direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D8⋊3D7, D8⋊12D14, C28.3C24, C56.32C23, Dic28⋊13C22, Dic14.1C23, (C14×D8)⋊8C2, (C2×D8)⋊13D7, C4.41(D4×D7), C14⋊2(C4○D8), (C4×D7).27D4, D14.9(C2×D4), C28.78(C2×D4), C7⋊C8.20C23, C4.3(C23×D7), (C2×C8).245D14, (C8×D7)⋊14C22, (C7×D8)⋊10C22, D4.D7⋊8C22, D4.1(C22×D7), (C7×D4).1C23, C8.38(C22×D7), (C2×Dic28)⋊19C2, (C2×D4).180D14, D4⋊2D7⋊6C22, (C2×C56).97C22, Dic7.68(C2×D4), (C22×D7).61D4, (C4×D7).24C23, C22.137(D4×D7), (C2×C28).520C23, (C2×Dic7).215D4, C14.104(C22×D4), (D4×C14).162C22, (C2×Dic14).195C22, (D7×C2×C8)⋊5C2, C7⋊2(C2×C4○D8), C2.77(C2×D4×D7), (C2×D4.D7)⋊26C2, (C2×D4⋊2D7)⋊24C2, (C2×C14).393(C2×D4), (C2×C7⋊C8).283C22, (C2×C4×D7).256C22, (C2×C4).610(C22×D7), SmallGroup(448,1209)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D8⋊3D7
G = < a,b,c,d,e | a2=b8=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >
Subgroups: 1188 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C14, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C2×C4○D8, C8×D7, Dic28, C2×C7⋊C8, D4.D7, C2×C56, C7×D8, C2×Dic14, C2×C4×D7, D4⋊2D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, D7×C2×C8, C2×Dic28, D8⋊3D7, C2×D4.D7, C14×D8, C2×D4⋊2D7, C2×D8⋊3D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C4○D8, C22×D4, C22×D7, C2×C4○D8, D4×D7, C23×D7, D8⋊3D7, C2×D4×D7, C2×D8⋊3D7
►On 224 points
Generators in S224
(1 196)(2 197)(3 198)(4 199)(5 200)(6 193)(7 194)(8 195)(9 180)(10 181)(11 182)(12 183)(13 184)(14 177)(15 178)(16 179)(17 174)(18 175)(19 176)(20 169)(21 170)(22 171)(23 172)(24 173)(25 221)(26 222)(27 223)(28 224)(29 217)(30 218)(31 219)(32 220)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 113)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 121)(56 122)(57 216)(58 209)(59 210)(60 211)(61 212)(62 213)(63 214)(64 215)(65 202)(66 203)(67 204)(68 205)(69 206)(70 207)(71 208)(72 201)(73 141)(74 142)(75 143)(76 144)(77 137)(78 138)(79 139)(80 140)(81 168)(82 161)(83 162)(84 163)(85 164)(86 165)(87 166)(88 167)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 145)(96 146)(97 190)(98 191)(99 192)(100 185)(101 186)(102 187)(103 188)(104 189)(105 156)(106 157)(107 158)(108 159)(109 160)(110 153)(111 154)(112 155)
(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)
(1 221)(2 220)(3 219)(4 218)(5 217)(6 224)(7 223)(8 222)(9 65)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 57)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 196)(26 195)(27 194)(28 193)(29 200)(30 199)(31 198)(32 197)(33 90)(34 89)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 111)(42 110)(43 109)(44 108)(45 107)(46 106)(47 105)(48 112)(49 81)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(73 99)(74 98)(75 97)(76 104)(77 103)(78 102)(79 101)(80 100)(113 149)(114 148)(115 147)(116 146)(117 145)(118 152)(119 151)(120 150)(121 162)(122 161)(123 168)(124 167)(125 166)(126 165)(127 164)(128 163)(129 154)(130 153)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 188)(138 187)(139 186)(140 185)(141 192)(142 191)(143 190)(144 189)(169 213)(170 212)(171 211)(172 210)(173 209)(174 216)(175 215)(176 214)(177 205)(178 204)(179 203)(180 202)(181 201)(182 208)(183 207)(184 206)
