metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Dic7⋊8C4, C28.33(C4⋊C4), (C2×C28).21Q8, C28.64(C2×Q8), C4⋊C4.232D14, (C2×C28).471D4, C28.Q8⋊27C2, C28.62(C22×C4), C4.Dic14⋊27C2, (C2×C4).14Dic14, C4.29(C2×Dic14), (C22×C14).71D4, C42⋊C2.4D7, C2.1(D4⋊D14), C7⋊4(M4(2)⋊C4), C4.25(Dic7⋊C4), (C2×C28).325C23, (C22×C4).107D14, C23.53(C7⋊D4), C14.104(C8⋊C22), C2.1(D4.9D14), C22.8(Dic7⋊C4), C4⋊Dic7.324C22, C14.104(C8.C22), (C22×C28).145C22, C7⋊C8⋊7(C2×C4), C4.87(C2×C4×D7), C14.39(C2×C4⋊C4), (C2×C4).43(C4×D7), (C2×C28).87(C2×C4), (C2×C7⋊C8).85C22, (C2×C14).11(C4⋊C4), (C2×C14).454(C2×D4), C2.13(C2×Dic7⋊C4), (C2×C4⋊Dic7).36C2, C22.70(C2×C7⋊D4), (C2×C4).240(C7⋊D4), (C7×C4⋊C4).263C22, (C7×C42⋊C2).4C2, (C2×C4).425(C22×D7), (C2×C4.Dic7).16C2, SmallGroup(448,529)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.(C2×Q8)
G = < a,b,c,d | a28=b2=c4=1, d2=a14c2, ab=ba, cac-1=a-1, dad-1=a15, bc=cb, dbd-1=a14b, dcd-1=a7c-1 >
Subgroups: 436 in 118 conjugacy classes, 63 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic7, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C7⋊C8, C2×Dic7, C2×C28, C2×C28, C2×C28, C22×C14, M4(2)⋊C4, C2×C7⋊C8, C4.Dic7, C4⋊Dic7, C4⋊Dic7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, C28.Q8, C4.Dic14, C2×C4.Dic7, C2×C4⋊Dic7, C7×C42⋊C2, C28.(C2×Q8)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C8⋊C22, C8.C22, Dic14, C4×D7, C7⋊D4, C22×D7, M4(2)⋊C4, Dic7⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, C2×Dic7⋊C4, D4⋊D14, D4.9D14, C28.(C2×Q8)
►On 224 points
Generators in S224
(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 55)(2 56)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(57 209)(58 210)(59 211)(60 212)(61 213)(62 214)(63 215)(64 216)(65 217)(66 218)(67 219)(68 220)(69 221)(70 222)(71 223)(72 224)(73 197)(74 198)(75 199)(76 200)(77 201)(78 202)(79 203)(80 204)(81 205)(82 206)(83 207)(84 208)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 168)(110 141)(111 142)(112 143)(113 188)(114 189)(115 190)(116 191)(117 192)(118 193)(119 194)(120 195)(121 196)(122 169)(123 170)(124 171)(125 172)(126 173)(127 174)(128 175)(129 176)(130 177)(131 178)(132 179)(133 180)(134 181)(135 182)(136 183)(137 184)(138 185)(139 186)(140 187)
(1 207 55 83)(2 206 56 82)(3 205 29 81)(4 204 30 80)(5 203 31 79)(6 202 32 78)(7 201 33 77)(8 200 34 76)(9 199 35 75)(10 198 36 74)(11 197 37 73)(12 224 38 72)(13 223 39 71)(14 222 40 70)(15 221 41 69)(16 220 42 68)(17 219 43 67)(18 218 44 66)(19 217 45 65)(20 216 46 64)(21 215 47 63)(22 214 48 62)(23 213 49 61)(24 212 50 60)(25 211 51 59)(26 210 52 58)(27 209 53 57)(28 208 54 84)(85 170 158 137)(86 169 159 136)(87 196 160 135)(88 195 161 134)(89 194 162 133)(90 193 163 132)(91 192 164 131)(92 191 165 130)(93 190 166 129)(94 189 167 128)(95 188 168 127)(96 187 141 126)(97 186 142 125)(98 185 143 124)(99 184 144 123)(100 183 145 122)(101 182 146 121)(102 181 147 120)(103 180 148 119)(104 179 149 118)(105 178 150 117)(106 177 151 116)(107 176 152 115)(108 175 153 114)(109 174 154 113)(110 173 155 140)(111 172 156 139)(112 171 157 138)
