metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2.D8⋊7D7, C4⋊C4.53D14, (C2×C8).30D14, D14⋊C8.11C2, C14.30(C4○D8), C4.83(C4○D28), C28.43(C4○D4), C14.Q16⋊22C2, D14⋊2Q8.9C2, C4.Dic14⋊22C2, (C22×D7).30D4, C22.234(D4×D7), C28.44D4⋊27C2, C2.15(D8⋊3D7), (C2×C28).304C23, (C2×C56).244C22, C4.31(Q8⋊2D7), (C2×Dic7).169D4, C7⋊5(C23.20D4), C2.24(Q16⋊D7), C14.72(C8.C22), C4⋊Dic7.127C22, C2.18(D14.5D4), (C2×Dic14).92C22, C14.48(C22.D4), (C7×C2.D8)⋊14C2, C4⋊C4⋊7D7.9C2, (C2×C7⋊C8).73C22, (C2×C4×D7).42C22, (C2×C14).309(C2×D4), (C7×C4⋊C4).97C22, (C2×C4).407(C22×D7), SmallGroup(448,422)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D8⋊7D7
G = < a,b,c,d,e | a2=b8=d7=e2=1, c2=a, ebe=ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, cd=dc, ece=ab4c, ede=d-1 >
Subgroups: 492 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C7⋊C8, C56, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C23.20D4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C4.Dic14, C14.Q16, C28.44D4, D14⋊C8, C7×C2.D8, C4⋊C4⋊7D7, D14⋊2Q8, C2.D8⋊7D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C4○D8, C8.C22, C22×D7, C23.20D4, C4○D28, D4×D7, Q8⋊2D7, D14.5D4, D8⋊3D7, Q16⋊D7, C2.D8⋊7D7
►On 224 points
Generators in S224
(1 38)(2 39)(3 40)(4 33)(5 34)(6 35)(7 36)(8 37)(9 149)(10 150)(11 151)(12 152)(13 145)(14 146)(15 147)(16 148)(17 144)(18 137)(19 138)(20 139)(21 140)(22 141)(23 142)(24 143)(25 125)(26 126)(27 127)(28 128)(29 121)(30 122)(31 123)(32 124)(41 183)(42 184)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 132)(50 133)(51 134)(52 135)(53 136)(54 129)(55 130)(56 131)(57 168)(58 161)(59 162)(60 163)(61 164)(62 165)(63 166)(64 167)(65 203)(66 204)(67 205)(68 206)(69 207)(70 208)(71 201)(72 202)(73 199)(74 200)(75 193)(76 194)(77 195)(78 196)(79 197)(80 198)(81 112)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 113)(96 114)(97 172)(98 173)(99 174)(100 175)(101 176)(102 169)(103 170)(104 171)(153 189)(154 190)(155 191)(156 192)(157 185)(158 186)(159 187)(160 188)(209 221)(210 222)(211 223)(212 224)(213 217)(214 218)(215 219)(216 220)
(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)
(1 192 38 156)(2 191 39 155)(3 190 40 154)(4 189 33 153)(5 188 34 160)(6 187 35 159)(7 186 36 158)(8 185 37 157)(9 183 149 41)(10 182 150 48)(11 181 151 47)(12 180 152 46)(13 179 145 45)(14 178 146 44)(15 177 147 43)(16 184 148 42)(17 175 144 100)(18 174 137 99)(19 173 138 98)(20 172 139 97)(21 171 140 104)(22 170 141 103)(23 169 142 102)(24 176 143 101)(25 67 125 205)(26 66 126 204)(27 65 127 203)(28 72 128 202)(29 71 121 201)(30 70 122 208)(31 69 123 207)(32 68 124 206)(49 60 132 163)(50 59 133 162)(51 58 134 161)(52 57 135 168)(53 64 136 167)(54 63 129 166)(55 62 130 165)(56 61 131 164)(73 119 199 93)(74 118 200 92)(75 117 193 91)(76 116 194 90)(77 115 195 89)(78 114 196 96)(79 113 197 95)(80 120 198 94)(81 223 112 211)(82 222 105 210)(83 221 106 209)(84 220 107 216)(85 219 108 215)(86 218 109 214)(87 217 110 213)(88 224 111 212)
