direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C2.D8, D14.11D8, D14.4Q16, C56⋊5(C2×C4), (C8×D7)⋊1C4, C8⋊13(C4×D7), C2.4(D7×D8), (C4×D7).5Q8, C4.29(Q8×D7), C2.4(D7×Q16), C56⋊1C4⋊20C2, C14.28(C2×D8), C28.20(C2×Q8), C4⋊C4.169D14, (C2×C8).229D14, C14.23(C2×Q16), C22.89(D4×D7), D14.11(C4⋊C4), C28.Q8⋊19C2, Dic7.6(C4⋊C4), C28.48(C22×C4), (C2×C56).81C22, (C2×C28).295C23, (C2×Dic7).102D4, (C22×D7).107D4, C4⋊Dic7.121C22, C7⋊1(C2×C2.D8), C7⋊C8⋊22(C2×C4), (D7×C2×C8).2C2, C4.79(C2×C4×D7), (D7×C4⋊C4).7C2, C2.14(D7×C4⋊C4), (C7×C2.D8)⋊3C2, C14.13(C2×C4⋊C4), (C4×D7).27(C2×C4), (C2×C14).300(C2×D4), (C7×C4⋊C4).88C22, (C2×C7⋊C8).233C22, (C2×C4×D7).234C22, (C2×C4).398(C22×D7), SmallGroup(448,413)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C2.D8
G = < a,b,c,d,e | a7=b2=c2=d8=1, e2=c, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 620 in 130 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic7, Dic7, C28, C28, D14, C2×C14, C2.D8, C2.D8, C2×C4⋊C4, C22×C8, C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C2×C2.D8, C8×D7, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×C4×D7, C28.Q8, C56⋊1C4, C7×C2.D8, D7×C4⋊C4, D7×C2×C8, D7×C2.D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, D14, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C4×D7, C22×D7, C2×C2.D8, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4, D7×D8, D7×Q16, D7×C2.D8
►On 224 points
Generators in S224
(1 138 45 27 36 199 83)(2 139 46 28 37 200 84)(3 140 47 29 38 193 85)(4 141 48 30 39 194 86)(5 142 41 31 40 195 87)(6 143 42 32 33 196 88)(7 144 43 25 34 197 81)(8 137 44 26 35 198 82)(9 222 108 168 169 74 190)(10 223 109 161 170 75 191)(11 224 110 162 171 76 192)(12 217 111 163 172 77 185)(13 218 112 164 173 78 186)(14 219 105 165 174 79 187)(15 220 106 166 175 80 188)(16 221 107 167 176 73 189)(17 113 124 64 52 184 133)(18 114 125 57 53 177 134)(19 115 126 58 54 178 135)(20 116 127 59 55 179 136)(21 117 128 60 56 180 129)(22 118 121 61 49 181 130)(23 119 122 62 50 182 131)(24 120 123 63 51 183 132)(65 202 94 215 99 148 155)(66 203 95 216 100 149 156)(67 204 96 209 101 150 157)(68 205 89 210 102 151 158)(69 206 90 211 103 152 159)(70 207 91 212 104 145 160)(71 208 92 213 97 146 153)(72 201 93 214 98 147 154)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 81)(8 82)(9 168)(10 161)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 124)(18 125)(19 126)(20 127)(21 128)(22 121)(23 122)(24 123)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)(49 181)(50 182)(51 183)(52 184)(53 177)(54 178)(55 179)(56 180)(57 134)(58 135)(59 136)(60 129)(61 130)(62 131)(63 132)(64 133)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 153)(72 154)(89 102)(90 103)(91 104)(92 97)(93 98)(94 99)(95 100)(96 101)(105 219)(106 220)(107 221)(108 222)(109 223)(110 224)(111 217)(112 218)(137 198)(138 199)(139 200)(140 193)(141 194)(142 195)(143 196)(144 197)(145 207)(146 208)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(169 190)(170 191)(171 192)(172 185)(173 186)(174 187)(175 188)(176 189)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 174)(18 175)(19 176)(20 169)(21 170)(22 171)(23 172)(24 173)(25 213)(26 214)(27 215)(28 216)(29 209)(30 210)(31 211)(32 212)(33 104)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 89)(49 224)(50 217)(51 218)(52 219)(53 220)(54 221)(55 222)(56 223)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 113)(80 114)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(105 184)(106 177)(107 178)(108 179)(109 180)(110 181)(111 182)(112 183)(121 192)(122 185)(123 186)(124 187)(125 188)(126 189)(127 190)(128 191)(129 161)(130 162)(131 163)(132 164)(133 165)(134 166)(135 167)(136 168)(137 201)(138 202)(139 203)(140 204)(141 205)(142 206)(143 207)(144 208)(145 196)(146 197)(147 198)(148 199)(149 200)(150 193)(151 194)(152 195)
(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)
(1 179 65 108)(2 178 66 107)(3 177 67 106)(4 184 68 105)(5 183 69 112)(6 182 70 111)(7 181 71 110)(8 180 72 109)(9 199 59 148)(10 198 60 147)(11 197 61 146)(12 196 62 145)(13 195 63 152)(14 194 64 151)(15 193 57 150)(16 200 58 149)(17 89 174 48)(18 96 175 47)(19 95 176 46)(20 94 169 45)(21 93 170 44)(22 92 171 43)(23 91 172 42)(24 90 173 41)(25 118 213 76)(26 117 214 75)(27 116 215 74)(28 115 216 73)(29 114 209 80)(30 113 210 79)(31 120 211 78)(32 119 212 77)(33 122 104 185)(34 121 97 192)(35 128 98 191)(36 127 99 190)(37 126 100 189)(38 125 101 188)(39 124 102 187)(40 123 103 186)(49 153 224 81)(50 160 217 88)(51 159 218 87)(52 158 219 86)(53 157 220 85)(54 156 221 84)(55 155 222 83)(56 154 223 82)(129 201 161 137)(130 208 162 144)(131 207 163 143)(132 206 164 142)(133 205 165 141)(134 204 166 140)(135 203 167 139)(136 202 168 138)
G:=sub<Sym(224)| (1,138,45,27,36,199,83)(2,139,46,28,37,200,84)(3,140,47,29,38,193,85)(4,141,48,30,39,194,86)(5,142,41,31,40,195,87)(6,143,42,32,33,196,88)(7,144,43,25,34,197,81)(8,137,44,26,35,198,82)(9,222,108,168,169,74,190)(10,223,109,161,170,75,191)(11,224,110,162,171,76,192)(12,217,111,163,172,77,185)(13,218,112,164,173,78,186)(14,219,105,165,174,79,187)(15,220,106,166,175,80,188)(16,221,107,167,176,73,189)(17,113,124,64,52,184,133)(18,114,125,57,53,177,134)(19,115,126,58,54,178,135)(20,116,127,59,55,179,136)(21,117,128,60,56,180,129)(22,118,121,61,49,181,130)(23,119,122,62,50,182,131)(24,120,123,63,51,183,132)(65,202,94,215,99,148,155)(66,203,95,216,100,149,156)(67,204,96,209,101,150,157)(68,205,89,210,102,151,158)(69,206,90,211,103,152,159)(70,207,91,212,104,145,160)(71,208,92,213,97,146,153)(72,201,93,214,98,147,154), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,81)(8,82)(9,168)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,124)(18,125)(19,126)(20,127)(21,128)(22,121)(23,122)(24,123)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41)(49,181)(50,182)(51,183)(52,184)(53,177)(54,178)(55,179)(56,180)(57,134)(58,135)(59,136)(60,129)(61,130)(62,131)(63,132)(64,133)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,153)(72,154)(89,102)(90,103)(91,104)(92,97)(93,98)(94,99)(95,100)(96,101)(105,219)(106,220)(107,221)(108,222)(109,223)(110,224)(111,217)(112,218)(137,198)(138,199)(139,200)(140,193)(141,194)(142,195)(143,196)(144,197)(145,207)(146,208)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(169,190)(170,191)(171,192)(172,185)(173,186)(174,187)(175,188)(176,189), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,57)(16,58)(17,174)(18,175)(19,176)(20,169)(21,170)(22,171)(23,172)(24,173)(25,213)(26,214)(27,215)(28,216)(29,209)(30,210)(31,211)(32,212)(33,104)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,89)(49,224)(50,217)(51,218)(52,219)(53,220)(54,221)(55,222)(56,223)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,113)(80,114)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(105,184)(106,177)(107,178)(108,179)(109,180)(110,181)(111,182)(112,183)(121,192)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)(144,208)(145,196)(146,197)(147,198)(148,199)(149,200)(150,193)(151,194)(152,195), (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), (1,179,65,108)(2,178,66,107)(3,177,67,106)(4,184,68,105)(5,183,69,112)(6,182,70,111)(7,181,71,110)(8,180,72,109)(9,199,59,148)(10,198,60,147)(11,197,61,146)(12,196,62,145)(13,195,63,152)(14,194,64,151)(15,193,57,150)(16,200,58,149)(17,89,174,48)(18,96,175,47)(19,95,176,46)(20,94,169,45)(21,93,170,44)(22,92,171,43)(23,91,172,42)(24,90,173,41)(25,118,213,76)(26,117,214,75)(27,116,215,74)(28,115,216,73)(29,114,209,80)(30,113,210,79)(31,120,211,78)(32,119,212,77)(33,122,104,185)(34,121,97,192)(35,128,98,191)(36,127,99,190)(37,126,100,189)(38,125,101,188)(39,124,102,187)(40,123,103,186)(49,153,224,81)(50,160,217,88)(51,159,218,87)(52,158,219,86)(53,157,220,85)(54,156,221,84)(55,155,222,83)(56,154,223,82)(129,201,161,137)(130,208,162,144)(131,207,163,143)(132,206,164,142)(133,205,165,141)(134,204,166,140)(135,203,167,139)(136,202,168,138)>;
G:=Group( (1,138,45,27,36,199,83)(2,139,46,28,37,200,84)(3,140,47,29,38,193,85)(4,141,48,30,39,194,86)(5,142,41,31,40,195,87)(6,143,42,32,33,196,88)(7,144,43,25,34,197,81)(8,137,44,26,35,198,82)(9,222,108,168,169,74,190)(10,223,109,161,170,75,191)(11,224,110,162,171,76,192)(12,217,111,163,172,77,185)(13,218,112,164,173,78,186)(14,219,105,165,174,79,187)(15,220,106,166,175,80,188)(16,221,107,167,176,73,189)(17,113,124,64,52,184,133)(18,114,125,57,53,177,134)(19,115,126,58,54,178,135)(20,116,127,59,55,179,136)(21,117,128,60,56,180,129)(22,118,121,61,49,181,130)(23,119,122,62,50,182,131)(24,120,123,63,51,183,132)(65,202,94,215,99,148,155)(66,203,95,216,100,149,156)(67,204,96,209,101,150,157)(68,205,89,210,102,151,158)(69,206,90,211,103,152,159)(70,207,91,212,104,145,160)(71,208,92,213,97,146,153)(72,201,93,214,98,147,154), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,81)(8,82)(9,168)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,124)(18,125)(19,126)(20,127)(21,128)(22,121)(23,122)(24,123)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41)(49,181)(50,182)(51,183)(52,184)(53,177)(54,178)(55,179)(56,180)(57,134)(58,135)(59,136)(60,129)(61,130)(62,131)(63,132)(64,133)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,153)(72,154)(89,102)(90,103)(91,104)(92,97)(93,98)(94,99)(95,100)(96,101)(105,219)(106,220)(107,221)(108,222)(109,223)(110,224)(111,217)(112,218)(137,198)(138,199)(139,200)(140,193)(141,194)(142,195)(143,196)(144,197)(145,207)(146,208)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(169,190)(170,191)(171,192)(172,185)(173,186)(174,187)(175,188)(176,189), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,57)(16,58)(17,174)(18,175)(19,176)(20,169)(21,170)(22,171)(23,172)(24,173)(25,213)(26,214)(27,215)(28,216)(29,209)(30,210)(31,211)(32,212)(33,104)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,89)(49,224)(50,217)(51,218)(52,219)(53,220)(54,221)(55,222)(56,223)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,113)(80,114)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(105,184)(106,177)(107,178)(108,179)(109,180)(110,181)(111,182)(112,183)(121,192)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)(144,208)(145,196)(146,197)(147,198)(148,199)(149,200)(150,193)(151,194)(152,195), (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), (1,179,65,108)(2,178,66,107)(3,177,67,106)(4,184,68,105)(5,183,69,112)(6,182,70,111)(7,181,71,110)(8,180,72,109)(9,199,59,148)(10,198,60,147)(11,197,61,146)(12,196,62,145)(13,195,63,152)(14,194,64,151)(15,193,57,150)(16,200,58,149)(17,89,174,48)(18,96,175,47)(19,95,176,46)(20,94,169,45)(21,93,170,44)(22,92,171,43)(23,91,172,42)(24,90,173,41)(25,118,213,76)(26,117,214,75)(27,116,215,74)(28,115,216,73)(29,114,209,80)(30,113,210,79)(31,120,211,78)(32,119,212,77)(33,122,104,185)(34,121,97,192)(35,128,98,191)(36,127,99,190)(37,126,100,189)(38,125,101,188)(39,124,102,187)(40,123,103,186)(49,153,224,81)(50,160,217,88)(51,159,218,87)(52,158,219,86)(53,157,220,85)(54,156,221,84)(55,155,222,83)(56,154,223,82)(129,201,161,137)(130,208,162,144)(131,207,163,143)(132,206,164,142)(133,205,165,141)(134,204,166,140)(135,203,167,139)(136,202,168,138) );
G=PermutationGroup([[(1,138,45,27,36,199,83),(2,139,46,28,37,200,84),(3,140,47,29,38,193,85),(4,141,48,30,39,194,86),(5,142,41,31,40,195,87),(6,143,42,32,33,196,88),(7,144,43,25,34,197,81),(8,137,44,26,35,198,82),(9,222,108,168,169,74,190),(10,223,109,161,170,75,191),(11,224,110,162,171,76,192),(12,217,111,163,172,77,185),(13,218,112,164,173,78,186),(14,219,105,165,174,79,187),(15,220,106,166,175,80,188),(16,221,107,167,176,73,189),(17,113,124,64,52,184,133),(18,114,125,57,53,177,134),(19,115,126,58,54,178,135),(20,116,127,59,55,179,136),(21,117,128,60,56,180,129),(22,118,121,61,49,181,130),(23,119,122,62,50,182,131),(24,120,123,63,51,183,132),(65,202,94,215,99,148,155),(66,203,95,216,100,149,156),(67,204,96,209,101,150,157),(68,205,89,210,102,151,158),(69,206,90,211,103,152,159),(70,207,91,212,104,145,160),(71,208,92,213,97,146,153),(72,201,93,214,98,147,154)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,81),(8,82),(9,168),(10,161),(11,162),(12,163),(13,164),(14,165),(15,166),(16,167),(17,124),(18,125),(19,126),(20,127),(21,128),(22,121),(23,122),(24,123),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41),(49,181),(50,182),(51,183),(52,184),(53,177),(54,178),(55,179),(56,180),(57,134),(58,135),(59,136),(60,129),(61,130),(62,131),(63,132),(64,133),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(71,153),(72,154),(89,102),(90,103),(91,104),(92,97),(93,98),(94,99),(95,100),(96,101),(105,219),(106,220),(107,221),(108,222),(109,223),(110,224),(111,217),(112,218),(137,198),(138,199),(139,200),(140,193),(141,194),(142,195),(143,196),(144,197),(145,207),(146,208),(147,201),(148,202),(149,203),(150,204),(151,205),(152,206),(169,190),(170,191),(171,192),(172,185),(173,186),(174,187),(175,188),(176,189)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,57),(16,58),(17,174),(18,175),(19,176),(20,169),(21,170),(22,171),(23,172),(24,173),(25,213),(26,214),(27,215),(28,216),(29,209),(30,210),(31,211),(32,212),(33,104),(34,97),(35,98),(36,99),(37,100),(38,101),(39,102),(40,103),(41,90),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,89),(49,224),(50,217),(51,218),(52,219),(53,220),(54,221),(55,222),(56,223),(73,115),(74,116),(75,117),(76,118),(77,119),(78,120),(79,113),(80,114),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(105,184),(106,177),(107,178),(108,179),(109,180),(110,181),(111,182),(112,183),(121,192),(122,185),(123,186),(124,187),(125,188),(126,189),(127,190),(128,191),(129,161),(130,162),(131,163),(132,164),(133,165),(134,166),(135,167),(136,168),(137,201),(138,202),(139,203),(140,204),(141,205),(142,206),(143,207),(144,208),(145,196),(146,197),(147,198),(148,199),(149,200),(150,193),(151,194),(152,195)], [(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)], [(1,179,65,108),(2,178,66,107),(3,177,67,106),(4,184,68,105),(5,183,69,112),(6,182,70,111),(7,181,71,110),(8,180,72,109),(9,199,59,148),(10,198,60,147),(11,197,61,146),(12,196,62,145),(13,195,63,152),(14,194,64,151),(15,193,57,150),(16,200,58,149),(17,89,174,48),(18,96,175,47),(19,95,176,46),(20,94,169,45),(21,93,170,44),(22,92,171,43),(23,91,172,42),(24,90,173,41),(25,118,213,76),(26,117,214,75),(27,116,215,74),(28,115,216,73),(29,114,209,80),(30,113,210,79),(31,120,211,78),(32,119,212,77),(33,122,104,185),(34,121,97,192),(35,128,98,191),(36,127,99,190),(37,126,100,189),(38,125,101,188),(39,124,102,187),(40,123,103,186),(49,153,224,81),(50,160,217,88),(51,159,218,87),(52,158,219,86),(53,157,220,85),(54,156,221,84),(55,155,222,83),(56,154,223,82),(129,201,161,137),(130,208,162,144),(131,207,163,143),(132,206,164,142),(133,205,165,141),(134,204,166,140),(135,203,167,139),(136,202,168,138)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D4 | D7 | D8 | Q16 | D14 | D14 | C4×D7 | Q8×D7 | D4×D7 | D7×D8 | D7×Q16 |
| kernel | D7×C2.D8 | C28.Q8 | C56⋊1C4 | C7×C2.D8 | D7×C4⋊C4 | D7×C2×C8 | C8×D7 | C4×D7 | C2×Dic7 | C22×D7 | C2.D8 | D14 | D14 | C4⋊C4 | C2×C8 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 3 | 4 | 4 | 6 | 3 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of D7×C2.D8 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 79 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 82 |
| 0 | 0 | 31 | 31 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 105 | 29 |
| 0 | 0 | 29 | 8 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,31,31,0,0,82,31],[15,0,0,0,0,15,0,0,0,0,105,29,0,0,29,8] >;
D7×C2.D8 in GAP, Magma, Sage, TeX
D_7\times C_2.D_8
% in TeX
G:=Group("D7xC2.D8"); // GroupNames label
G:=SmallGroup(448,413);
// by ID
G=gap.SmallGroup(448,413);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^8=1,e^2=c,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations