metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D8).3D7, (C7×D4).6D4, (D4×Dic7)⋊6C2, (C14×D8).8C2, (C2×C8).34D14, Dic7⋊C8⋊28C2, (C2×D4).61D14, C28.163(C2×D4), D4.1(C7⋊D4), C7⋊6(D4.2D4), C14.32(C4○D8), C28.92(C4○D4), D4⋊Dic7⋊27C2, C28.17D4⋊3C2, C4.8(D4⋊2D7), (C2×Dic7).63D4, C22.253(D4×D7), C2.27(D8⋊D7), C2.16(D8⋊3D7), C28.44D4⋊28C2, C14.48(C8⋊C22), (C2×C28).430C23, (C2×C56).248C22, (D4×C14).80C22, C14.108(C4⋊D4), C4⋊Dic7.164C22, (C4×Dic7).44C22, C2.23(Dic7⋊D4), (C2×Dic14).119C22, C4.35(C2×C7⋊D4), (C2×D4.D7)⋊16C2, (C2×C14).343(C2×D4), (C2×C7⋊C8).147C22, (C2×C4).520(C22×D7), SmallGroup(448,687)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D8).D7
G = < a,b,c,d,e | a2=b8=c2=d7=1, e2=ab4, ebe-1=ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 580 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C7×D4, C22×C14, D4.2D4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D4.D7, C23.D7, C2×C56, C7×D8, C2×Dic14, C22×Dic7, D4×C14, Dic7⋊C8, C28.44D4, D4⋊Dic7, C2×D4.D7, D4×Dic7, C28.17D4, C14×D8, (C2×D8).D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C4○D8, C8⋊C22, C7⋊D4, C22×D7, D4.2D4, D4×D7, D4⋊2D7, C2×C7⋊D4, D8⋊D7, D8⋊3D7, Dic7⋊D4, (C2×D8).D7
►On 224 points
Generators in S224
(1 101)(2 102)(3 103)(4 104)(5 97)(6 98)(7 99)(8 100)(9 110)(10 111)(11 112)(12 105)(13 106)(14 107)(15 108)(16 109)(17 173)(18 174)(19 175)(20 176)(21 169)(22 170)(23 171)(24 172)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 153)(32 154)(33 198)(34 199)(35 200)(36 193)(37 194)(38 195)(39 196)(40 197)(41 165)(42 166)(43 167)(44 168)(45 161)(46 162)(47 163)(48 164)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 177)(56 178)(57 186)(58 187)(59 188)(60 189)(61 190)(62 191)(63 192)(64 185)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(73 133)(74 134)(75 135)(76 136)(77 129)(78 130)(79 131)(80 132)(81 148)(82 149)(83 150)(84 151)(85 152)(86 145)(87 146)(88 147)(89 124)(90 125)(91 126)(92 127)(93 128)(94 121)(95 122)(96 123)(113 203)(114 204)(115 205)(116 206)(117 207)(118 208)(119 201)(120 202)(209 224)(210 217)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)
(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 97)(2 104)(3 103)(4 102)(5 101)(6 100)(7 99)(8 98)(9 108)(10 107)(11 106)(12 105)(13 112)(14 111)(15 110)(16 109)(17 171)(18 170)(19 169)(20 176)(21 175)(22 174)(23 173)(24 172)(25 153)(26 160)(27 159)(28 158)(29 157)(30 156)(31 155)(32 154)(33 194)(34 193)(35 200)(36 199)(37 198)(38 197)(39 196)(40 195)(41 165)(42 164)(43 163)(44 162)(45 161)(46 168)(47 167)(48 166)(49 179)(50 178)(51 177)(52 184)(53 183)(54 182)(55 181)(56 180)(57 192)(58 191)(59 190)(60 189)(61 188)(62 187)(63 186)(64 185)(65 141)(66 140)(67 139)(68 138)(69 137)(70 144)(71 143)(72 142)(73 135)(74 134)(75 133)(76 132)(77 131)(78 130)(79 129)(80 136)(81 146)(82 145)(83 152)(84 151)(85 150)(86 149)(87 148)(88 147)(89 124)(90 123)(91 122)(92 121)(93 128)(94 127)(95 126)(96 125)(113 201)(114 208)(115 207)(116 206)(117 205)(118 204)(119 203)(120 202)(209 218)(210 217)(211 224)(212 223)(213 222)(214 221)(215 220)(216 219)