(1 25 43 94 55 174 111)(2 26 44 95 56 175 112)(3 27 45 96 49 176 105)(4 28 46 89 50 169 106)(5 29 47 90 51 170 107)(6 30 48 91 52 171 108)(7 31 41 92 53 172 109)(8 32 42 93 54 173 110)(9 74 64 139 214 186 69)(10 75 57 140 215 187 70)(11 76 58 141 216 188 71)(12 77 59 142 209 189 72)(13 78 60 143 210 190 65)(14 79 61 144 211 191 66)(15 80 62 137 212 192 67)(16 73 63 138 213 185 68)(17 223 155 204 146 197 164)(18 224 156 205 147 198 165)(19 217 157 206 148 199 166)(20 218 158 207 149 200 167)(21 219 159 208 150 193 168)(22 220 160 201 151 194 161)(23 221 153 202 152 195 162)(24 222 154 203 145 196 163)(33 128 180 115 133 102 83)(34 121 181 116 134 103 84)(35 122 182 117 135 104 85)(36 123 183 118 136 97 86)(37 124 184 119 129 98 87)(38 125 177 120 130 99 88)(39 126 178 113 131 100 81)(40 127 179 114 132 101 82)
(1 111)(2 81)(3 105)(4 83)(5 107)(6 85)(7 109)(8 87)(9 163)(10 61)(11 165)(12 63)(13 167)(14 57)(15 161)(16 59)(17 208)(18 71)(19 202)(20 65)(21 204)(22 67)(23 206)(24 69)(25 174)(26 100)(27 176)(28 102)(29 170)(30 104)(31 172)(32 98)(33 106)(34 84)(35 108)(36 86)(37 110)(38 88)(39 112)(40 82)(41 53)(42 129)(43 55)(44 131)(45 49)(46 133)(47 51)(48 135)(50 180)(52 182)(54 184)(56 178)(58 147)(60 149)(62 151)(64 145)(66 140)(68 142)(70 144)(72 138)(73 77)(74 196)(75 79)(76 198)(78 200)(80 194)(89 115)(91 117)(93 119)(95 113)(97 123)(99 125)(101 127)(103 121)(122 171)(124 173)(126 175)(128 169)(130 177)(132 179)(134 181)(136 183)(137 201)(139 203)(141 205)(143 207)(146 168)(148 162)(150 164)(152 166)(153 217)(154 214)(155 219)(156 216)(157 221)(158 210)(159 223)(160 212)(185 209)(186 222)(187 211)(188 224)(189 213)(190 218)(191 215)(192 220)(193 197)(195 199)
G:=sub<Sym(224)| (1,38)(2,39)(3,40)(4,33)(5,34)(6,35)(7,36)(8,37)(9,149)(10,150)(11,151)(12,152)(13,145)(14,146)(15,147)(16,148)(17,144)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,125)(26,126)(27,127)(28,128)(29,121)(30,122)(31,123)(32,124)(41,183)(42,184)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,132)(50,133)(51,134)(52,135)(53,136)(54,129)(55,130)(56,131)(57,168)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,203)(66,204)(67,205)(68,206)(69,207)(70,208)(71,201)(72,202)(73,199)(74,200)(75,193)(76,194)(77,195)(78,196)(79,197)(80,198)(81,112)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114)(97,172)(98,173)(99,174)(100,175)(101,176)(102,169)(103,170)(104,171)(153,189)(154,190)(155,191)(156,192)(157,185)(158,186)(159,187)(160,188)(209,221)(210,222)(211,223)(212,224)(213,217)(214,218)(215,219)(216,220), (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), (1,192,38,156)(2,191,39,155)(3,190,40,154)(4,189,33,153)(5,188,34,160)(6,187,35,159)(7,186,36,158)(8,185,37,157)(9,183,149,41)(10,182,150,48)(11,181,151,47)(12,180,152,46)(13,179,145,45)(14,178,146,44)(15,177,147,43)(16,184,148,42)(17,175,144,100)(18,174,137,99)(19,173,138,98)(20,172,139,97)(21,171,140,104)(22,170,141,103)(23,169,142,102)(24,176,143,101)(25,67,125,205)(26,66,126,204)(27,65,127,203)(28,72,128,202)(29,71,121,201)(30,70,122,208)(31,69,123,207)(32,68,124,206)(49,60,132,163)(50,59,133,162)(51,58,134,161)(52,57,135,168)(53,64,136,167)(54,63,129,166)(55,62,130,165)(56,61,131,164)(73,119,199,93)(74,118,200,92)(75,117,193,91)(76,116,194,90)(77,115,195,89)(78,114,196,96)(79,113,197,95)(80,120,198,94)(81,223,112,211)(82,222,105,210)(83,221,106,209)(84,220,107,216)(85,219,108,215)(86,218,109,214)(87,217,110,213)(88,224,111,212), (1,25,43,94,55,174,111)(2,26,44,95,56,175,112)(3,27,45,96,49,176,105)(4,28,46,89,50,169,106)(5,29,47,90,51,170,107)(6,30,48,91,52,171,108)(7,31,41,92,53,172,109)(8,32,42,93,54,173,110)(9,74,64,139,214,186,69)(10,75,57,140,215,187,70)(11,76,58,141,216,188,71)(12,77,59,142,209,189,72)(13,78,60,143,210,190,65)(14,79,61,144,211,191,66)(15,80,62,137,212,192,67)(16,73,63,138,213,185,68)(17,223,155,204,146,197,164)(18,224,156,205,147,198,165)(19,217,157,206,148,199,166)(20,218,158,207,149,200,167)(21,219,159,208,150,193,168)(22,220,160,201,151,194,161)(23,221,153,202,152,195,162)(24,222,154,203,145,196,163)(33,128,180,115,133,102,83)(34,121,181,116,134,103,84)(35,122,182,117,135,104,85)(36,123,183,118,136,97,86)(37,124,184,119,129,98,87)(38,125,177,120,130,99,88)(39,126,178,113,131,100,81)(40,127,179,114,132,101,82), (1,111)(2,81)(3,105)(4,83)(5,107)(6,85)(7,109)(8,87)(9,163)(10,61)(11,165)(12,63)(13,167)(14,57)(15,161)(16,59)(17,208)(18,71)(19,202)(20,65)(21,204)(22,67)(23,206)(24,69)(25,174)(26,100)(27,176)(28,102)(29,170)(30,104)(31,172)(32,98)(33,106)(34,84)(35,108)(36,86)(37,110)(38,88)(39,112)(40,82)(41,53)(42,129)(43,55)(44,131)(45,49)(46,133)(47,51)(48,135)(50,180)(52,182)(54,184)(56,178)(58,147)(60,149)(62,151)(64,145)(66,140)(68,142)(70,144)(72,138)(73,77)(74,196)(75,79)(76,198)(78,200)(80,194)(89,115)(91,117)(93,119)(95,113)(97,123)(99,125)(101,127)(103,121)(122,171)(124,173)(126,175)(128,169)(130,177)(132,179)(134,181)(136,183)(137,201)(139,203)(141,205)(143,207)(146,168)(148,162)(150,164)(152,166)(153,217)(154,214)(155,219)(156,216)(157,221)(158,210)(159,223)(160,212)(185,209)(186,222)(187,211)(188,224)(189,213)(190,218)(191,215)(192,220)(193,197)(195,199)>;
G:=Group( (1,38)(2,39)(3,40)(4,33)(5,34)(6,35)(7,36)(8,37)(9,149)(10,150)(11,151)(12,152)(13,145)(14,146)(15,147)(16,148)(17,144)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,125)(26,126)(27,127)(28,128)(29,121)(30,122)(31,123)(32,124)(41,183)(42,184)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,132)(50,133)(51,134)(52,135)(53,136)(54,129)(55,130)(56,131)(57,168)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,203)(66,204)(67,205)(68,206)(69,207)(70,208)(71,201)(72,202)(73,199)(74,200)(75,193)(76,194)(77,195)(78,196)(79,197)(80,198)(81,112)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114)(97,172)(98,173)(99,174)(100,175)(101,176)(102,169)(103,170)(104,171)(153,189)(154,190)(155,191)(156,192)(157,185)(158,186)(159,187)(160,188)(209,221)(210,222)(211,223)(212,224)(213,217)(214,218)(215,219)(216,220), (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), (1,192,38,156)(2,191,39,155)(3,190,40,154)(4,189,33,153)(5,188,34,160)(6,187,35,159)(7,186,36,158)(8,185,37,157)(9,183,149,41)(10,182,150,48)(11,181,151,47)(12,180,152,46)(13,179,145,45)(14,178,146,44)(15,177,147,43)(16,184,148,42)(17,175,144,100)(18,174,137,99)(19,173,138,98)(20,172,139,97)(21,171,140,104)(22,170,141,103)(23,169,142,102)(24,176,143,101)(25,67,125,205)(26,66,126,204)(27,65,127,203)(28,72,128,202)(29,71,121,201)(30,70,122,208)(31,69,123,207)(32,68,124,206)(49,60,132,163)(50,59,133,162)(51,58,134,161)(52,57,135,168)(53,64,136,167)(54,63,129,166)(55,62,130,165)(56,61,131,164)(73,119,199,93)(74,118,200,92)(75,117,193,91)(76,116,194,90)(77,115,195,89)(78,114,196,96)(79,113,197,95)(80,120,198,94)(81,223,112,211)(82,222,105,210)(83,221,106,209)(84,220,107,216)(85,219,108,215)(86,218,109,214)(87,217,110,213)(88,224,111,212), (1,25,43,94,55,174,111)(2,26,44,95,56,175,112)(3,27,45,96,49,176,105)(4,28,46,89,50,169,106)(5,29,47,90,51,170,107)(6,30,48,91,52,171,108)(7,31,41,92,53,172,109)(8,32,42,93,54,173,110)(9,74,64,139,214,186,69)(10,75,57,140,215,187,70)(11,76,58,141,216,188,71)(12,77,59,142,209,189,72)(13,78,60,143,210,190,65)(14,79,61,144,211,191,66)(15,80,62,137,212,192,67)(16,73,63,138,213,185,68)(17,223,155,204,146,197,164)(18,224,156,205,147,198,165)(19,217,157,206,148,199,166)(20,218,158,207,149,200,167)(21,219,159,208,150,193,168)(22,220,160,201,151,194,161)(23,221,153,202,152,195,162)(24,222,154,203,145,196,163)(33,128,180,115,133,102,83)(34,121,181,116,134,103,84)(35,122,182,117,135,104,85)(36,123,183,118,136,97,86)(37,124,184,119,129,98,87)(38,125,177,120,130,99,88)(39,126,178,113,131,100,81)(40,127,179,114,132,101,82), (1,111)(2,81)(3,105)(4,83)(5,107)(6,85)(7,109)(8,87)(9,163)(10,61)(11,165)(12,63)(13,167)(14,57)(15,161)(16,59)(17,208)(18,71)(19,202)(20,65)(21,204)(22,67)(23,206)(24,69)(25,174)(26,100)(27,176)(28,102)(29,170)(30,104)(31,172)(32,98)(33,106)(34,84)(35,108)(36,86)(37,110)(38,88)(39,112)(40,82)(41,53)(42,129)(43,55)(44,131)(45,49)(46,133)(47,51)(48,135)(50,180)(52,182)(54,184)(56,178)(58,147)(60,149)(62,151)(64,145)(66,140)(68,142)(70,144)(72,138)(73,77)(74,196)(75,79)(76,198)(78,200)(80,194)(89,115)(91,117)(93,119)(95,113)(97,123)(99,125)(101,127)(103,121)(122,171)(124,173)(126,175)(128,169)(130,177)(132,179)(134,181)(136,183)(137,201)(139,203)(141,205)(143,207)(146,168)(148,162)(150,164)(152,166)(153,217)(154,214)(155,219)(156,216)(157,221)(158,210)(159,223)(160,212)(185,209)(186,222)(187,211)(188,224)(189,213)(190,218)(191,215)(192,220)(193,197)(195,199) );
G=PermutationGroup([[(1,38),(2,39),(3,40),(4,33),(5,34),(6,35),(7,36),(8,37),(9,149),(10,150),(11,151),(12,152),(13,145),(14,146),(15,147),(16,148),(17,144),(18,137),(19,138),(20,139),(21,140),(22,141),(23,142),(24,143),(25,125),(26,126),(27,127),(28,128),(29,121),(30,122),(31,123),(32,124),(41,183),(42,184),(43,177),(44,178),(45,179),(46,180),(47,181),(48,182),(49,132),(50,133),(51,134),(52,135),(53,136),(54,129),(55,130),(56,131),(57,168),(58,161),(59,162),(60,163),(61,164),(62,165),(63,166),(64,167),(65,203),(66,204),(67,205),(68,206),(69,207),(70,208),(71,201),(72,202),(73,199),(74,200),(75,193),(76,194),(77,195),(78,196),(79,197),(80,198),(81,112),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,113),(96,114),(97,172),(98,173),(99,174),(100,175),(101,176),(102,169),(103,170),(104,171),(153,189),(154,190),(155,191),(156,192),(157,185),(158,186),(159,187),(160,188),(209,221),(210,222),(211,223),(212,224),(213,217),(214,218),(215,219),(216,220)], [(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)], [(1,192,38,156),(2,191,39,155),(3,190,40,154),(4,189,33,153),(5,188,34,160),(6,187,35,159),(7,186,36,158),(8,185,37,157),(9,183,149,41),(10,182,150,48),(11,181,151,47),(12,180,152,46),(13,179,145,45),(14,178,146,44),(15,177,147,43),(16,184,148,42),(17,175,144,100),(18,174,137,99),(19,173,138,98),(20,172,139,97),(21,171,140,104),(22,170,141,103),(23,169,142,102),(24,176,143,101),(25,67,125,205),(26,66,126,204),(27,65,127,203),(28,72,128,202),(29,71,121,201),(30,70,122,208),(31,69,123,207),(32,68,124,206),(49,60,132,163),(50,59,133,162),(51,58,134,161),(52,57,135,168),(53,64,136,167),(54,63,129,166),(55,62,130,165),(56,61,131,164),(73,119,199,93),(74,118,200,92),(75,117,193,91),(76,116,194,90),(77,115,195,89),(78,114,196,96),(79,113,197,95),(80,120,198,94),(81,223,112,211),(82,222,105,210),(83,221,106,209),(84,220,107,216),(85,219,108,215),(86,218,109,214),(87,217,110,213),(88,224,111,212)], [(1,25,43,94,55,174,111),(2,26,44,95,56,175,112),(3,27,45,96,49,176,105),(4,28,46,89,50,169,106),(5,29,47,90,51,170,107),(6,30,48,91,52,171,108),(7,31,41,92,53,172,109),(8,32,42,93,54,173,110),(9,74,64,139,214,186,69),(10,75,57,140,215,187,70),(11,76,58,141,216,188,71),(12,77,59,142,209,189,72),(13,78,60,143,210,190,65),(14,79,61,144,211,191,66),(15,80,62,137,212,192,67),(16,73,63,138,213,185,68),(17,223,155,204,146,197,164),(18,224,156,205,147,198,165),(19,217,157,206,148,199,166),(20,218,158,207,149,200,167),(21,219,159,208,150,193,168),(22,220,160,201,151,194,161),(23,221,153,202,152,195,162),(24,222,154,203,145,196,163),(33,128,180,115,133,102,83),(34,121,181,116,134,103,84),(35,122,182,117,135,104,85),(36,123,183,118,136,97,86),(37,124,184,119,129,98,87),(38,125,177,120,130,99,88),(39,126,178,113,131,100,81),(40,127,179,114,132,101,82)], [(1,111),(2,81),(3,105