(1 43 40 172 183 163 187)(2 44 33 173 184 164 188)(3 45 34 174 177 165 189)(4 46 35 175 178 166 190)(5 47 36 176 179 167 191)(6 48 37 169 180 168 192)(7 41 38 170 181 161 185)(8 42 39 171 182 162 186)(9 81 99 193 136 118 20)(10 82 100 194 129 119 21)(11 83 101 195 130 120 22)(12 84 102 196 131 113 23)(13 85 103 197 132 114 24)(14 86 104 198 133 115 17)(15 87 97 199 134 116 18)(16 88 98 200 135 117 19)(25 160 149 59 70 54 78)(26 153 150 60 71 55 79)(27 154 151 61 72 56 80)(28 155 152 62 65 49 73)(29 156 145 63 66 50 74)(30 157 146 64 67 51 75)(31 158 147 57 68 52 76)(32 159 148 58 69 53 77)(89 216 205 126 144 219 107)(90 209 206 127 137 220 108)(91 210 207 128 138 221 109)(92 211 208 121 139 222 110)(93 212 201 122 140 223 111)(94 213 202 123 141 224 112)(95 214 203 124 142 217 105)(96 215 204 125 143 218 106)
(1 102)(2 103)(3 104)(4 97)(5 98)(6 99)(7 100)(8 101)(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 174)(18 175)(19 176)(20 169)(21 170)(22 171)(23 172)(24 173)(25 142)(26 143)(27 144)(28 137)(29 138)(30 139)(31 140)(32 141)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 81)(49 108)(50 109)(51 110)(52 111)(53 112)(54 105)(55 106)(56 107)(57 212)(58 213)(59 214)(60 215)(61 216)(62 209)(63 210)(64 211)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 89)(73 220)(74 221)(75 222)(76 223)(77 224)(78 217)(79 218)(80 219)(113 183)(114 184)(115 177)(116 178)(117 179)(118 180)(119 181)(120 182)(121 157)(122 158)(123 159)(124 160)(125 153)(126 154)(127 155)(128 156)(129 161)(130 162)(131 163)(132 164)(133 165)(134 166)(135 167)(136 168)(145 207)(146 208)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(185 194)(186 195)(187 196)(188 197)(189 198)(190 199)(191 200)(192 193)
G:=sub<Sym(224)| (1,196)(2,197)(3,198)(4,199)(5,200)(6,193)(7,194)(8,195)(9,180)(10,181)(11,182)(12,183)(13,184)(14,177)(15,178)(16,179)(17,174)(18,175)(19,176)(20,169)(21,170)(22,171)(23,172)(24,173)(25,221)(26,222)(27,223)(28,224)(29,217)(30,218)(31,219)(32,220)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,113)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,121)(56,122)(57,216)(58,209)(59,210)(60,211)(61,212)(62,213)(63,214)(64,215)(65,202)(66,203)(67,204)(68,205)(69,206)(70,207)(71,208)(72,201)(73,141)(74,142)(75,143)(76,144)(77,137)(78,138)(79,139)(80,140)(81,168)(82,161)(83,162)(84,163)(85,164)(86,165)(87,166)(88,167)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,145)(96,146)(97,190)(98,191)(99,192)(100,185)(101,186)(102,187)(103,188)(104,189)(105,156)(106,157)(107,158)(108,159)(109,160)(110,153)(111,154)(112,155), (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), (1,221)(2,220)(3,219)(4,218)(5,217)(6,224)(7,223)(8,222)(9,65)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,57)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,196)(26,195)(27,194)(28,193)(29,200)(30,199)(31,198)(32,197)(33,90)(34,89)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,112)(49,81)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(73,99)(74,98)(75,97)(76,104)(77,103)(78,102)(79,101)(80,100)(113,149)(114,148)(115,147)(116,146)(117,145)(118,152)(119,151)(120,150)(121,162)(122,161)(123,168)(124,167)(125,166)(126,165)(127,164)(128,163)(129,154)(130,153)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189)(169,213)(170,212)(171,211)(172,210)(173,209)(174,216)(175,215)(176,214)(177,205)(178,204)(179,203)(180,202)(181,201)(182,208)(183,207)(184,206), (1,43,40,172,183,163,187)(2,44,33,173,184,164,188)(3,45,34,174,177,165,189)(4,46,35,175,178,166,190)(5,47,36,176,179,167,191)(6,48,37,169,180,168,192)(7,41,38,170,181,161,185)(8,42,39,171,182,162,186)(9,81,99,193,136,118,20)(10,82,100,194,129,119,21)(11,83,101,195,130,120,22)(12,84,102,196,131,113,23)(13,85,103,197,132,114,24)(14,86,104,198,133,115,17)(15,87,97,199,134,116,18)(16,88,98,200,135,117,19)(25,160,149,59,70,54,78)(26,153,150,60,71,55,79)(27,154,151,61,72,56,80)(28,155,152,62,65,49,73)(29,156,145,63,66,50,74)(30,157,146,64,67,51,75)(31,158,147,57,68,52,76)(32,159,148,58,69,53,77)(89,216,205,126,144,219,107)(90,209,206,127,137,220,108)(91,210,207,128,138,221,109)(92,211,208,121,139,222,110)(93,212,201,122,140,223,111)(94,213,202,123,141,224,112)(95,214,203,124,142,217,105)(96,215,204,125,143,218,106), (1,102)(2,103)(3,104)(4,97)(5,98)(6,99)(7,100)(8,101)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,174)(18,175)(19,176)(20,169)(21,170)(22,171)(23,172)(24,173)(25,142)(26,143)(27,144)(28,137)(29,138)(30,139)(31,140)(32,141)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,81)(49,108)(50,109)(51,110)(52,111)(53,112)(54,105)(55,106)(56,107)(57,212)(58,213)(59,214)(60,215)(61,216)(62,209)(63,210)(64,211)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,89)(73,220)(74,221)(75,222)(76,223)(77,224)(78,217)(79,218)(80,219)(113,183)(114,184)(115,177)(116,178)(117,179)(118,180)(119,181)(120,182)(121,157)(122,158)(123,159)(124,160)(125,153)(126,154)(127,155)(128,156)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(145,207)(146,208)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(185,194)(186,195)(187,196)(188,197)(189,198)(190,199)(191,200)(192,193)>;
G:=Group( (1,196)(2,197)(3,198)(4,199)(5,200)(6,193)(7,194)(8,195)(9,180)(10,181)(11,182)(12,183)(13,184)(14,177)(15,178)(16,179)(17,174)(18,175)(19,176)(20,169)(21,170)(22,171)(23,172)(24,173)(25,221)(26,222)(27,223)(28,224)(29,217)(30,218)(31,219)(32,220)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,113)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,121)(56,122)(57,216)(58,209)(59,210)(60,211)(61,212)(62,213)(63,214)(64,215)(65,202)(66,203)(67,204)(68,205)(69,206)(70,207)(71,208)(72,201)(73,141)(74,142)(75,143)(76,144)(77,137)(78,138)(79,139)(80,140)(81,168)(82,161)(83,162)(84,163)(85,164)(86,165)(87,166)(88,167)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,145)(96,146)(97,190)(98,191)(99,192)(100,185)(101,186)(102,187)(103,188)(104,189)(105,156)(106,157)(107,158)(108,159)(109,160)(110,153)(111,154)(112,155), (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), (1,221)(2,220)(3,219)(4,218)(5,217)(6,224)(7,223)(8,222)(9,65)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,57)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,196)(26,195)(27,194)(28,193)(29,200)(30,199)(31,198)(32,197)(33,90)(34,89)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,112)(49,81)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(73,99)(74,98)(75,97)(76,104)(77,103)(78,102)(79,101)(80,100)(113,149)(114,148)(115,147)(116,146)(117,145)(118,152)(119,151)(120,150)(121,162)(122,161)(123,168)(124,167)(125,166)(126,165)(127,164)(128,163)(129,154)(130,153)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189)(169,213)(170,212)(171,211)(172,210)(173,209)(174,216)(175,215)(176,214)(177,205)(178,204)(179,203)(180,202)(181,201)(182,208)(183,207)(184,206), (1,43,40,172,183,163,187)(2,44,33,173,184,164,188)(3,45,34,174,177,165,189)(4,46,35,175,178,166,190)(5,47,36,176,179,167,191)(6,48,37,169,180,168,192)(7,41,38,170,181,161,185)(8,42,39,171,182,162,186)(9,81,99,193,136,118,20)(10,82,100,194,129,119,21)(11,83,101,195,130,120,22)(12,84,102,196,131,113,23)(13,85,103,197,132,114,24)(14,86,104,198,133,115,17)(15,87,97,199,134,116,18)(16,88,98,200,135,117,19)(25,160,149,59,70,54,78)(26,153,150,60,71,55,79)(27,154,151,61,72,56,80)(28,155,152,62,65,49,73)(29,156,145,63,66,50,74)(30,157,146,64,67,51,75)(31,158,147,57,68,52,76)(32,159,148,58,69,53,77)(89,216,205,126,144,219,107)(90,209,206,127,137,220,108)(91,210,207,128,138,221,109)(92,211,208,121,139,222,110)(93,212,201,122,140,223,111)(94,213,202,123,141,224,112)(95,214,203,124,142,217,105)(96,215,204,125,143,218,106), (1,102)(2,103)(3,104)(4,97)(5,98)(6,99)(7,100)(8,101)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,174)(18,175)(19,176)(20,169)(21,170)(22,171)(23,172)(24,173)(25,142)(26,143)(27,144)(28,137)(29,138)(30,139)(31,140)(32,141)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,81)(49,108)(50,109)(51,110)(52,111)(53,112)(54,105)(55,106)(56,107)(57,212)(58,213)(59,214)(60,215)(61,216)(62,209)(63,210)(64,211)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,89)(73,220)(74,221)(75,222)(76,223)(77,224)(78,217)(79,218)(80,219)(113,183)(114,184)(115,177)(116,178)(117,179)(118,180)(119,181)(120,182)(121,157)(122,158)(123,159)(124,160)(125,153)(126,154)(127,155)(128,156)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(145,207)(146,208)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(185,194)(186,195)(187,196)(188,197)(189,198)(190,199)(191,200)(192,193) );
G=PermutationGroup([[(1,196),(2,197),(3,198),(4,199),(5,200),(6,193),(7,194),(8,195),(9,180),(10,181),(11,182),(12,183),(13,184),(14,177),(15,178),(16,179),(17,174),(18,175),(19,176),(20,169),(21,170),(22,171),(23,172),(24,173),(25,221),(26,222),(27,223),(28,224),(29,217),(30,218),(31,219),(32,220),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,113),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,121),(56,122),(57,216),(58,209),(59,210),(60,211),(61,212),(62,213),(63,214),(64,215),(65,202),(66,203),(67,204),(68,205),(69,206),(70,207),(71,208),(72,201),(73,141),(74,142),(75,143),(76,144),(77,137),(78,138),(79,139),(80,140),(81,168),(82,161),(83,162),(84,163),(85,164),(86,165),(87,166),(88,167),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,145),(96,146),(97,190),(98,191),(99,192),(100,185),(101,186),(102,187),(103,188),(104,189),(105,156),(106,157),(107,158),(108,159),(109,160),(110,153),(111,154),(112,155)], [(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)], [(1,221),(2,220),(3,219),(4,218),(5,217),(6,224),(7,223),(8,222),(9,65),(10,72),(11,71),(12,70),(13,69),(14,68),(15,67),(16,66),(17,57),(18,64),(19,63),(20,62),(21,61),(22,60),(23,59),(24,58),(25,196),(26,195),(27,194),(28,193),(29,200),(30,199),(31,198),(32,197),(33,90),(34,89),(35,96),(36,95),(37,94),(38,93),(39,92),(40,91),(41,111),(42,110),(43,109),(44,108),(45,107),(46,106),(47,105),(48,112),(49,81),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(73,99),(74,98),(75,97),(76,104),(77,103),(78,102),(79,101),(80,100),(113,149),(114,148),(115,147),(116,146),(117,145),(118,152),(119,151),(120,150),(121,162),(122,161),(123,168),(124,167),(125,166),(126,165),(127,164),(128,163),(129,154),(130,153),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,188),(138,187),(139,186),(140,185),(141,192),(142,191),(143,190),(144,189),(169,213),(170,212),(171,211),(172,210),(173,209),(174,216),(175,215),(176,214),(177,205),(178,204),(179,203),(180,202),(181,201),(182,208),(183,207),(184,206)], [(1,43,40,172,183,163,187),(2,44,33,173,184,164,188),(3,45,34,174,177,165,189),(4,46,35,175,178,166,190),(5,47,36,176,179,167,191),(6,48,37,169,180,168,192),(7,41,38,170,181,161,185),(8,42,39,171,182,162,186),(9,81,99,193,136,118,20),(10,82,100,194,129,119,21),(11,83,101,195,130,120,22),(12,84,102,196,131,113,23),(13,85,103,197,132,114,24),(14,86,104,198,133,115,17),(15,87,97,199,134,116,18),(16,88,98,200,135,117,19),(25,160,149,59,70,54,78),(26,153,150,60,71,55,79),(27,154,151,61,72,56,80),(28,155,152,62,65,49,73),(29,156,145,63,66,50,74),(30,157,146,64,67,51,75),(31,158,147,57,68,52,76),(32,159,148,58,69,53,77),(89,216,205,126,144,219,107),(90,209,206,127,137,220,108),(91,210,207,128,138,221,109),(92,211,208,121,139,222,110),(93,212,201,122,140,223,111),(94,213,202,123,141,224,112),(95,214,203,124,142,217,105),(96,215,204,125,143,218,106)], [(1,102),(2,103),(3,104),(4,97),(5,98),(6,99),(7,100),(8,101),(9,37),(10,38),(11,39),(12,40),(13,33),(14,34),(15,35),(16,36),(17,174),(18,175),(19,176),(20,169),(21,170),(22,171),(23,172),(24,173),(25,142),(26,143),(27,144),(28,137),(29,138),(30,139),(31,140),(32,141),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,81),(49,108),(50,109),(51,110),(52,111),(53,112),(54,105),(55,106),(56,107),(57,212),(58,213),(59,214),(60,215),(61,216),(62,209),(63,210),(64,211),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,89),(73,220),(74,221),(75,222),(76,223),(77,224),(78,217),(79,218),(80,219),(113,183),(114,184),(115,177),(116,178),(117,179),(118,180),(119,181),(120,182),(121,157),(122,158),(123,159),(124,160),(125,153),(126,154),(127,155),(128,156),(129,161),(130,162),(131,163),(132,164),(133,165),(134,166),(135,167),(136,168),(145,207),(146,208),(147,201),(148,202),(149,203),(150,204),(151,205),(152,206),(185,194),(186,195),(187,196),(188,197),(189,198),(190,199),(191,200),(192,193)]])
70 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 | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| 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 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 14 | 14 | 2 | 2 | 7 | 7 | 7 | 7 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | D4×D7 | D4×D7 | D8⋊3D7 |
| kernel | C2×D8⋊3D7 | D7×C2×C8 | C2×Dic28 | D8⋊3D7 | C2×D4.D7 | C14×D8 | C2×D4⋊2D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×D8 | C2×C8 | D8 | C2×D4 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 12 | 6 | 8 | 3 | 3 | 12 |
Matrix representation of C2×D8⋊3D7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 32 | 44 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 111 | 69 |
| 0 | 0 | 54 | 2 |
| 9 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 104 | 1 | 0 | 0 |
| 33 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 41 | 112 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,18,32,0,0,0,44],[1,0,0,0,0,1,0,0,0,0,111,54,0,0,69,2],[9,1,0,0,112,0,0,0,0,0,1,0,0,0,0,1],[104,33,0,0,1,9,0,0,0,0,1,41,0,0,0,112] >;
C2×D8⋊3D7 in GAP, Magma, Sage, TeX
C_2\times D_8\rtimes_3D_7
% in TeX
G:=Group("C2xD8:3D7"); // GroupNames label
G:=SmallGroup(448,1209);
// by ID
G=gap.SmallGroup(448,1209);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,1123,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations