direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C2×C14, C56⋊13C23, C28.78C24, (C22×C8)⋊7C14, C8⋊2(C22×C14), C4.18(D4×C14), (C2×C56)⋊50C22, (C22×C56)⋊21C2, C28.325(C2×D4), (C2×C28).432D4, D4⋊1(C22×C14), (C7×D4)⋊12C23, C4.1(C23×C14), C23.60(C7×D4), (D4×C14)⋊65C22, (C22×D4)⋊10C14, C22.65(D4×C14), (C2×C28).971C23, (C22×C14).221D4, C14.199(C22×D4), (C22×C28).601C22, (D4×C2×C14)⋊25C2, C2.23(D4×C2×C14), (C2×C8)⋊12(C2×C14), (C2×C4).88(C7×D4), (C2×D4)⋊14(C2×C14), (C2×C14).686(C2×D4), (C22×C4).128(C2×C14), (C2×C4).141(C22×C14), SmallGroup(448,1352)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 658 in 338 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×8], C4, C4 [×3], C22 [×7], C22 [×32], C7, C8 [×4], C2×C4 [×6], D4 [×8], D4 [×12], C23, C23 [×20], C14, C14 [×6], C14 [×8], C2×C8 [×6], D8 [×16], C22×C4, C2×D4 [×12], C2×D4 [×6], C24 [×2], C28, C28 [×3], C2×C14 [×7], C2×C14 [×32], C22×C8, C2×D8 [×12], C22×D4 [×2], C56 [×4], C2×C28 [×6], C7×D4 [×8], C7×D4 [×12], C22×C14, C22×C14 [×20], C22×D8, C2×C56 [×6], C7×D8 [×16], C22×C28, D4×C14 [×12], D4×C14 [×6], C23×C14 [×2], C22×C56, C14×D8 [×12], D4×C2×C14 [×2], D8×C2×C14
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], D8 [×4], C2×D4 [×6], C24, C2×C14 [×35], C2×D8 [×6], C22×D4, C7×D4 [×4], C22×C14 [×15], C22×D8, C7×D8 [×4], D4×C14 [×6], C23×C14, C14×D8 [×6], D4×C2×C14, D8×C2×C14
Generators and relations
G = < a,b,c,d | a2=b14=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 224 points
Generators in S224
(1 151)(2 152)(3 153)(4 154)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 186)(16 187)(17 188)(18 189)(19 190)(20 191)(21 192)(22 193)(23 194)(24 195)(25 196)(26 183)(27 184)(28 185)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 85)(40 86)(41 87)(42 88)(43 158)(44 159)(45 160)(46 161)(47 162)(48 163)(49 164)(50 165)(51 166)(52 167)(53 168)(54 155)(55 156)(56 157)(57 224)(58 211)(59 212)(60 213)(61 214)(62 215)(63 216)(64 217)(65 218)(66 219)(67 220)(68 221)(69 222)(70 223)(71 122)(72 123)(73 124)(74 125)(75 126)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 121)(99 129)(100 130)(101 131)(102 132)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 139)(110 140)(111 127)(112 128)(169 204)(170 205)(171 206)(172 207)(173 208)(174 209)(175 210)(176 197)(177 198)(178 199)(179 200)(180 201)(181 202)(182 203)
(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)
(1 165 85 217 137 126 197 27)(2 166 86 218 138 113 198 28)(3 167 87 219 139 114 199 15)(4 168 88 220 140 115 200 16)(5 155 89 221 127 116 201 17)(6 156 90 222 128 117 202 18)(7 157 91 223 129 118 203 19)(8 158 92 224 130 119 204 20)(9 159 93 211 131 120 205 21)(10 160 94 212 132 121 206 22)(11 161 95 213 133 122 207 23)(12 162 96 214 134 123 208 24)(13 163 97 215 135 124 209 25)(14 164 98 216 136 125 210 26)(29 68 111 79 180 188 141 54)(30 69 112 80 181 189 142 55)(31 70 99 81 182 190 143 56)(32 57 100 82 169 191 144 43)(33 58 101 83 170 192 145 44)(34 59 102 84 171 193 146 45)(35 60 103 71 172 194 147 46)(36 61 104 72 173 195 148 47)(37 62 105 73 174 196 149 48)(38 63 106 74 175 183 150 49)(39 64 107 75 176 184 151 50)(40 65 108 76 177 185 152 51)(41 66 109 77 178 186 153 52)(42 67 110 78 179 187 154 53)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 99)(15 84)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 96)(30 97)(31 98)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 217)(44 218)(45 219)(46 220)(47 221)(48 222)(49 223)(50 224)(51 211)(52 212)(53 213)(54 214)(55 215)(56 216)(57 165)(58 166)(59 167)(60 168)(61 155)(62 156)(63 157)(64 158)(65 159)(66 160)(67 161)(68 162)(69 163)(70 164)(113 192)(114 193)(115 194)(116 195)(117 196)(118 183)(119 184)(120 185)(121 186)(122 187)(123 188)(124 189)(125 190)(126 191)(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)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)
