metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.65D14, C22⋊Q8.4D7, (C2×C28).265D4, (C2×Q8).27D14, C14.99(C4○D8), C28.Q8⋊39C2, Q8⋊Dic7⋊14C2, C4.Dic14⋊38C2, (C22×C14).91D4, C28.189(C4○D4), C4.95(D4⋊2D7), (C2×C28).364C23, C28.55D4.8C2, (C22×C4).126D14, C23.26(C7⋊D4), C7⋊7(C23.20D4), (Q8×C14).45C22, C14.89(C8.C22), C4⋊Dic7.339C22, C2.18(D4.8D14), C2.10(C28.C23), (C22×C28).168C22, C23.21D14.14C2, C14.82(C22.D4), C2.16(C23.18D14), (C7×C22⋊Q8).3C2, (C2×C14).495(C2×D4), (C2×C7⋊C8).114C22, (C2×C4).172(C7⋊D4), (C7×C4⋊C4).112C22, (C2×C4).464(C22×D7), C22.170(C2×C7⋊D4), SmallGroup(448,579)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.(C4○D8)
G = < a,b,c,d,e | a2=b4=1, c2=d14=b2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ab2c, ece-1=b-1c, ede-1=d13 >
Subgroups: 348 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×C14, C23.20D4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×C28, Q8×C14, C28.Q8, C4.Dic14, C28.55D4, Q8⋊Dic7, C23.21D14, C7×C22⋊Q8, C14.(C4○D8)
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C4○D8, C8.C22, C7⋊D4, C22×D7, C23.20D4, D4⋊2D7, C2×C7⋊D4, C23.18D14, C28.C23, D4.8D14, C14.(C4○D8)
►On 224 points
Generators in S224
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 159)(42 160)(43 161)(44 162)(45 163)(46 164)(47 165)(48 166)(49 167)(50 168)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(85 178)(86 179)(87 180)(88 181)(89 182)(90 183)(91 184)(92 185)(93 186)(94 187)(95 188)(96 189)(97 190)(98 191)(99 192)(100 193)(101 194)(102 195)(103 196)(104 169)(105 170)(106 171)(107 172)(108 173)(109 174)(110 175)(111 176)(112 177)(113 203)(114 204)(115 205)(116 206)(117 207)(118 208)(119 209)(120 210)(121 211)(122 212)(123 213)(124 214)(125 215)(126 216)(127 217)(128 218)(129 219)(130 220)(131 221)(132 222)(133 223)(134 224)(135 197)(136 198)(137 199)(138 200)(139 201)(140 202)
(1 63 15 77)(2 64 16 78)(3 65 17 79)(4 66 18 80)(5 67 19 81)(6 68 20 82)(7 69 21 83)(8 70 22 84)(9 71 23 57)(10 72 24 58)(11 73 25 59)(12 74 26 60)(13 75 27 61)(14 76 28 62)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)(85 185 99 171)(86 186 100 172)(87 187 101 173)(88 188 102 174)(89 189 103 175)(90 190 104 176)(91 191 105 177)(92 192 106 178)(93 193 107 179)(94 194 108 180)(95 195 109 181)(96 196 110 182)(97 169 111 183)(98 170 112 184)(113 120 127 134)(114 121 128 135)(115 122 129 136)(116 123 130 137)(117 124 131 138)(118 125 132 139)(119 126 133 140)(141 162 155 148)(142 163 156 149)(143 164 157 150)(144 165 158 151)(145 166 159 152)(146 167 160 153)(147 168 161 154)(197 204 211 218)(198 205 212 219)(199 206 213 220)(200 207 214 221)(201 208 215 222)(202 209 216 223)(203 210 217 224)
(1 42 15 56)(2 147 16 161)(3 44 17 30)(4 149 18 163)(5 46 19 32)(6 151 20 165)(7 48 21 34)(8 153 22 167)(9 50 23 36)(10 155 24 141)(11 52 25 38)(12 157 26 143)(13 54 27 40)(14 159 28 145)(29 57 43 71)(31 59 45 73)(33 61 47 75)(35 63 49 77)(37 65 51 79)(39 67 53 81)(41 69 55 83)(58 148 72 162)(60 150 74 164)(62 152 76 166)(64 154 78 168)(66 156 80 142)(68 158 82 144)(70 160 84 146)(85 214 99 200)(86 139 100 125)(87 216 101 202)(88 113 102 127)(89 218 103 204)(90 115 104 129)(91 220 105 206)(92 117 106 131)(93 222 107 208)(94 119 108 133)(95 224 109 210)(96 121 110 135)(97 198 111 212)(98 123 112 137)(114 182 128 196)(116 184 130 170)(118 186 132 172)(120 188 134 174)(122 190 136 176)(124 192 138 178)(126 194 140 180)(169 219 183 205)(171 221 185 207)(173 223 187 209)(175 197 189 211)(177 199 191 213)(179 201 193 215)(181 203 195 217)
(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 93 70 186)(2 106 71 171)(3 91 72 184)(4 104 73 169)(5 89 74 182)(6 102 75 195)(7 87 76 180)(8 100 77 193)(9 85 78 178)(10 98 79 191)(11 111 80 176)(12 96 81 189)(13 109 82 174)(14 94 83 187)(15 107 84 172)(16 92 57 185)(17 105 58 170)(18 90 59 183)(19 103 60 196)(20 88 61 181)(21 101 62 194)(22 86 63 179)(23 99 64 192)(24 112 65 177)(25 97 66 190)(26 110 67 175)(27 95 68 188)(28 108 69 173)(29 200 147 138)(30 213 148 123)(31 198 149 136)(32 211 150 121)(33 224 151 134)(34 209 152 119)(35 222 153 132)(36 207 154 117)(37 220 155 130)(38 205 156 115)(39 218 157 128)(40 203 158 113)(41 216 159 126)(42 201 160 139)(43 214 161 124)(44 199 162 137)(45 212 163 122)(46 197 164 135)(47 210 165 120)(48 223 166 133)(49 208 167 118)(50 221 168 131)(51 206 141 116)(52 219 142 129)(53 204 143 114)(54 217 144 127)(55 202 145 140)(56 215 146 125)
G:=sub<Sym(224)| (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(85,178)(86,179)(87,180)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(97,190)(98,191)(99,192)(100,193)(101,194)(102,195)(103,196)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,203)(114,204)(115,205)(116,206)(117,207)(118,208)(119,209)(120,210)(121,211)(122,212)(123,213)(124,214)(125,215)(126,216)(127,217)(128,218)(129,219)(130,220)(131,221)(132,222)(133,223)(134,224)(135,197)(136,198)(137,199)(138,200)(139,201)(140,202), (1,63,15,77)(2,64,16,78)(3,65,17,79)(4,66,18,80)(5,67,19,81)(6,68,20,82)(7,69,21,83)(8,70,22,84)(9,71,23,57)(10,72,24,58)(11,73,25,59)(12,74,26,60)(13,75,27,61)(14,76,28,62)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(85,185,99,171)(86,186,100,172)(87,187,101,173)(88,188,102,174)(89,189,103,175)(90,190,104,176)(91,191,105,177)(92,192,106,178)(93,193,107,179)(94,194,108,180)(95,195,109,181)(96,196,110,182)(97,169,111,183)(98,170,112,184)(113,120,127,134)(114,121,128,135)(115,122,129,136)(116,123,130,137)(117,124,131,138)(118,125,132,139)(119,126,133,140)(141,162,155,148)(142,163,156,149)(143,164,157,150)(144,165,158,151)(145,166,159,152)(146,167,160,153)(147,168,161,154)(197,204,211,218)(198,205,212,219)(199,206,213,220)(200,207,214,221)(201,208,215,222)(202,209,216,223)(203,210,217,224), (1,42,15,56)(2,147,16,161)(3,44,17,30)(4,149,18,163)(5,46,19,32)(6,151,20,165)(7,48,21,34)(8,153,22,167)(9,50,23,36)(10,155,24,141)(11,52,25,38)(12,157,26,143)(13,54,27,40)(14,159,28,145)(29,57,43,71)(31,59,45,73)(33,61,47,75)(35,63,49,77)(37,65,51,79)(39,67,53,81)(41,69,55,83)(58,148,72,162)(60,150,74,164)(62,152,76,166)(64,154,78,168)(66,156,80,142)(68,158,82,144)(70,160,84,146)(85,214,99,200)(86,139,100,125)(87,216,101,202)(88,113,102,127)(89,218,103,204)(90,115,104,129)(91,220,105,206)(92,117,106,131)(93,222,107,208)(94,119,108,133)(95,224,109,210)(96,121,110,135)(97,198,111,212)(98,123,112,137)(114,182,128,196)(116,184,130,170)(118,186,132,172)(120,188,134,174)(122,190,136,176)(124,192,138,178)(126,194,140,180)(169,219,183,205)(171,221,185,207)(173,223,187,209)(175,197,189,211)(177,199,191,213)(179,201,193,215)(181,203,195,217), (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), (1,93,70,186)(2,106,71,171)(3,91,72,184)(4,104,73,169)(5,89,74,182)(6,102,75,195)(7,87,76,180)(8,100,77,193)(9,85,78,178)(10,98,79,191)(11,111,80,176)(12,96,81,189)(13,109,82,174)(14,94,83,187)(15,107,84,172)(16,92,57,185)(17,105,58,170)(18,90,59,183)(19,103,60,196)(20,88,61,181)(21,101,62,194)(22,86,63,179)(23,99,64,192)(24,112,65,177)(25,97,66,190)(26,110,67,175)(27,95,68,188)(28,108,69,173)(29,200,147,138)(30,213,148,123)(31,198,149,136)(32,211,150,121)(33,224,151,134)(34,209,152,119)(35,222,153,132)(36,207,154,117)(37,220,155,130)(38,205,156,115)(39,218,157,128)(40,203,158,113)(41,216,159,126)(42,201,160,139)(43,214,161,124)(44,199,162,137)(45,212,163,122)(46,197,164,135)(47,210,165,120)(48,223,166,133)(49,208,167,118)(50,221,168,131)(51,206,141,116)(52,219,142,129)(53,204,143,114)(54,217,144,127)(55,202,145,140)(56,215,146,125)>;
G:=Group( (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(85,178)(86,179)(87,180)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(97,190)(98,191)(99,192)(100,193)(101,194)(102,195)(103,196)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,203)(114,204)(115,205)(116,206)(117,207)(118,208)(119,209)(120,210)(121,211)(122,212)(123,213)(124,214)(125,215)(126,216)(127,217)(128,218)(129,219)(130,220)(131,221)(132,222)(133,223)(134,224)(135,197)(136,198)(137,199)(138,200)(139,201)(140,202), (1,63,15,77)(2,64,16,78)(3,65,17,79)(4,66,18,80)(5,67,19,81)(6,68,20,82)(7,69,21,83)(8,70,22,84)(9,71,23,57)(10,72,24,58)(11,73,25,59)(12,74,26,60)(13,75,27,61)(14,76,28,62)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(85,185,99,171)(86,186,100,172)(87,187,101,173)(88,188,102,174)(89,189,103,175)(90,190,104,176)(91,191,105,177)(92,192,106,178)(93,193,107,179)(94,194,108,180)(95,195,109,181)(96,196,110,182)(97,169,111,183)(98,170,112,184)(113,120,127,134)(114,121,128,135)(115,122,129,136)(116,123,130,137)(117,124,131,138)(118,125,132,139)(119,126,133,140)(141,162,155,148)(142,163,156,149)(143,164,157,150)(144,165,158,151)(145,166,159,152)(146,167,160,153)(147,168,161,154)(197,204,211,218)(198,205,212,219)(199,206,213,220)(200,207,214,221)(201,208,215,222)(202,209,216,223)(203,210,217,224), (1,42,15,56)(2,147,16,161)(3,44,17,30)(4,149,18,163)(5,46,19,32)(6,151,20,165)(7,48,21,34)(8,153,22,167)(9,50,23,36)(10,155,24,141)(11,52,25,38)(12,157,26,143)(13,54,27,40)(14,159,28,145)(29,57,43,71)(31,59,45,73)(33,61,47,75)(35,63,49,77)(37,65,51,79)(39,67,53,81)(41,69,55,83)(58,148,72,162)(60,150,74,164)(62,152,76,166)(64,154,78,168)(66,156,80,142)(68,158,82,144)(70,160,84,146)(85,214,99,200)(86,139,100,125)(87,216,101,202)(88,113,102,127)(89,218,103,204)(90,115,104,129)(91,220,105,206)(92,117,106,131)(93,222,107,208)(94,119,108,133)(95,224,109,210)(96,121,110,135)(97,198,111,212)(98,123,112,137)(114,182,128,196)(116,184,130,170)(118,186,132,172)(120,188,134,174)(122,190,136,176)(124,192,138,178)(126,194,140,180)(169,219,183,205)(171,221,185,207)(173,223,187,209)(175,197,189,211)(177,199,191,213)(179,201,193,215)(181,203,195,217), (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), (1,93,70,186)(2,106,71,171)(3,91,72,184)(4,104,73,169)(5,89,74,182)(6,102,75,195)(7,87,76,180)(8,100,77,193)(9,85,78,178)(10,98,79,191)(11,111,80,176)(12,96,81,189)(13,109,82,174)(14,94,83,187)(15,107,84,172)(16,92,57,185)(17,105,58,170)(18,90,59,183)(19,103,60,196)(20,88,61,181)(21,101,62,194)(22,86,63,179)(23,99,64,192)(24,112,65,177)(25,97,66,190)(26,110,67,175)(27,95,68,188)(28,108,69,173)(29,200,147,138)(30,213,148,123)(31,198,149,136)(32,211,150,121)(33,224,151,134)(34,209,152,119)(35,222,153,132)(36,207,154,117)(37,220,155,130)(38,205,156,115)(39,218,157,128)(40,203,158,113)(41,216,159,126)(42,201,160,139)(43,214,161,124)(44,199,162,137)(45,212,163,122)(46,197,164,135)(47,210,165,120)(48,223,166,133)(49,208,167,118)(50,221,168,131)(51,206,141,116)(52,219,142,129)(53,204,143,114)(54,217,144,127)(55,202,145,140)(56,215,146,125) );
G=PermutationGroup([[(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,84),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,147),(30,148),(31,149),(32,150),(33,151),(34,152),(35,153),(36,154),(37,155),(38,156),(39,157),(40,158),(41,159),(42,160),(43,161),(44,162),(45,163),(46,164),(47,165),(48,166),(49,167),(50,168),(51,141),(52,142),(53,143),(54,144),(55,145),(56,146),(85,178),(86,179),(87,180),(88,181),(89,182),(90,183),(91,184),(92,185),(93,186),(94,187),(95,188),(96,189),(97,190),(98,191),(99,192),(100,193),(101,194),(102,195),(103,196),(104,169),(105,170),(106,171),(107,172),(108,173),(109,174),(110,175),(111,176),(112,177),(113,203),(114,204),(115,205),(116,206),(117,207),(118,208),(119,209),(120,210),(121,211),(122,212),(123,213),(124,214),(125,215),(126,216),(127,217),(128,218),(129,219),(130,220),(131,221),(132,222),(133,223),(134,224),(135,197),(136,198),(137,199),(138,200),(139,201),(140,202)], [(1,63,15,77),(2,64,16,78),(3,65,17,79),(4,66,18,80),(5,67,19,81),(6,68,20,82),(7,69,21,83),(8,70,22,84),(9,71,23,57),(10,72,24,58),(11,73,25,59),(12,74,26,60),(13,75,27,61),(14,76,28,62),(29,50,43,36),(30,51,44,37),(31,52,45,38),(32,53,46,39),(33,54,47,40),(34,55,48,41),(35,56,49,42),(85,185,99,171),(86,186,100,172),(87,187,101,173),(88,188,102,174),(89,189,103,175),(90,190,104,176),(91,191,105,177),(92,192,106,178),(93,193,107,179),(94,194,108,180),(95,195,109,181),(96,196,110,182),(97,169,111,183),(98,170,112,184),(113,120,127,134),(114,121,128,135),(115,122,129,136),(116,123,130,137),(117,124,131,138),(118,125,132,139),(119,126,133,140),(141,162,155,148),(142,163,156,149),(143,164,157,150),(144,165,158,151),(145,166,159,152),(146,167,160,153),(147,168,161,154),(197,204,211,218),(198,205,212,219),(199,206,213,220),(200,207,214,221),(201,208,215,222),(202,209,216,223),(203,210,217,224)], [(1,42,15,56),(2,147,16,161),(3,44,17,30),(4,149,18,163),(5,46,19,32),(6,151,20,165),(7,48,21,34),(8,153,22,167),(9,50,23,36),(10,155,24,141),(11,52,25,38),(12,157,26,143),(13,54,27,40),(14,159,28,145),(29,57,43,71),(31,59,45,73),(33,61,47,75),(35,63,49,77),(37,65,51,79),(39,67,53,81),(41,69,55,83),(58,148,72,162),(60,150,74,164),(62,152,76,166),(64,154,78,168),(66,156,80,142),(68,158,82,144),(70,160,84,146),(85,214,99,200),(86,139,100,125),(87,216,101,202),(88,113,102,127),(89,218,103,204),(90,115,104,129),(91,220,105,206),(92,117,106,131),(93,222,107,208),(94,119,108,133),(95,224,109,210),(96,121,110,135),(97,198,111,212),(98,123,112,137),(114,182,128,196),(116,184,130,170),(118,186,132,172),(120,188,134,174),(122,190,136,176),(124,192,138,178),(126,194,140,180),(169,219,183,205),(171,221,185,207),(173,223,187,209),(175,197,189,211),(177,199,191,213),(179,201,193,215),(181,203,195,217)], [(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)], [(1,93,70,186),(2,106,71,171),(3,91,72,184),(4,104,73,169),(5,89,74,182),(6,102,75,195),(7,87,76,180),(8,100,77,193),(9,85,78,178),(10,98,79,191),(11,111,80,176),(12,96,81,189),(13,109,82,174),(14,94,83,187),(15,107,84,172),(16,92,57,185),(17,105,58,170),(18,90,59,183),(19,103,60,196),(20,88,61,181),(21,101,62,194),(22,86,63,179),(23,99,64,192),(24,112,65,177),(25,97,66,190),(26,110,67,175),(27,95,68,188),(28,108,69,173),(29,200,147,138),(30,213,148,123),(31,198,149,136),(32,211,150,121),(33,224,151,134),(34,209,152,119),(35,222,153,132),(36,207,154,117),(37,220,155,130),(38,205,156,115),(39,218,157,128),(40,203,158,113),(41,216,159,126),(42,201,160,139),(43,214,161,124),(44,199,162,137),(45,212,163,122),(46,197,164,135),(47,210,165,120),(48,223,166,133),(49,208,167,118),(50,221,168,131),(51,206,141,116),(52,219,142,129),(53,204,143,114),(54,217,144,127),(55,202,145,140),(56,215,146,125)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
61 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 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C7⋊D4 | C8.C22 | D4⋊2D7 | C28.C23 | D4.8D14 |
| kernel | C14.(C4○D8) | C28.Q8 | C4.Dic14 | C28.55D4 | Q8⋊Dic7 | C23.21D14 | C7×C22⋊Q8 | C2×C28 | C22×C14 | C22⋊Q8 | C28 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C2×C4 | C23 | C14 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 4 | 6 | 6 | 1 | 6 | 6 | 6 |
Matrix representation of C14.(C4○D8) ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 11 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 91 | 101 | 0 | 0 | 0 | 0 |
| 12 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 56 | 50 | 0 | 0 |
| 0 | 0 | 48 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 94 | 2 |
| 0 | 0 | 0 | 0 | 46 | 19 |
| 79 | 25 | 0 | 0 | 0 | 0 |
| 88 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 94 | 1 |
| 40 | 8 | 0 | 0 | 0 | 0 |
| 12 | 73 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 4 | 0 | 0 |
| 0 | 0 | 36 | 104 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,98,11,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[91,12,0,0,0,0,101,22,0,0,0,0,0,0,56,48,0,0,0,0,50,57,0,0,0,0,0,0,94,46,0,0,0,0,2,19],[79,88,0,0,0,0,25,25,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,112,94,0,0,0,0,0,1],[40,12,0,0,0,0,8,73,0,0,0,0,0,0,9,36,0,0,0,0,4,104,0,0,0,0,0,0,98,0,0,0,0,0,0,98] >;
C14.(C4○D8) in GAP, Magma, Sage, TeX
C_{14}.(C_4\circ D_8) % in TeX
G:=Group("C14.(C4oD8)"); // GroupNames label
G:=SmallGroup(448,579);
// by ID
G=gap.SmallGroup(448,579);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,254,219,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^14=b^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^13>;
// generators/relations