direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D28.2C4, C28.67C24, C56.71C23, (C2×C8)⋊37D14, C14⋊1(C8○D4), C4○D28.7C4, (C22×C8)⋊13D7, C7⋊C8.31C23, (C22×C56)⋊19C2, (C2×C56)⋊49C22, (C2×D28).19C4, D28.30(C2×C4), (C8×D7)⋊19C22, C8.65(C22×D7), C23.39(C4×D7), C4.66(C23×D7), C8⋊D7⋊21C22, C14.30(C23×C4), (C4×D7).34C23, (C2×C28).880C23, C28.121(C22×C4), Dic14.32(C2×C4), (C2×Dic14).19C4, C4○D28.58C22, D14.12(C22×C4), (C22×C4).441D14, C4.Dic7⋊39C22, Dic7.12(C22×C4), (C22×C28).544C22, C7⋊1(C2×C8○D4), (D7×C2×C8)⋊25C2, C4.121(C2×C4×D7), C7⋊D4.5(C2×C4), (C2×C8⋊D7)⋊29C2, C2.31(D7×C22×C4), C22.11(C2×C4×D7), (C4×D7).23(C2×C4), (C2×C4).119(C4×D7), (C2×C7⋊D4).17C4, (C2×C28).236(C2×C4), (C2×C4○D28).27C2, (C2×C7⋊C8).327C22, (C2×C4.Dic7)⋊33C2, (C2×C4×D7).302C22, (C2×Dic7).71(C2×C4), (C22×D7).45(C2×C4), (C2×C4).824(C22×D7), (C22×C14).103(C2×C4), (C2×C14).156(C22×C4), SmallGroup(448,1191)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 932 in 266 conjugacy classes, 151 normal (29 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 [×4], C2×C8 [×10], M4(2) [×12], 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, C22×C8 [×2], C2×M4(2) [×3], 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], C4.Dic7 [×4], C2×C56 [×2], C2×C56 [×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], D28.2C4 [×8], C2×C4.Dic7, C22×C56, C2×C4○D28, C2×D28.2C4
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, D28.2C4 [×2], D7×C22×C4, C2×D28.2C4
Generators and relations
G = < a,b,c,d | a2=b28=c2=1, d4=b14, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >
►On 224 points
Generators in S224
(1 83)(2 84)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 220)(30 221)(31 222)(32 223)(33 224)(34 197)(35 198)(36 199)(37 200)(38 201)(39 202)(40 203)(41 204)(42 205)(43 206)(44 207)(45 208)(46 209)(47 210)(48 211)(49 212)(50 213)(51 214)(52 215)(53 216)(54 217)(55 218)(56 219)(85 186)(86 187)(87 188)(88 189)(89 190)(90 191)(91 192)(92 193)(93 194)(94 195)(95 196)(96 169)(97 170)(98 171)(99 172)(100 173)(101 174)(102 175)(103 176)(104 177)(105 178)(106 179)(107 180)(108 181)(109 182)(110 183)(111 184)(112 185)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 161)(130 162)(131 163)(132 164)(133 165)(134 166)(135 167)(136 168)(137 141)(138 142)(139 143)(140 144)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 32)(30 31)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(81 84)(82 83)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(107 112)(108 111)(109 110)(113 116)(114 115)(117 140)(118 139)(119 138)(120 137)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(153 168)(154 167)(155 166)(156 165)(157 164)(158 163)(159 162)(160 161)(169 196)(170 195)(171 194)(172 193)(173 192)(174 191)(175 190)(176 189)(177 188)(178 187)(179 186)(180 185)(181 184)(182 183)(197 218)(198 217)(199 216)(200 215)(201 214)(202 213)(203 212)(204 211)(205 210)(206 209)(207 208)(219 224)(220 223)(221 222)
(1 31 169 161 15 45 183 147)(2 32 170 162 16 46 184 148)(3 33 171 163 17 47 185 149)(4 34 172 164 18 48 186 150)(5 35 173 165 19 49 187 151)(6 36 174 166 20 50 188 152)(7 37 175 167 21 51 189 153)(8 38 176 168 22 52 190 154)(9 39 177 141 23 53 191 155)(10 40 178 142 24 54 192 156)(11 41 179 143 25 55 193 157)(12 42 180 144 26 56 194 158)(13 43 181 145 27 29 195 159)(14 44 182 146 28 30 196 160)(57 224 98 131 71 210 112 117)(58 197 99 132 72 211 85 118)(59 198 100 133 73 212 86 119)(60 199 101 134 74 213 87 120)(61 200 102 135 75 214 88 121)(62 201 103 136 76 215 89 122)(63 202 104 137 77 216 90 123)(64 203 105 138 78 217 91 124)(65 204 106 139 79 218 92 125)(66 205 107 140 80 219 93 126)(67 206 108 113 81 220 94 127)(68 207 109 114 82 221 95 128)(69 208 110 115 83 222 96 129)(70 209 111 116 84 223 97 130)