(1 160 41 101)(2 147 42 88)(3 162 43 103)(4 149 44 90)(5 164 45 105)(6 151 46 92)(7 166 47 107)(8 153 48 94)(9 168 49 109)(10 155 50 96)(11 142 51 111)(12 157 52 98)(13 144 53 85)(14 159 54 100)(15 146 55 87)(16 161 56 102)(17 148 29 89)(18 163 30 104)(19 150 31 91)(20 165 32 106)(21 152 33 93)(22 167 34 108)(23 154 35 95)(24 141 36 110)(25 156 37 97)(26 143 38 112)(27 158 39 99)(28 145 40 86)(57 130 223 177)(58 117 224 192)(59 132 197 179)(60 119 198 194)(61 134 199 181)(62 121 200 196)(63 136 201 183)(64 123 202 170)(65 138 203 185)(66 125 204 172)(67 140 205 187)(68 127 206 174)(69 114 207 189)(70 129 208 176)(71 116 209 191)(72 131 210 178)(73 118 211 193)(74 133 212 180)(75 120 213 195)(76 135 214 182)(77 122 215 169)(78 137 216 184)(79 124 217 171)(80 139 218 186)(81 126 219 173)(82 113 220 188)(83 128 221 175)(84 115 222 190)
G:=sub<Sym(224)| (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), (1,55)(2,56)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(57,209)(58,210)(59,211)(60,212)(61,213)(62,214)(63,215)(64,216)(65,217)(66,218)(67,219)(68,220)(69,221)(70,222)(71,223)(72,224)(73,197)(74,198)(75,199)(76,200)(77,201)(78,202)(79,203)(80,204)(81,205)(82,206)(83,207)(84,208)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,141)(111,142)(112,143)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187), (1,207,55,83)(2,206,56,82)(3,205,29,81)(4,204,30,80)(5,203,31,79)(6,202,32,78)(7,201,33,77)(8,200,34,76)(9,199,35,75)(10,198,36,74)(11,197,37,73)(12,224,38,72)(13,223,39,71)(14,222,40,70)(15,221,41,69)(16,220,42,68)(17,219,43,67)(18,218,44,66)(19,217,45,65)(20,216,46,64)(21,215,47,63)(22,214,48,62)(23,213,49,61)(24,212,50,60)(25,211,51,59)(26,210,52,58)(27,209,53,57)(28,208,54,84)(85,170,158,137)(86,169,159,136)(87,196,160,135)(88,195,161,134)(89,194,162,133)(90,193,163,132)(91,192,164,131)(92,191,165,130)(93,190,166,129)(94,189,167,128)(95,188,168,127)(96,187,141,126)(97,186,142,125)(98,185,143,124)(99,184,144,123)(100,183,145,122)(101,182,146,121)(102,181,147,120)(103,180,148,119)(104,179,149,118)(105,178,150,117)(106,177,151,116)(107,176,152,115)(108,175,153,114)(109,174,154,113)(110,173,155,140)(111,172,156,139)(112,171,157,138), (1,160,41,101)(2,147,42,88)(3,162,43,103)(4,149,44,90)(5,164,45,105)(6,151,46,92)(7,166,47,107)(8,153,48,94)(9,168,49,109)(10,155,50,96)(11,142,51,111)(12,157,52,98)(13,144,53,85)(14,159,54,100)(15,146,55,87)(16,161,56,102)(17,148,29,89)(18,163,30,104)(19,150,31,91)(20,165,32,106)(21,152,33,93)(22,167,34,108)(23,154,35,95)(24,141,36,110)(25,156,37,97)(26,143,38,112)(27,158,39,99)(28,145,40,86)(57,130,223,177)(58,117,224,192)(59,132,197,179)(60,119,198,194)(61,134,199,181)(62,121,200,196)(63,136,201,183)(64,123,202,170)(65,138,203,185)(66,125,204,172)(67,140,205,187)(68,127,206,174)(69,114,207,189)(70,129,208,176)(71,116,209,191)(72,131,210,178)(73,118,211,193)(74,133,212,180)(75,120,213,195)(76,135,214,182)(77,122,215,169)(78,137,216,184)(79,124,217,171)(80,139,218,186)(81,126,219,173)(82,113,220,188)(83,128,221,175)(84,115,222,190)>;
G:=Group( (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), (1,55)(2,56)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(57,209)(58,210)(59,211)(60,212)(61,213)(62,214)(63,215)(64,216)(65,217)(66,218)(67,219)(68,220)(69,221)(70,222)(71,223)(72,224)(73,197)(74,198)(75,199)(76,200)(77,201)(78,202)(79,203)(80,204)(81,205)(82,206)(83,207)(84,208)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,141)(111,142)(112,143)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187), (1,207,55,83)(2,206,56,82)(3,205,29,81)(4,204,30,80)(5,203,31,79)(6,202,32,78)(7,201,33,77)(8,200,34,76)(9,199,35,75)(10,198,36,74)(11,197,37,73)(12,224,38,72)(13,223,39,71)(14,222,40,70)(15,221,41,69)(16,220,42,68)(17,219,43,67)(18,218,44,66)(19,217,45,65)(20,216,46,64)(21,215,47,63)(22,214,48,62)(23,213,49,61)(24,212,50,60)(25,211,51,59)(26,210,52,58)(27,209,53,57)(28,208,54,84)(85,170,158,137)(86,169,159,136)(87,196,160,135)(88,195,161,134)(89,194,162,133)(90,193,163,132)(91,192,164,131)(92,191,165,130)(93,190,166,129)(94,189,167,128)(95,188,168,127)(96,187,141,126)(97,186,142,125)(98,185,143,124)(99,184,144,123)(100,183,145,122)(101,182,146,121)(102,181,147,120)(103,180,148,119)(104,179,149,118)(105,178,150,117)(106,177,151,116)(107,176,152,115)(108,175,153,114)(109,174,154,113)(110,173,155,140)(111,172,156,139)(112,171,157,138), (1,160,41,101)(2,147,42,88)(3,162,43,103)(4,149,44,90)(5,164,45,105)(6,151,46,92)(7,166,47,107)(8,153,48,94)(9,168,49,109)(10,155,50,96)(11,142,51,111)(12,157,52,98)(13,144,53,85)(14,159,54,100)(15,146,55,87)(16,161,56,102)(17,148,29,89)(18,163,30,104)(19,150,31,91)(20,165,32,106)(21,152,33,93)(22,167,34,108)(23,154,35,95)(24,141,36,110)(25,156,37,97)(26,143,38,112)(27,158,39,99)(28,145,40,86)(57,130,223,177)(58,117,224,192)(59,132,197,179)(60,119,198,194)(61,134,199,181)(62,121,200,196)(63,136,201,183)(64,123,202,170)(65,138,203,185)(66,125,204,172)(67,140,205,187)(68,127,206,174)(69,114,207,189)(70,129,208,176)(71,116,209,191)(72,131,210,178)(73,118,211,193)(74,133,212,180)(75,120,213,195)(76,135,214,182)(77,122,215,169)(78,137,216,184)(79,124,217,171)(80,139,218,186)(81,126,219,173)(82,113,220,188)(83,128,221,175)(84,115,222,190) );
G=PermutationGroup([[(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)], [(1,55),(2,56),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(57,209),(58,210),(59,211),(60,212),(61,213),(62,214),(63,215),(64,216),(65,217),(66,218),(67,219),(68,220),(69,221),(70,222),(71,223),(72,224),(73,197),(74,198),(75,199),(76,200),(77,201),(78,202),(79,203),(80,204),(81,205),(82,206),(83,207),(84,208),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163),(105,164),(106,165),(107,166),(108,167),(109,168),(110,141),(111,142),(112,143),(113,188),(114,189),(115,190),(116,191),(117,192),(118,193),(119,194),(120,195),(121,196),(122,169),(123,170),(124,171),(125,172),(126,173),(127,174),(128,175),(129,176),(130,177),(131,178),(132,179),(133,180),(134,181),(135,182),(136,183),(137,184),(138,185),(139,186),(140,187)], [(1,207,55,83),(2,206,56,82),(3,205,29,81),(4,204,30,80),(5,203,31,79),(6,202,32,78),(7,201,33,77),(8,200,34,76),(9,199,35,75),(10,198,36,74),(11,197,37,73),(12,224,38,72),(13,223,39,71),(14,222,40,70),(15,221,41,69),(16,220,42,68),(17,219,43,67),(18,218,44,66),(19,217,45,65),(20,216,46,64),(21,215,47,63),(22,214,48,62),(23,213,49,61),(24,212,50,60),(25,211,51,59),(26,210,52,58),(27,209,53,57),(28,208,54,84),(85,170,158,137),(86,169,159,136),(87,196,160,135),(88,195,161,134),(89,194,162,133),(90,193,163,132),(91,192,164,131),(92,191,165,130),(93,190,166,129),(94,189,167,128),(95,188,168,127),(96,187,141,126),(97,186,142,125),(98,185,143,124),(99,184,144,123),(100,183,145,122),(101,182,146,121),(102,181,147,120),(103,180,148,119),(104,179,149,118),(105,178,150,117),(106,177,151,116),(107,176,152,115),(108,175,153,114),(109,174,154,113),(110,173,155,140),(111,172,156,139),(112,171,157,138)], [(1,160,41,101),(2,147,42,88),(3,162,43,103),(4,149,44,90),(5,164,45,105),(6,151,46,92),(7,166,47,107),(8,153,48,94),(9,168,49,109),(10,155,50,96),(11,142,51,111),(12,157,52,98),(13,144,53,85),(14,159,54,100),(15,146,55,87),(16,161,56,102),(17,148,29,89),(18,163,30,104),(19,150,31,91),(20,165,32,106),(21,152,33,93),(22,167,34,108),(23,154,35,95),(24,141,36,110),(25,156,37,97),(26,143,38,112),(27,158,39,99),(28,145,40,86),(57,130,223,177),(58,117,224,192),(59,132,197,179),(60,119,198,194),(61,134,199,181),(62,121,200,196),(63,136,201,183),(64,123,202,170),(65,138,203,185),(66,125,204,172),(67,140,205,187),(68,127,206,174),(69,114,207,189),(70,129,208,176),(71,116,209,191),(72,131,210,178),(73,118,211,193),(74,133,212,180),(75,120,213,195),(76,135,214,182),(77,122,215,169),(78,137,216,184),(79,124,217,171),(80,139,218,186),(81,126,219,173),(82,113,220,188),(83,128,221,175),(84,115,222,190)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D7 | D14 | D14 | Dic14 | C4×D7 | C7⋊D4 | C7⋊D4 | C8⋊C22 | C8.C22 | D4⋊D14 | D4.9D14 |
| kernel | C28.(C2×Q8) | C28.Q8 | C4.Dic14 | C2×C4.Dic7 | C2×C4⋊Dic7 | C7×C42⋊C2 | C4.Dic7 | C2×C28 | C2×C28 | C22×C14 | C42⋊C2 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 3 | 6 | 3 | 12 | 12 | 6 | 6 | 1 | 1 | 6 | 6 |
Matrix representation of C28.(C2×Q8) ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 78 | 32 | 0 | 0 |
| 0 | 0 | 49 | 94 | 0 | 0 |
| 0 | 0 | 40 | 45 | 58 | 81 |
| 0 | 0 | 80 | 56 | 32 | 109 |
| 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 | 86 | 83 | 112 | 0 |
| 0 | 0 | 30 | 0 | 0 | 112 |
| 23 | 2 | 0 | 0 | 0 | 0 |
| 74 | 90 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 6 | 0 | 0 |
| 0 | 0 | 94 | 62 | 0 | 0 |
| 0 | 0 | 64 | 98 | 89 | 64 |
| 0 | 0 | 48 | 70 | 74 | 24 |
| 98 | 104 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 98 |
| 0 | 0 | 0 | 112 | 15 | 70 |
| 0 | 0 | 27 | 30 | 1 | 0 |
| 0 | 0 | 83 | 0 | 0 | 1 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,78,49,40,80,0,0,32,94,45,56,0,0,0,0,58,32,0,0,0,0,81,109],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,86,30,0,0,0,1,83,0,0,0,0,0,112,0,0,0,0,0,0,112],[23,74,0,0,0,0,2,90,0,0,0,0,0,0,51,94,64,48,0,0,6,62,98,70,0,0,0,0,89,74,0,0,0,0,64,24],[98,0,0,0,0,0,104,15,0,0,0,0,0,0,112,0,27,83,0,0,0,112,30,0,0,0,0,15,1,0,0,0,98,70,0,1] >;
C28.(C2×Q8) in GAP, Magma, Sage, TeX
C_{28}.(C_2\times Q_8) % in TeX
G:=Group("C28.(C2xQ8)"); // GroupNames label
G:=SmallGroup(448,529);
// by ID
G=gap.SmallGroup(448,529);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,254,387,100,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=a^14*c^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^15,b*c=c*b,d*b*d^-1=a^14*b,d*c*d^-1=a^7*c^-1>;
// generators/relations