G:=sub<Sym(224)| (1,151)(2,152)(3,153)(4,154)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,183)(27,184)(28,185)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,85)(40,86)(41,87)(42,88)(43,158)(44,159)(45,160)(46,161)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,168)(54,155)(55,156)(56,157)(57,224)(58,211)(59,212)(60,213)(61,214)(62,215)(63,216)(64,217)(65,218)(66,219)(67,220)(68,221)(69,222)(70,223)(71,122)(72,123)(73,124)(74,125)(75,126)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,135)(106,136)(107,137)(108,138)(109,139)(110,140)(111,127)(112,128)(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203), (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), (1,165,85,217,137,126,197,27)(2,166,86,218,138,113,198,28)(3,167,87,219,139,114,199,15)(4,168,88,220,140,115,200,16)(5,155,89,221,127,116,201,17)(6,156,90,222,128,117,202,18)(7,157,91,223,129,118,203,19)(8,158,92,224,130,119,204,20)(9,159,93,211,131,120,205,21)(10,160,94,212,132,121,206,22)(11,161,95,213,133,122,207,23)(12,162,96,214,134,123,208,24)(13,163,97,215,135,124,209,25)(14,164,98,216,136,125,210,26)(29,68,111,79,180,188,141,54)(30,69,112,80,181,189,142,55)(31,70,99,81,182,190,143,56)(32,57,100,82,169,191,144,43)(33,58,101,83,170,192,145,44)(34,59,102,84,171,193,146,45)(35,60,103,71,172,194,147,46)(36,61,104,72,173,195,148,47)(37,62,105,73,174,196,149,48)(38,63,106,74,175,183,150,49)(39,64,107,75,176,184,151,50)(40,65,108,76,177,185,152,51)(41,66,109,77,178,186,153,52)(42,67,110,78,179,187,154,53), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,99)(15,84)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,96)(30,97)(31,98)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,217)(44,218)(45,219)(46,220)(47,221)(48,222)(49,223)(50,224)(51,211)(52,212)(53,213)(54,214)(55,215)(56,216)(57,165)(58,166)(59,167)(60,168)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,161)(68,162)(69,163)(70,164)(113,192)(114,193)(115,194)(116,195)(117,196)(118,183)(119,184)(120,185)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(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)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)>;
G:=Group( (1,151)(2,152)(3,153)(4,154)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,183)(27,184)(28,185)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,85)(40,86)(41,87)(42,88)(43,158)(44,159)(45,160)(46,161)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,168)(54,155)(55,156)(56,157)(57,224)(58,211)(59,212)(60,213)(61,214)(62,215)(63,216)(64,217)(65,218)(66,219)(67,220)(68,221)(69,222)(70,223)(71,122)(72,123)(73,124)(74,125)(75,126)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,135)(106,136)(107,137)(108,138)(109,139)(110,140)(111,127)(112,128)(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203), (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), (1,165,85,217,137,126,197,27)(2,166,86,218,138,113,198,28)(3,167,87,219,139,114,199,15)(4,168,88,220,140,115,200,16)(5,155,89,221,127,116,201,17)(6,156,90,222,128,117,202,18)(7,157,91,223,129,118,203,19)(8,158,92,224,130,119,204,20)(9,159,93,211,131,120,205,21)(10,160,94,212,132,121,206,22)(11,161,95,213,133,122,207,23)(12,162,96,214,134,123,208,24)(13,163,97,215,135,124,209,25)(14,164,98,216,136,125,210,26)(29,68,111,79,180,188,141,54)(30,69,112,80,181,189,142,55)(31,70,99,81,182,190,143,56)(32,57,100,82,169,191,144,43)(33,58,101,83,170,192,145,44)(34,59,102,84,171,193,146,45)(35,60,103,71,172,194,147,46)(36,61,104,72,173,195,148,47)(37,62,105,73,174,196,149,48)(38,63,106,74,175,183,150,49)(39,64,107,75,176,184,151,50)(40,65,108,76,177,185,152,51)(41,66,109,77,178,186,153,52)(42,67,110,78,179,187,154,53), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,99)(15,84)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,96)(30,97)(31,98)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,217)(44,218)(45,219)(46,220)(47,221)(48,222)(49,223)(50,224)(51,211)(52,212)(53,213)(54,214)(55,215)(56,216)(57,165)(58,166)(59,167)(60,168)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,161)(68,162)(69,163)(70,164)(113,192)(114,193)(115,194)(116,195)(117,196)(118,183)(119,184)(120,185)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(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)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210) );