G:=sub<Sym(224)| (1,83)(2,84)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,220)(30,221)(31,222)(32,223)(33,224)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,209)(47,210)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(85,186)(86,187)(87,188)(88,189)(89,190)(90,191)(91,192)(92,193)(93,194)(94,195)(95,196)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175)(103,176)(104,177)(105,178)(106,179)(107,180)(108,181)(109,182)(110,183)(111,184)(112,185)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,141)(138,142)(139,143)(140,144), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,32)(30,31)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(81,84)(82,83)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(107,112)(108,111)(109,110)(113,116)(114,115)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(153,168)(154,167)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161)(169,196)(170,195)(171,194)(172,193)(173,192)(174,191)(175,190)(176,189)(177,188)(178,187)(179,186)(180,185)(181,184)(182,183)(197,218)(198,217)(199,216)(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208)(219,224)(220,223)(221,222), (1,31,169,161,15,45,183,147)(2,32,170,162,16,46,184,148)(3,33,171,163,17,47,185,149)(4,34,172,164,18,48,186,150)(5,35,173,165,19,49,187,151)(6,36,174,166,20,50,188,152)(7,37,175,167,21,51,189,153)(8,38,176,168,22,52,190,154)(9,39,177,141,23,53,191,155)(10,40,178,142,24,54,192,156)(11,41,179,143,25,55,193,157)(12,42,180,144,26,56,194,158)(13,43,181,145,27,29,195,159)(14,44,182,146,28,30,196,160)(57,224,98,131,71,210,112,117)(58,197,99,132,72,211,85,118)(59,198,100,133,73,212,86,119)(60,199,101,134,74,213,87,120)(61,200,102,135,75,214,88,121)(62,201,103,136,76,215,89,122)(63,202,104,137,77,216,90,123)(64,203,105,138,78,217,91,124)(65,204,106,139,79,218,92,125)(66,205,107,140,80,219,93,126)(67,206,108,113,81,220,94,127)(68,207,109,114,82,221,95,128)(69,208,110,115,83,222,96,129)(70,209,111,116,84,223,97,130)>;
G:=Group( (1,83)(2,84)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,220)(30,221)(31,222)(32,223)(33,224)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,209)(47,210)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(85,186)(86,187)(87,188)(88,189)(89,190)(90,191)(91,192)(92,193)(93,194)(94,195)(95,196)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175)(103,176)(104,177)(105,178)(106,179)(107,180)(108,181)(109,182)(110,183)(111,184)(112,185)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,141)(138,142)(139,143)(140,144), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,32)(30,31)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(81,84)(82,83)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(107,112)(108,111)(109,110)(113,116)(114,115)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(153,168)(154,167)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161)(169,196)(170,195)(171,194)(172,193)(173,192)(174,191)(175,190)(176,189)(177,188)(178,187)(179,186)(180,185)(181,184)(182,183)(197,218)(198,217)(199,216)(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208)(219,224)(220,223)(221,222), (1,31,169,161,15,45,183,147)(2,32,170,162,16,46,184,148)(3,33,171,163,17,47,185,149)(4,34,172,164,18,48,186,150)(5,35,173,165,19,49,187,151)(6,36,174,166,20,50,188,152)(7,37,175,167,21,51,189,153)(8,38,176,168,22,52,190,154)(9,39,177,141,23,53,191,155)(10,40,178,142,24,54,192,156)(11,41,179,143,25,55,193,157)(12,42,180,144,26,56,194,158)(13,43,181,145,27,29,195,159)(14,44,182,146,28,30,196,160)(57,224,98,131,71,210,112,117)(58,197,99,132,72,211,85,118)(59,198,100,133,73,212,86,119)(60,199,101,134,74,213,87,120)(61,200,102,135,75,214,88,121)(62,201,103,136,76,215,89,122)(63,202,104,137,77,216,90,123)(64,203,105,138,78,217,91,124)(65,204,106,139,79,218,92,125)(66,205,107,140,80,219,93,126)(67,206,108,113,81,220,94,127)(68,207,109,114,82,221,95,128)(69,208,110,115,83,222,96,129)(70,209,111,116,84,223,97,130) );
G=PermutationGroup([(1,83),(2,84),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,81),(28,82),(29,220),(30,221),(31,222),(32,223),(33,224),(34,197),(35,198),(36,199),(37,200),(38,201),(39,202),(40,203),(41,204),(42,205),(43,206),(44,207),(45,208),(46,209),(47,210),(48,211),(49,212),(50,213),(51,214),(52,215),(53,216),(54,217),(55,218),(56,219),(85,186),(86,187),(87,188),(88,189),(89,190),(90,191),(91,192),(92,193),(93,194),(94,195),(95,196),(96,169),(97,170),(98,171),(99,172),(100,173),(101,174),(102,175),(103,176),(104,177),(105,178),(106,179),(107,180),(108,181),(109,182),(110,183),(111,184),(112,185),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152),(121,153),(122,154),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,161),(130,162),(131,163),(132,164),(133,165),(134,166),(135,167),(136,168),(137,141),(138,142),(139,143),(140,144)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,32),(30,31),(33,56),(34,55),(35,54),(36,53),(37,52),(38,51),(39,50),(40,49),(41,48),(42,47),(43,46),(44,45),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(81,84),(82,83),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96),(107,112),(108,111),(109,110),(113,116),(114,115),(117,140),(118,139),(119,138),(120,137),(121,136),(122,135),(123,134),(124,133),(125,132),(126,131),(127,130),(128,129),(141,152),(142,151),(143,150),(144,149),(145,148),(146,147),(153,168),(154,167),(155,166),(156,165),(157,164),(158,163),(159,162),(160,161),(169,196),(170,195),(171,194),(172,193),(173,192),(174,191),(175,190),(176,189),(177,188),(178,187),(179,186),(180,185),(181,184),(182,183),(197,218),(198,217),(199,216),(200,215),(201,214),(202,213),(203,212),(204,211),(205,210),(206,209),(207,208),(219,224),(220,223),(221,222)], [(1,31,169,161,15,45,183,147),(2,32,170,162,16,46,184,148),(3,33,171,163,17,47,185,149),(4,34,172,164,18,48,186,150),(5,35,173,165,19,49,187,151),(6,36,174,166,20,50,188,152),(7,37,175,167,21,51,189,153),(8,38,176,168,22,52,190,154),(9,39,177,141,23,53,191,155),(10,40,178,142,24,54,192,156),(11,41,179,143,25,55,193,157),(12,42,180,144,26,56,194,158),(13,43,181,145,27,29,195,159),(14,44,182,146,28,30,196,160),(57,224,98,131,71,210,112,117),(58,197,99,132,72,211,85,118),(59,198,100,133,73,212,86,119),(60,199,101,134,74,213,87,120),(61,200,102,135,75,214,88,121),(62,201,103,136,76,215,89,122),(63,202,104,137,77,216,90,123),(64,203,105,138,78,217,91,124),(65,204,106,139,79,218,92,125),(66,205,107,140,80,219,93,126),(67,206,108,113,81,220,94,127),(68,207,109,114,82,221,95,128),(69,208,110,115,83,222,96,129),(70,209,111,116,84,223,97,130)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 10 | 24 | 0 | 0 |
| 89 | 112 | 0 | 0 |
| 0 | 0 | 78 | 32 |
| 0 | 0 | 49 | 94 |
| 112 | 0 | 0 | 0 |
| 24 | 1 | 0 | 0 |
| 0 | 0 | 47 | 38 |
| 0 | 0 | 43 | 66 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 0 | 44 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[10,89,0,0,24,112,0,0,0,0,78,49,0,0,32,94],[112,24,0,0,0,1,0,0,0,0,47,43,0,0,38,66],[98,0,0,0,0,98,0,0,0,0,44,0,0,0,0,44] >;
136 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 | 8J | 8K | 8L | 8M | ··· | 8T | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| 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 | 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 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D7 | D14 | D14 | C8○D4 | C4×D7 | C4×D7 | D28.2C4 |
| kernel | C2×D28.2C4 | D7×C2×C8 | C2×C8⋊D7 | D28.2C4 | C2×C4.Dic7 | C22×C56 | C2×C4○D28 | C2×Dic14 | C2×D28 | C4○D28 | C2×C7⋊D4 | C22×C8 | C2×C8 | C22×C4 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 3 | 18 | 3 | 8 | 18 | 6 | 48 |
In GAP, Magma, Sage, TeX
C_2\times D_{28}._2C_4 % in TeX
G:=Group("C2xD28.2C4"); // GroupNames label
G:=SmallGroup(448,1191);
// by ID
G=gap.SmallGroup(448,1191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,80,102,18822]);
// 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,b*d=d*b,c*d=d*c>;
// generators/relations
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