,185)(172,184)(173,183)(174,182)(175,181)(176,180)(177,179)(188,196)(189,195)(190,194)(191,193)(197,211)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,69,201,90,15,83,215,104)(2,84,202,105,16,70,216,91)(3,71,203,92,17,57,217,106)(4,58,204,107,18,72,218,93)(5,73,205,94,19,59,219,108)(6,60,206,109,20,74,220,95)(7,75,207,96,21,61,221,110)(8,62,208,111,22,76,222,97)(9,77,209,98,23,63,223,112)(10,64,210,85,24,78,224,99)(11,79,211,100,25,65,197,86)(12,66,212,87,26,80,198,101)(13,81,213,102,27,67,199,88)(14,68,214,89,28,82,200,103)(29,191,120,153,43,177,134,167)(30,178,121,168,44,192,135,154)(31,193,122,155,45,179,136,141)(32,180,123,142,46,194,137,156)(33,195,124,157,47,181,138,143)(34,182,125,144,48,196,139,158)(35,169,126,159,49,183,140,145)(36,184,127,146,50,170,113,160)(37,171,128,161,51,185,114,147)(38,186,129,148,52,172,115,162)(39,173,130,163,53,187,116,149)(40,188,131,150,54,174,117,164)(41,175,132,165,55,189,118,151)(42,190,133,152,56,176,119,166) );
G=PermutationGroup([(1,165),(2,166),(3,167),(4,168),(5,141),(6,142),(7,143),(8,144),(9,145),(10,146),(11,147),(12,148),(13,149),(14,150),(15,151),(16,152),(17,153),(18,154),(19,155),(20,156),(21,157),(22,158),(23,159),(24,160),(25,161),(26,162),(27,163),(28,164),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70),(85,113),(86,114),(87,115),(88,116),(89,117),(90,118),(91,119),(92,120),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128),(101,129),(102,130),(103,131),(104,132),(105,133),(106,134),(107,135),(108,136),(109,137),(110,138),(111,139),(112,140),(169,209),(170,210),(171,211),(172,212),(173,213),(174,214),(175,215),(176,216),(177,217),(178,218),(179,219),(180,220),(181,221),(182,222),(183,223),(184,224),(185,197),(186,198),(187,199),(188,200),(189,201),(190,202),(191,203),(192,204),(193,205),(194,206),(195,207),(196,208)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(46,56),(47,55),(48,54),(49,53),(50,52),(57,73),(58,72),(59,71),(60,70),(61,69),(62,68),(63,67),(64,66),(74,84),(75,83),(76,82),(77,81),(78,80),(85,87),(88,112),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101),(113,115),(116,140),(117,139),(118,138),(119,137),(120,136),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(141,167),(142,166),(143,165),(144,164),(145,163),(146,162),(147,161),(148,160),(149,159),(150,158),(151,157),(152,156),(153,155),(169,187),(170,186),(171,185),(172,184),(173,183),(174,182),(175,181),(176,180),(177,179),(188,196),(189,195),(190,194),(191,193),(197,211),(198,210),(199,209),(200,208),(201,207),(202,206),(203,205),(212,224),(213,223),(214,222),(215,221),(216,220),(217,219)], [(1,69,201,90,15,83,215,104),(2,84,202,105,16,70,216,91),(3,71,203,92,17,57,217,106),(4,58,204,107,18,72,218,93),(5,73,205,94,19,59,219,108),(6,60,206,109,20,74,220,95),(7,75,207,96,21,61,221,110),(8,62,208,111,22,76,222,97),(9,77,209,98,23,63,223,112),(10,64,210,85,24,78,224,99),(11,79,211,100,25,65,197,86),(12,66,212,87,26,80,198,101),(13,81,213,102,27,67,199,88),(14,68,214,89,28,82,200,103),(29,191,120,153,43,177,134,167),(30,178,121,168,44,192,135,154),(31,193,122,155,45,179,136,141),(32,180,123,142,46,194,137,156),(33,195,124,157,47,181,138,143),(34,182,125,144,48,196,139,158),(35,169,126,159,49,183,140,145),(36,184,127,146,50,170,113,160),(37,171,128,161,51,185,114,147),(38,186,129,148,52,172,115,162),(39,173,130,163,53,187,116,149),(40,188,131,150,54,174,117,164),(41,175,132,165,55,189,118,151),(42,190,133,152,56,176,119,166)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 15 | 89 | 0 | 0 |
| 48 | 89 | 0 | 0 |
| 0 | 0 | 85 | 18 |
| 0 | 0 | 38 | 28 |
| 79 | 1 | 0 | 0 |
| 88 | 34 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 22 | 1 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 102 | 112 |
| 0 | 0 | 106 | 11 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[15,48,0,0,89,89,0,0,0,0,85,38,0,0,18,28],[79,88,0,0,1,34,0,0,0,0,112,22,0,0,0,1],[15,0,0,0,0,15,0,0,0,0,102,106,0,0,112,11] >;
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D7 | D14 | D14 | D14 | C8○D4 | C4×D7 | C4×D7 | D28.C4 |
| kernel | C2×D28.C4 | D7×C2×C8 | C2×C8⋊D7 | D28.C4 | C22×C7⋊C8 | C14×M4(2) | C2×C4○D28 | C2×Dic14 | C2×D28 | C4○D28 | C2×C7⋊D4 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 3 | 6 | 12 | 3 | 8 | 18 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_{28}.C_4 % in TeX
G:=Group("C2xD28.C4"); // GroupNames label
G:=SmallGroup(448,1197);
// by ID
G=gap.SmallGroup(448,1197);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,297,80,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^2=1,d^4=b^14,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^15,d*c*d^-1=b^14*c>;
// generators/relations
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