G=PermutationGroup([(1,151),(2,152),(3,153),(4,154),(5,141),(6,142),(7,143),(8,144),(9,145),(10,146),(11,147),(12,148),(13,149),(14,150),(15,186),(16,187),(17,188),(18,189),(19,190),(20,191),(21,192),(22,193),(23,194),(24,195),(25,196),(26,183),(27,184),(28,185),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,85),(40,86),(41,87),(42,88),(43,158),(44,159),(45,160),(46,161),(47,162),(48,163),(49,164),(50,165),(51,166),(52,167),(53,168),(54,155),(55,156),(56,157),(57,224),(58,211),(59,212),(60,213),(61,214),(62,215),(63,216),(64,217),(65,218),(66,219),(67,220),(68,221),(69,222),(70,223),(71,122),(72,123),(73,124),(74,125),(75,126),(76,113),(77,114),(78,115),(79,116),(80,117),(81,118),(82,119),(83,120),(84,121),(99,129),(100,130),(101,131),(102,132),(103,133),(104,134),(105,135),(106,136),(107,137),(108,138),(109,139),(110,140),(111,127),(112,128),(169,204),(170,205),(171,206),(172,207),(173,208),(174,209),(175,210),(176,197),(177,198),(178,199),(179,200),(180,201),(181,202),(182,203)], [(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)], [(1,165,85,217,137,126,197,27),(2,166,86,218,138,113,198,28),(3,167,87,219,139,114,199,15),(4,168,88,220,140,115,200,16),(5,155,89,221,127,116,201,17),(6,156,90,222,128,117,202,18),(7,157,91,223,129,118,203,19),(8,158,92,224,130,119,204,20),(9,159,93,211,131,120,205,21),(10,160,94,212,132,121,206,22),(11,161,95,213,133,122,207,23),(12,162,96,214,134,123,208,24),(13,163,97,215,135,124,209,25),(14,164,98,216,136,125,210,26),(29,68,111,79,180,188,141,54),(30,69,112,80,181,189,142,55),(31,70,99,81,182,190,143,56),(32,57,100,82,169,191,144,43),(33,58,101,83,170,192,145,44),(34,59,102,84,171,193,146,45),(35,60,103,71,172,194,147,46),(36,61,104,72,173,195,148,47),(37,62,105,73,174,196,149,48),(38,63,106,74,175,183,150,49),(39,64,107,75,176,184,151,50),(40,65,108,76,177,185,152,51),(41,66,109,77,178,186,153,52),(42,67,110,78,179,187,154,53)], [(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,99),(15,84),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,96),(30,97),(31,98),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,217),(44,218),(45,219),(46,220),(47,221),(48,222),(49,223),(50,224),(51,211),(52,212),(53,213),(54,214),(55,215),(56,216),(57,165),(58,166),(59,167),(60,168),(61,155),(62,156),(63,157),(64,158),(65,159),(66,160),(67,161),(68,162),(69,163),(70,164),(113,192),(114,193),(115,194),(116,195),(117,196),(118,183),(119,184),(120,185),(121,186),(122,187),(123,188),(124,189),(125,190),(126,191),(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),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 97 | 0 |
| 0 | 0 | 0 | 97 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 31 | 51 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 112 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,112,0,0,0,0,97,0,0,0,0,97],[112,0,0,0,0,1,0,0,0,0,0,31,0,0,51,51],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,112] >;
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14AP | 14AQ | ··· | 14CL | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D4 | D8 | C7×D4 | C7×D4 | C7×D8 |
| kernel | D8×C2×C14 | C22×C56 | C14×D8 | D4×C2×C14 | C22×D8 | C22×C8 | C2×D8 | C22×D4 | C2×C28 | C22×C14 | C2×C14 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 2 | 6 | 6 | 72 | 12 | 3 | 1 | 8 | 18 | 6 | 48 |
In GAP, Magma, Sage, TeX
D_8\times C_2\times C_{14} % in TeX
G:=Group("D8xC2xC14"); // GroupNames label
G:=SmallGroup(448,1352);
// by ID
G=gap.SmallGroup(448,1352);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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