),(4,83),(5,107),(6,85),(7,109),(8,87),(9,163),(10,61),(11,165),(12,63),(13,167),(14,57),(15,161),(16,59),(17,208),(18,71),(19,202),(20,65),(21,204),(22,67),(23,206),(24,69),(25,174),(26,100),(27,176),(28,102),(29,170),(30,104),(31,172),(32,98),(33,106),(34,84),(35,108),(36,86),(37,110),(38,88),(39,112),(40,82),(41,53),(42,129),(43,55),(44,131),(45,49),(46,133),(47,51),(48,135),(50,180),(52,182),(54,184),(56,178),(58,147),(60,149),(62,151),(64,145),(66,140),(68,142),(70,144),(72,138),(73,77),(74,196),(75,79),(76,198),(78,200),(80,194),(89,115),(91,117),(93,119),(95,113),(97,123),(99,125),(101,127),(103,121),(122,171),(124,173),(126,175),(128,169),(130,177),(132,179),(134,181),(136,183),(137,201),(139,203),(141,205),(143,207),(146,168),(148,162),(150,164),(152,166),(153,217),(154,214),(155,219),(156,216),(157,221),(158,210),(159,223),(160,212),(185,209),(186,222),(187,211),(188,224),(189,213),(190,218),(191,215),(192,220),(193,197),(195,199)]])
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 | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| 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 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C4○D8 | C4○D28 | C8.C22 | Q8⋊2D7 | D4×D7 | D8⋊3D7 | Q16⋊D7 |
| kernel | C2.D8⋊7D7 | C4.Dic14 | C14.Q16 | C28.44D4 | D14⋊C8 | C7×C2.D8 | C4⋊C4⋊7D7 | D14⋊2Q8 | C2×Dic7 | C22×D7 | C2.D8 | C28 | C4⋊C4 | C2×C8 | C14 | C4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 6 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C2.D8⋊7D7 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 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 | 0 | 0 | 1 |
| 55 | 7 | 0 | 0 | 0 | 0 |
| 20 | 58 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 40 | 44 |
| 34 | 105 | 0 | 0 | 0 | 0 |
| 74 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 28 |
| 0 | 0 | 0 | 0 | 22 | 96 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 79 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 65 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 23 | 112 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[55,20,0,0,0,0,7,58,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,40,0,0,0,0,0,44],[34,74,0,0,0,0,105,79,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,22,0,0,0,0,28,96],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,1,79,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,65,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,23,0,0,0,0,0,112] >;
C2.D8⋊7D7 in GAP, Magma, Sage, TeX
C_2.D_8\rtimes_7D_7
% in TeX
G:=Group("C2.D8:7D7"); // GroupNames label
G:=SmallGroup(448,422);
// by ID
G=gap.SmallGroup(448,422);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,64,254,219,268,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=d^7=e^2=1,c^2=a,e*b*e=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e=a*b^4*c,e*d*e=d^-1>;
// generators/relations