(1 80 22 177 167 65 14)(2 73 23 178 168 66 15)(3 74 24 179 161 67 16)(4 75 17 180 162 68 9)(5 76 18 181 163 69 10)(6 77 19 182 164 70 11)(7 78 20 183 165 71 12)(8 79 21 184 166 72 13)(25 197 186 113 211 125 85)(26 198 187 114 212 126 86)(27 199 188 115 213 127 87)(28 200 189 116 214 128 88)(29 193 190 117 215 121 81)(30 194 191 118 216 122 82)(31 195 192 119 209 123 83)(32 196 185 120 210 124 84)(33 58 204 219 91 145 156)(34 59 205 220 92 146 157)(35 60 206 221 93 147 158)(36 61 207 222 94 148 159)(37 62 208 223 95 149 160)(38 63 201 224 96 150 153)(39 64 202 217 89 151 154)(40 57 203 218 90 152 155)(41 143 105 99 130 176 53)(42 144 106 100 131 169 54)(43 137 107 101 132 170 55)(44 138 108 102 133 171 56)(45 139 109 103 134 172 49)(46 140 110 104 135 173 50)(47 141 111 97 136 174 51)(48 142 112 98 129 175 52)
(1 191 97 58)(2 63 98 188)(3 185 99 60)(4 57 100 190)(5 187 101 62)(6 59 102 192)(7 189 103 64)(8 61 104 186)(9 203 106 117)(10 114 107 208)(11 205 108 119)(12 116 109 202)(13 207 110 113)(14 118 111 204)(15 201 112 115)(16 120 105 206)(17 155 169 29)(18 26 170 160)(19 157 171 31)(20 28 172 154)(21 159 173 25)(22 30 174 156)(23 153 175 27)(24 32 176 158)(33 80 194 136)(34 133 195 77)(35 74 196 130)(36 135 197 79)(37 76 198 132)(38 129 199 73)(39 78 200 134)(40 131 193 75)(41 93 161 124)(42 121 162 90)(43 95 163 126)(44 123 164 92)(45 89 165 128)(46 125 166 94)(47 91 167 122)(48 127 168 96)(49 151 183 88)(50 85 184 148)(51 145 177 82)(52 87 178 150)(53 147 179 84)(54 81 180 152)(55 149 181 86)(56 83 182 146)(65 216 141 219)(66 224 142 213)(67 210 143 221)(68 218 144 215)(69 212 137 223)(70 220 138 209)(71 214 139 217)(72 222 140 211)
G:=sub<Sym(224)| (1,101)(2,102)(3,103)(4,104)(5,97)(6,98)(7,99)(8,100)(9,110)(10,111)(11,112)(12,105)(13,106)(14,107)(15,108)(16,109)(17,173)(18,174)(19,175)(20,176)(21,169)(22,170)(23,171)(24,172)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,153)(32,154)(33,198)(34,199)(35,200)(36,193)(37,194)(38,195)(39,196)(40,197)(41,165)(42,166)(43,167)(44,168)(45,161)(46,162)(47,163)(48,164)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,177)(56,178)(57,186)(58,187)(59,188)(60,189)(61,190)(62,191)(63,192)(64,185)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(73,133)(74,134)(75,135)(76,136)(77,129)(78,130)(79,131)(80,132)(81,148)(82,149)(83,150)(84,151)(85,152)(86,145)(87,146)(88,147)(89,124)(90,125)(91,126)(92,127)(93,128)(94,121)(95,122)(96,123)(113,203)(114,204)(115,205)(116,206)(117,207)(118,208)(119,201)(120,202)(209,224)(210,217)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223), (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), (1,97)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,108)(10,107)(11,106)(12,105)(13,112)(14,111)(15,110)(16,109)(17,171)(18,170)(19,169)(20,176)(21,175)(22,174)(23,173)(24,172)(25,153)(26,160)(27,159)(28,158)(29,157)(30,156)(31,155)(32,154)(33,194)(34,193)(35,200)(36,199)(37,198)(38,197)(39,196)(40,195)(41,165)(42,164)(43,163)(44,162)(45,161)(46,168)(47,167)(48,166)(49,179)(50,178)(51,177)(52,184)(53,183)(54,182)(55,181)(56,180)(57,192)(58,191)(59,190)(60,189)(61,188)(62,187)(63,186)(64,185)(65,141)(66,140)(67,139)(68,138)(69,137)(70,144)(71,143)(72,142)(73,135)(74,134)(75,133)(76,132)(77,131)(78,130)(79,129)(80,136)(81,146)(82,145)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,124)(90,123)(91,122)(92,121)(93,128)(94,127)(95,126)(96,125)(113,201)(114,208)(115,207)(116,206)(117,205)(118,204)(119,203)(120,202)(209,218)(210,217)(211,224)(212,223)(213,222)(214,221)(215,220)(216,219), (1,80,22,177,167,65,14)(2,73,23,178,168,66,15)(3,74,24,179,161,67,16)(4,75,17,180,162,68,9)(5,76,18,181,163,69,10)(6,77,19,182,164,70,11)(7,78,20,183,165,71,12)(8,79,21,184,166,72,13)(25,197,186,113,211,125,85)(26,198,187,114,212,126,86)(27,199,188,115,213,127,87)(28,200,189,116,214,128,88)(29,193,190,117,215,121,81)(30,194,191,118,216,122,82)(31,195,192,119,209,123,83)(32,196,185,120,210,124,84)(33,58,204,219,91,145,156)(34,59,205,220,92,146,157)(35,60,206,221,93,147,158)(36,61,207,222,94,148,159)(37,62,208,223,95,149,160)(38,63,201,224,96,150,153)(39,64,202,217,89,151,154)(40,57,203,218,90,152,155)(41,143,105,99,130,176,53)(42,144,106,100,131,169,54)(43,137,107,101,132,170,55)(44,138,108,102,133,171,56)(45,139,109,103,134,172,49)(46,140,110,104,135,173,50)(47,141,111,97,136,174,51)(48,142,112,98,129,175,52), (1,191,97,58)(2,63,98,188)(3,185,99,60)(4,57,100,190)(5,187,101,62)(6,59,102,192)(7,189,103,64)(8,61,104,186)(9,203,106,117)(10,114,107,208)(11,205,108,119)(12,116,109,202)(13,207,110,113)(14,118,111,204)(15,201,112,115)(16,120,105,206)(17,155,169,29)(18,26,170,160)(19,157,171,31)(20,28,172,154)(21,159,173,25)(22,30,174,156)(23,153,175,27)(24,32,176,158)(33,80,194,136)(34,133,195,77)(35,74,196,130)(36,135,197,79)(37,76,198,132)(38,129,199,73)(39,78,200,134)(40,131,193,75)(41,93,161,124)(42,121,162,90)(43,95,163,126)(44,123,164,92)(45,89,165,128)(46,125,166,94)(47,91,167,122)(48,127,168,96)(49,151,183,88)(50,85,184,148)(51,145,177,82)(52,87,178,150)(53,147,179,84)(54,81,180,152)(55,149,181,86)(56,83,182,146)(65,216,141,219)(66,224,142,213)(67,210,143,221)(68,218,144,215)(69,212,137,223)(70,220,138,209)(71,214,139,217)(72,222,140,211)>;
G:=Group( (1,101)(2,102)(3,103)(4,104)(5,97)(6,98)(7,99)(8,100)(9,110)(10,111)(11,112)(12,105)(13,106)(14,107)(15,108)(16,109)(17,173)(18,174)(19,175)(20,176)(21,169)(22,170)(23,171)(24,172)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,153)(32,154)(33,198)(34,199)(35,200)(36,193)(37,194)(38,195)(39,196)(40,197)(41,165)(42,166)(43,167)(44,168)(45,161)(46,162)(47,163)(48,164)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,177)(56,178)(57,186)(58,187)(59,188)(60,189)(61,190)(62,191)(63,192)(64,185)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(73,133)(74,134)(75,135)(76,136)(77,129)(78,130)(79,131)(80,132)(81,148)(82,149)(83,150)(84,151)(85,152)(86,145)(87,146)(88,147)(89,124)(90,125)(91,126)(92,127)(93,128)(94,121)(95,122)(96,123)(113,203)(114,204)(115,205)(116,206)(117,207)(118,208)(119,201)(120,202)(209,224)(210,217)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223), (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), (1,97)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,108)(10,107)(11,106)(12,105)(13,112)(14,111)(15,110)(16,109)(17,171)(18,170)(19,169)(20,176)(21,175)(22,174)(23,173)(24,172)(25,153)(26,160)(27,159)(28,158)(29,157)(30,156)(31,155)(32,154)(33,194)(34,193)(35,200)(36,199)(37,198)(38,197)(39,196)(40,195)(41,165)(42,164)(43,163)(44,162)(45,161)(46,168)(47,167)(48,166)(49,179)(50,178)(51,177)(52,184)(53,183)(54,182)(55,181)(56,180)(57,192)(58,191)(59,190)(60,189)(61,188)(62,187)(63,186)(64,185)(65,141)(66,140)(67,139)(68,138)(69,137)(70,144)(71,143)(72,142)(73,135)(74,134)(75,133)(76,132)(77,131)(78,130)(79,129)(80,136)(81,146)(82,145)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,124)(90,123)(91,122)(92,121)(93,128)(94,127)(95,126)(96,125)(113,201)(114,208)(115,207)(116,206)(117,205)(118,204)(119,203)(120,202)(209,218)(210,217)(211,224)(212,223)(213,222)(214,221)(215,220)(216,219), (1,80,22,177,167,65,14)(2,73,23,178,168,66,15)(3,74,24,179,161,67,16)(4,75,17,180,162,68,9)(5,76,18,181,163,69,10)(6,77,19,182,164,70,11)(7,78,20,183,165,71,12)(8,79,21,184,166,72,13)(25,197,186,113,211,125,85)(26,198,187,114,212,126,86)(27,199,188,115,213,127,87)(28,200,189,116,214,128,88)(29,193,190,117,215,121,81)(30,194,191,118,216,122,82)(31,195,192,119,209,123,83)(32,196,185,120,210,124,84)(33,58,204,219,91,145,156)(34,59,205,220,92,146,157)(35,60,206,221,93,147,158)(36,61,207,222,94,148,159)(37,62,208,223,95,149,160)(38,63,201,224,96,150,153)(39,64,202,217,89,151,154)(40,57,203,218,90,152,155)(41,143,105,99,130,176,53)(42,144,106,100,131,169,54)(43,137,107,101,132,170,55)(44,138,108,102,133,171,56)(45,139,109,103,134,172,49)(46,140,110,104,135,173,50)(47,141,111,97,136,174,51)(48,142,112,98,129,175,52), (1,191,97,58)(2,63,98,188)(3,185,99,60)(4,57,100,190)(5,187,101,62)(6,59,102,192)(7,189,103,64)(8,61,104,186)(9,203,106,117)(10,114,107,208)(11,205,108,119)(12,116,109,202)(13,207,110,113)(14,118,111,204)(15,201,112,115)(16,120,105,206)(17,155,169,29)(18,26,170,160)(19,157,171,31)(20,28,172,154)(21,159,173,25)(22,30,174,156)(23,153,175,27)(24,32,176,158)(33,80,194,136)(34,133,195,77)(35,74,196,130)(36,135,197,79)(37,76,198,132)(38,129,199,73)(39,78,200,134)(40,131,193,75)(41,93,161,124)(42,121,162,90)(43,95,163,126)(44,123,164,92)(45,89,165,128)(46,125,166,94)(47,91,167,122)(48,127,168,96)(49,151,183,88)(50,85,184,148)(51,145,177,82)(52,87,178,150)(53,147,179,84)(54,81,180,152)(55,149,181,86)(56,83,182,146)(65,216,141,219)(66,224,142,213)(67,210,143,221)(68,218,144,215)(69,212,137,223)(70,220,138,209)(71,214,139,217)(72,222,140,211) );
G=PermutationGroup([[(1,101),(2,102),(3,103),(4,104),(5,97),(6,98),(7,99),(8,100),(9,110),(10,111),(11,112),(12,105),(13,106),(14,107),(15,108),(16,109),(17,173),(18,174),(19,175),(20,176),(21,169),(22,170),(23,171),(24,172),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,153),(32,154),(33,198),(34,199),(35,200),(36,193),(37,194),(38,195),(39,196),(40,197),(41,165),(42,166),(43,167),(44,168),(45,161),(46,162),(47,163),(48,164),(49,179),(50,180),(51,181),(52,182),(53,183),(54,184),(55,177),(56,178),(57,186),(58,187),(59,188),(60,189),(61,190),(62,191),(63,192),(64,185),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144),(73,133),(74,134),(75,135),(76,136),(77,129),(78,130),(79,131),(80,132),(81,148),(82,149),(83,150),(84,151),(85,152),(86,145),(87,146),(88,147),(89,124),(90,125),(91,126),(92,127),(93,128),(94,121),(95,122),(96,123),(113,203),(114,204),(115,205),(116,206),(117,207),(118,208),(119,201),(120,202),(209,224),(210,217),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223)], [(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)], [(1,97),(2,104),(3,103),(4,102),(5,101),(6,100),(7,99),(8,98),(9,108),(10,107),(11,106),(12,105),(13,112),(14,111),(15,110),(16,109),(17,171),(18,170),(19,169),(20,176),(21,175),(22,174),(23,173),(24,172),(25,153),(26,160),(27,159),(28,158),(29,157),(30,156),(31,155),(32,154),(33,194),(34,193),(35,200),(36,199),(37,198),(38,197),(39,196),(40,195),(41,165),(42,164),(43,163),(44,162),(45,161),(46,168),(47,167),(48,166),(49,179),(50,178),(51,177),(52,184),(53,183),(54,182),(55,181),(56,180),(57,192),(58,191),(59,190),(60,189),(61,188),(62,187),(63,186),(64,185),(65,141),(66,140),(67,139),(68,138),(69,137),(70,144),(71,143),(72,142),(73,135),(74,134),(75,133),(76,132),(77,131),(78,130),(79,129),(80,136),(81,146),(82,145),(83,152),(84,151),(85,150),(86,149),(87,148),(88,147),(89,124),(90,123),(91,122),(92,121),(93,128),(94,127),(95,126),(96,125),(113,201),(114,208),(115,207),(116,206),(117,205),(118,204),(119,203),(120,202),(209,218),(210,217),(211,224),(212,223),(213,222),(214,221),(215,220),(216,219)], [(1,80,22,177,167,65,14),(2,73,23,178,168,66,15),(3,74,24,179,161,67,16),(4,75,17,180,162,68,9),(5,76,18,181,163,69,10),(6,77,19,182,164,70,11),(7,78,20,183,165,71,12),(8,79,21,184,166,72,13),(25,197,186,113,211,125,85),(26,198,187,114,212,126,86),(27,199,188,115,213,127,87),(28,200,189,116,214,128,88),(29,193,190,117,215,121,81),(30,194,191,118,216,122,82),(31,195,192,119,209,123,83),(32,196,185,120,210,124,84),(33,58,204,219,91,145,156),(34,59,205,220,92,146,157),(35,60,206,221,93,147,158),(36,61,207,222,94,148,159),(37,62,208,223,95,149,160),(38,63,201,224,96,150,153),(39,64,202,217,89,151,154),(40,57,203,218,90,152,155),(41,143,105,99,130,176,53),(42,144,106,100,131,169,54),(43,137,107,101,132,170,55),(44,138,108,102,133,171,56),(45,139,109,103,134,172,49),(46,140,110,104,135,173,50),(47,141,111,97,136,174,51),(48,142,112,98,129,175,52)], [(1,191,97,58),(2,63,98,188),(3,185,99,60),(4,57,100,190),(5,187,101,62),(6,59,102,192),(7,189,103,64),(8,61,104,186),(9,203,106,117),(10,114,107,208),(11,205,108,119),(12,116,109,202),(13,207,110,113),(14,118,111,204),(15,201,112,115),(16,120,105,206),(17,155,169,29),(18,26,170,160),(19,157,171,31),(20,28,172,154),(21,159,173,25),(22,30,174,156),(23,153,175,27),(24,32,176,158),(33,80,194,136),(34,133,195,77),(35,74,196,130),(36,135,197,79),(37,76,198,132),(38,129,199,73),(39,78,200,134),(40,131,193,75),(41,93,161,124),(42,121,162,90),(43,95,163,126),(44,123,164,92),(45,89,165,128),(46,125,166,94),(47,91,167,122),(48,127,168,96),(49,151,183,88),(50,85,184,148),(51,145,177,82),(52,87,178,150),(53,147,179,84),(54,81,180,152),(55,149,181,86),(56,83,182,146),(65,216,141,219),(66,224,142,213),(67,210,143,221),(68,218,144,215),(69,212,137,223),(70,220,138,209),(71,214,139,217),(72,222,140,211)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 2 | 2 | 14 | 14 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 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 | C7⋊D4 | C8⋊C22 | D4⋊2D7 | D4×D7 | D8⋊D7 | D8⋊3D7 |
| kernel | (C2×D8).D7 | Dic7⋊C8 | C28.44D4 | D4⋊Dic7 | C2×D4.D7 | D4×Dic7 | C28.17D4 | C14×D8 | C2×Dic7 | C7×D4 | C2×D8 | C28 | C2×C8 | C2×D4 | C14 | D4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 6 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of (C2×D8).D7 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 62 | 85 | 0 | 0 |
| 109 | 0 | 0 | 0 |
| 0 | 0 | 79 | 5 |
| 0 | 0 | 108 | 34 |
| 112 | 41 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 1 |
| 0 | 0 | 112 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 23 | 100 |
| 0 | 0 | 6 | 90 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[62,109,0,0,85,0,0,0,0,0,79,108,0,0,5,34],[112,0,0,0,41,1,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,9,112,0,0,1,0],[98,0,0,0,0,98,0,0,0,0,23,6,0,0,100,90] >;
(C2×D8).D7 in GAP, Magma, Sage, TeX
(C_2\times D_8).D_7
% in TeX
G:=Group("(C2xD8).D7"); // GroupNames label
G:=SmallGroup(448,687);
// by ID
G=gap.SmallGroup(448,687);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,1094,135,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=1,e^2=a*b^4,e*b*e^-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,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations