direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q16⋊D7, Q16⋊8D14, C28.11C24, C56.41C23, D28.6C23, Dic14.7C23, C4.47(D4×D7), C7⋊C8.4C23, (C2×Q16)⋊11D7, (C4×D7).17D4, C28.86(C2×D4), Q8⋊D7⋊9C22, (Q8×D7)⋊7C22, (C14×Q16)⋊11C2, D14.52(C2×D4), (C2×C8).104D14, C7⋊Q16⋊9C22, (C4×D7).6C23, C8.13(C22×D7), C4.11(C23×D7), (C7×Q8).5C23, Q8.5(C22×D7), C56⋊C2⋊15C22, C8⋊D7⋊14C22, C14⋊3(C8.C22), (C2×Q8).153D14, Dic7.57(C2×D4), (C7×Q16)⋊12C22, C22.143(D4×D7), (C2×C28).528C23, (C2×C56).152C22, (C2×Dic7).194D4, (C22×D7).100D4, C14.112(C22×D4), Q8⋊2D7.4C22, (C2×D28).179C22, (Q8×C14).150C22, (C2×Dic14).199C22, (C2×Q8×D7)⋊16C2, C2.85(C2×D4×D7), C7⋊3(C2×C8.C22), (C2×C8⋊D7)⋊9C2, (C2×Q8⋊D7)⋊27C2, (C2×C56⋊C2)⋊25C2, (C2×C7⋊Q16)⋊28C2, (C2×C14).401(C2×D4), (C2×C7⋊C8).181C22, (C2×Q8⋊2D7).8C2, (C2×C4×D7).158C22, (C2×C4).616(C22×D7), SmallGroup(448,1217)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q16⋊D7
G = < a,b,c,d,e | a2=b8=d7=e2=1, c2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b5, cd=dc, ece=b4c, ede=d-1 >
Subgroups: 1252 in 258 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×M4(2), C2×SD16, C2×Q16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×D7, C2×C8.C22, C8⋊D7, C56⋊C2, C2×C7⋊C8, Q8⋊D7, C7⋊Q16, C2×C56, C7×Q16, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, Q8×D7, Q8×D7, Q8⋊2D7, Q8⋊2D7, Q8×C14, C2×C8⋊D7, C2×C56⋊C2, Q16⋊D7, C2×Q8⋊D7, C2×C7⋊Q16, C14×Q16, C2×Q8×D7, C2×Q8⋊2D7, C2×Q16⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8.C22, C22×D4, C22×D7, C2×C8.C22, D4×D7, C23×D7, Q16⋊D7, C2×D4×D7, C2×Q16⋊D7
►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)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 221)(26 222)(27 223)(28 224)(29 217)(30 218)(31 219)(32 220)(33 213)(34 214)(35 215)(36 216)(37 209)(38 210)(39 211)(40 212)(41 144)(42 137)(43 138)(44 139)(45 140)(46 141)(47 142)(48 143)(49 116)(50 117)(51 118)(52 119)(53 120)(54 113)(55 114)(56 115)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 129)(64 130)(65 202)(66 203)(67 204)(68 205)(69 206)(70 207)(71 208)(72 201)(73 127)(74 128)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 188)(82 189)(83 190)(84 191)(85 192)(86 185)(87 186)(88 187)(89 161)(90 162)(91 163)(92 164)(93 165)(94 166)(95 167)(96 168)(97 155)(98 156)(99 157)(100 158)(101 159)(102 160)(103 153)(104 154)(105 170)(106 171)(107 172)(108 173)(109 174)(110 175)(111 176)(112 169)
(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)
(1 219 5 223)(2 218 6 222)(3 217 7 221)(4 224 8 220)(9 67 13 71)(10 66 14 70)(11 65 15 69)(12 72 16 68)(17 38 21 34)(18 37 22 33)(19 36 23 40)(20 35 24 39)(25 198 29 194)(26 197 30 193)(27 196 31 200)(28 195 32 199)(41 102 45 98)(42 101 46 97)(43 100 47 104)(44 99 48 103)(49 94 53 90)(50 93 54 89)(51 92 55 96)(52 91 56 95)(57 107 61 111)(58 106 62 110)(59 105 63 109)(60 112 64 108)(73 83 77 87)(74 82 78 86)(75 81 79 85)(76 88 80 84)(113 161 117 165)(114 168 118 164)(115 167 119 163)(116 166 120 162)(121 188 125 192)(122 187 126 191)(123 186 127 190)(124 185 128 189)(129 174 133 170)(130 173 134 169)(131 172 135 176)(132 171 136 175)(137 159 141 155)(138 158 142 154)(139 157 143 153)(140 156 144 160)(145 210 149 214)(146 209 150 213)(147 216 151 212)(148 215 152 211)(177 207 181 203)(178 206 182 202)(179 205 183 201)(180 204 184 208)
(1 47 72 163 174 151 189)(2 48 65 164 175 152 190)(3 41 66 165 176 145 191)(4 42 67 166 169 146 192)(5 43 68 167 170 147 185)(6 44 69 168 171 148 186)(7 45 70 161 172 149 187)(8 46 71 162 173 150 188)(9 116 134 213 125 220 97)(10 117 135 214 126 221 98)(11 118 136 215 127 222 99)(12 119 129 216 128 223 100)(13 120 130 209 121 224 101)(14 113 131 210 122 217 102)(15 114 132 211 123 218 103)(16 115 133 212 124 219 104)(17 84 198 144 203 93 111)(18 85 199 137 204 94 112)(19 86 200 138 205 95 105)(20 87 193 139 206 96 106)(21 88 194 140 207 89 107)(22 81 195 141 208 90 108)(23 82 196 142 201 91 109)(24 83 197 143 202 92 110)(25 156 181 50 61 34 80)(26 157 182 51 62 35 73)(27 158 183 52 63 36 74)(28 159 184 53 64 37 75)(29 160 177 54 57 38 76)(30 153 178 55 58 39 77)(31 154 179 56 59 40 78)(32 155 180 49 60 33 79)
(1 189)(2 186)(3 191)(4 188)(5 185)(6 190)(7 187)(8 192)(9 134)(10 131)(11 136)(12 133)(13 130)(14 135)(15 132)(16 129)(17 144)(18 141)(19 138)(20 143)(21 140)(22 137)(23 142)(24 139)(25 76)(26 73)(27 78)(28 75)(29 80)(30 77)(31 74)(32 79)(33 155)(34 160)(35 157)(36 154)(37 159)(38 156)(39 153)(40 158)(41 145)(42 150)(43 147)(44 152)(45 149)(46 146)(47 151)(48 148)(50 54)(52 56)(57 181)(58 178)(59 183)(60 180)(61 177)(62 182)(63 179)(64 184)(65 171)(66 176)(67 173)(68 170)(69 175)(70 172)(71 169)(72 174)(81 199)(82 196)(83 193)(84 198)(85 195)(86 200)(87 197)(88 194)(90 94)(92 96)(97 213)(98 210)(99 215)(100 212)(101 209)(102 214)(103 211)(104 216)(105 205)(106 202)(107 207)(108 204)(109 201)(110 206)(111 203)(112 208)(113 117)(115 119)(121 224)(122 221)(123 218)(124 223)(125 220)(126 217)(127 222)(128 219)(162 166)(164 168)
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)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,221)(26,222)(27,223)(28,224)(29,217)(30,218)(31,219)(32,220)(33,213)(34,214)(35,215)(36,216)(37,209)(38,210)(39,211)(40,212)(41,144)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,116)(50,117)(51,118)(52,119)(53,120)(54,113)(55,114)(56,115)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,129)(64,130)(65,202)(66,203)(67,204)(68,205)(69,206)(70,207)(71,208)(72,201)(73,127)(74,128)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,188)(82,189)(83,190)(84,191)(85,192)(86,185)(87,186)(88,187)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,153)(104,154)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,169), (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), (1,219,5,223)(2,218,6,222)(3,217,7,221)(4,224,8,220)(9,67,13,71)(10,66,14,70)(11,65,15,69)(12,72,16,68)(17,38,21,34)(18,37,22,33)(19,36,23,40)(20,35,24,39)(25,198,29,194)(26,197,30,193)(27,196,31,200)(28,195,32,199)(41,102,45,98)(42,101,46,97)(43,100,47,104)(44,99,48,103)(49,94,53,90)(50,93,54,89)(51,92,55,96)(52,91,56,95)(57,107,61,111)(58,106,62,110)(59,105,63,109)(60,112,64,108)(73,83,77,87)(74,82,78,86)(75,81,79,85)(76,88,80,84)(113,161,117,165)(114,168,118,164)(115,167,119,163)(116,166,120,162)(121,188,125,192)(122,187,126,191)(123,186,127,190)(124,185,128,189)(129,174,133,170)(130,173,134,169)(131,172,135,176)(132,171,136,175)(137,159,141,155)(138,158,142,154)(139,157,143,153)(140,156,144,160)(145,210,149,214)(146,209,150,213)(147,216,151,212)(148,215,152,211)(177,207,181,203)(178,206,182,202)(179,205,183,201)(180,204,184,208), (1,47,72,163,174,151,189)(2,48,65,164,175,152,190)(3,41,66,165,176,145,191)(4,42,67,166,169,146,192)(5,43,68,167,170,147,185)(6,44,69,168,171,148,186)(7,45,70,161,172,149,187)(8,46,71,162,173,150,188)(9,116,134,213,125,220,97)(10,117,135,214,126,221,98)(11,118,136,215,127,222,99)(12,119,129,216,128,223,100)(13,120,130,209,121,224,101)(14,113,131,210,122,217,102)(15,114,132,211,123,218,103)(16,115,133,212,124,219,104)(17,84,198,144,203,93,111)(18,85,199,137,204,94,112)(19,86,200,138,205,95,105)(20,87,193,139,206,96,106)(21,88,194,140,207,89,107)(22,81,195,141,208,90,108)(23,82,196,142,201,91,109)(24,83,197,143,202,92,110)(25,156,181,50,61,34,80)(26,157,182,51,62,35,73)(27,158,183,52,63,36,74)(28,159,184,53,64,37,75)(29,160,177,54,57,38,76)(30,153,178,55,58,39,77)(31,154,179,56,59,40,78)(32,155,180,49,60,33,79), (1,189)(2,186)(3,191)(4,188)(5,185)(6,190)(7,187)(8,192)(9,134)(10,131)(11,136)(12,133)(13,130)(14,135)(15,132)(16,129)(17,144)(18,141)(19,138)(20,143)(21,140)(22,137)(23,142)(24,139)(25,76)(26,73)(27,78)(28,75)(29,80)(30,77)(31,74)(32,79)(33,155)(34,160)(35,157)(36,154)(37,159)(38,156)(39,153)(40,158)(41,145)(42,150)(43,147)(44,152)(45,149)(46,146)(47,151)(48,148)(50,54)(52,56)(57,181)(58,178)(59,183)(60,180)(61,177)(62,182)(63,179)(64,184)(65,171)(66,176)(67,173)(68,170)(69,175)(70,172)(71,169)(72,174)(81,199)(82,196)(83,193)(84,198)(85,195)(86,200)(87,197)(88,194)(90,94)(92,96)(97,213)(98,210)(99,215)(100,212)(101,209)(102,214)(103,211)(104,216)(105,205)(106,202)(107,207)(108,204)(109,201)(110,206)(111,203)(112,208)(113,117)(115,119)(121,224)(122,221)(123,218)(124,223)(125,220)(126,217)(127,222)(128,219)(162,166)(164,168)>;
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)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,221)(26,222)(27,223)(28,224)(29,217)(30,218)(31,219)(32,220)(33,213)(34,214)(35,215)(36,216)(37,209)(38,210)(39,211)(40,212)(41,144)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,116)(50,117)(51,118)(52,119)(53,120)(54,113)(55,114)(56,115)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,129)(64,130)(65,202)(66,203)(67,204)(68,205)(69,206)(70,207)(71,208)(72,201)(73,127)(74,128)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,188)(82,189)(83,190)(84,191)(85,192)(86,185)(87,186)(88,187)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,153)(104,154)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,169), (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), (1,219,5,223)(2,218,6,222)(3,217,7,221)(4,224,8,220)(9,67,13,71)(10,66,14,70)(11,65,15,69)(12,72,16,68)(17,38,21,34)(18,37,22,33)(19,36,23,40)(20,35,24,39)(25,198,29,194)(26,197,30,193)(27,196,31,200)(28,195,32,199)(41,102,45,98)(42,101,46,97)(43,100,47,104)(44,99,48,103)(49,94,53,90)(50,93,54,89)(51,92,55,96)(52,91,56,95)(57,107,61,111)(58,106,62,110)(59,105,63,109)(60,112,64,108)(73,83,77,87)(74,82,78,86)(75,81,79,85)(76,88,80,84)(113,161,117,165)(114,168,118,164)(115,167,119,163)(116,166,120,162)(121,188,125,192)(122,187,126,191)(123,186,127,190)(124,185,128,189)(129,174,133,170)(130,173,134,169)(131,172,135,176)(132,171,136,175)(137,159,141,155)(138,158,142,154)(139,157,143,153)(140,156,144,160)(145,210,149,214)(146,209,150,213)(147,216,151,212)(148,215,152,211)(177,207,181,203)(178,206,182,202)(179,205,183,201)(180,204,184,208), (1,47,72,163,174,151,189)(2,48,65,164,175,152,190)(3,41,66,165,176,145,191)(4,42,67,166,169,146,192)(5,43,68,167,170,147,185)(6,44,69,168,171,148,186)(7,45,70,161,172,149,187)(8,46,71,162,173,150,188)(9,116,134,213,125,220,97)(10,117,135,214,126,221,98)(11,118,136,215,127,222,99)(12,119,129,216,128,223,100)(13,120,130,209,121,224,101)(14,113,131,210,122,217,102)(15,114,132,211,123,218,103)(16,115,133,212,124,219,104)(17,84,198,144,203,93,111)(18,85,199,137,204,94,112)(19,86,200,138,205,95,105)(20,87,193,139,206,96,106)(21,88,194,140,207,89,107)(22,81,195,141,208,90,108)(23,82,196,142,201,91,109)(24,83,197,143,202,92,110)(25,156,181,50,61,34,80)(26,157,182,51,62,35,73)(27,158,183,52,63,36,74)(28,159,184,53,64,37,75)(29,160,177,54,57,38,76)(30,153,178,55,58,39,77)(31,154,179,56,59,40,78)(32,155,180,49,60,33,79), (1,189)(2,186)(3,191)(4,188)(5,185)(6,190)(7,187)(8,192)(9,134)(10,131)(11,136)(12,133)(13,130)(14,135)(15,132)(16,129)(17,144)(18,141)(19,138)(20,143)(21,140)(22,137)(23,142)(24,139)(25,76)(26,73)(27,78)(28,75)(29,80)(30,77)(31,74)(32,79)(33,155)(34,160)(35,157)(36,154)(37,159)(38,156)(39,153)(40,158)(41,145)(42,150)(43,147)(44,152)(45,149)(46,146)(47,151)(48,148)(50,54)(52,56)(57,181)(58,178)(59,183)(60,180)(61,177)(62,182)(63,179)(64,184)(65,171)(66,176)(67,173)(68,170)(69,175)(70,172)(71,169)(72,174)(81,199)(82,196)(83,193)(84,198)(85,195)(86,200)(87,197)(88,194)(90,94)(92,96)(97,213)(98,210)(99,215)(100,212)(101,209)(102,214)(103,211)(104,216)(105,205)(106,202)(107,207)(108,204)(109,201)(110,206)(111,203)(112,208)(113,117)(115,119)(121,224)(122,221)(123,218)(124,223)(125,220)(126,217)(127,222)(128,219)(162,166)(164,168) );
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),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,151),(24,152),(25,221),(26,222),(27,223),(28,224),(29,217),(30,218),(31,219),(32,220),(33,213),(34,214),(35,215),(36,216),(37,209),(38,210),(39,211),(40,212),(41,144),(42,137),(43,138),(44,139),(45,140),(46,141),(47,142),(48,143),(49,116),(50,117),(51,118),(52,119),(53,120),(54,113),(55,114),(56,115),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,129),(64,130),(65,202),(66,203),(67,204),(68,205),(69,206),(70,207),(71,208),(72,201),(73,127),(74,128),(75,121),(76,122),(77,123),(78,124),(79,125),(80,126),(81,188),(82,189),(83,190),(84,191),(85,192),(86,185),(87,186),(88,187),(89,161),(90,162),(91,163),(92,164),(93,165),(94,166),(95,167),(96,168),(97,155),(98,156),(99,157),(100,158),(101,159),(102,160),(103,153),(104,154),(105,170),(106,171),(107,172),(108,173),(109,174),(110,175),(111,176),(112,169)], [(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)], [(1,219,5,223),(2,218,6,222),(3,217,7,221),(4,224,8,220),(9,67,13,71),(10,66,14,70),(11,65,15,69),(12,72,16,68),(17,38,21,34),(18,37,22,33),(19,36,23,40),(20,35,24,39),(25,198,29,194),(26,197,30,193),(27,196,31,200),(28,195,32,199),(41,102,45,98),(42,101,46,97),(43,100,47,104),(44,99,48,103),(49,94,53,90),(50,93,54,89),(51,92,55,96),(52,91,56,95),(57,107,61,111),(58,106,62,110),(59,105,63,109),(60,112,64,108),(73,83,77,87),(74,82,78,86),(75,81,79,85),(76,88,80,84),(113,161,117,165),(114,168,118,164),(115,167,119,163),(116,166,120,162),(121,188,125,192),(122,187,126,191),(123,186,127,190),(124,185,128,189),(129,174,133,170),(130,173,134,169),(131,172,135,176),(132,171,136,175),(137,159,141,155),(138,158,142,154),(139,157,143,153),(140,156,144,160),(145,210,149,214),(146,209,150,213),(147,216,151,212),(148,215,152,211),(177,207,181,203),(178,206,182,202),(179,205,183,201),(180,204,184,208)], [(1,47,72,163,174,151,189),(2,48,65,164,175,152,190),(3,41,66,165,176,145,191),(4,42,67,166,169,146,192),(5,43,68,167,170,147,185),(6,44,69,168,171,148,186),(7,45,70,161,172,149,187),(8,46,71,162,173,150,188),(9,116,134,213,125,220,97),(10,117,135,214,126,221,98),(11,118,136,215,127,222,99),(12,119,129,216,128,223,100),(13,120,130,209,121,224,101),(14,113,131,210,122,217,102),(15,114,132,211,123,218,103),(16,115,133,212,124,219,104),(17,84,198,144,203,93,111),(18,85,199,137,204,94,112),(19,86,200,138,205,95,105),(20,87,193,139,206,96,106),(21,88,194,140,207,89,107),(22,81,195,141,208,90,108),(23,82,196,142,201,91,109),(24,83,197,143,202,92,110),(25,156,181,50,61,34,80),(26,157,182,51,62,35,73),(27,158,183,52,63,36,74),(28,159,184,53,64,37,75),(29,160,177,54,57,38,76),(30,153,178,55,58,39,77),(31,154,179,56,59,40,78),(32,155,180,49,60,33,79)], [(1,189),(2,186),(3,191),(4,188),(5,185),(6,190),(7,187),(8,192),(9,134),(10,131),(11,136),(12,133),(13,130),(14,135),(15,132),(16,129),(17,144),(18,141),(19,138),(20,143),(21,140),(22,137),(23,142),(24,139),(25,76),(26,73),(27,78),(28,75),(29,80),(30,77),(31,74),(32,79),(33,155),(34,160),(35,157),(36,154),(37,159),(38,156),(39,153),(40,158),(41,145),(42,150),(43,147),(44,152),(45,149),(46,146),(47,151),(48,148),(50,54),(52,56),(57,181),(58,178),(59,183),(60,180),(61,177),(62,182),(63,179),(64,184),(65,171),(66,176),(67,173),(68,170),(69,175),(70,172),(71,169),(72,174),(81,199),(82,196),(83,193),(84,198),(85,195),(86,200),(87,197),(88,194),(90,94),(92,96),(97,213),(98,210),(99,215),(100,212),(101,209),(102,214),(103,211),(104,216),(105,205),(106,202),(107,207),(108,204),(109,201),(110,206),(111,203),(112,208),(113,117),(115,119),(121,224),(122,221),(123,218),(124,223),(125,220),(126,217),(127,222),(128,219),(162,166),(164,168)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 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 | 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 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C8.C22 | D4×D7 | D4×D7 | Q16⋊D7 |
| kernel | C2×Q16⋊D7 | C2×C8⋊D7 | C2×C56⋊C2 | Q16⋊D7 | C2×Q8⋊D7 | C2×C7⋊Q16 | C14×Q16 | C2×Q8×D7 | C2×Q8⋊2D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 12 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of C2×Q16⋊D7 ►in GL8(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 44 | 110 | 27 | 14 | 0 | 0 | 0 | 0 |
| 6 | 63 | 95 | 103 | 0 | 0 | 0 | 0 |
| 4 | 30 | 66 | 3 | 0 | 0 | 0 | 0 |
| 57 | 70 | 10 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 80 | 109 | 33 |
| 0 | 0 | 0 | 0 | 33 | 109 | 80 | 4 |
| 0 | 0 | 0 | 0 | 4 | 80 | 4 | 80 |
| 0 | 0 | 0 | 0 | 33 | 109 | 33 | 109 |
| 95 | 66 | 84 | 31 | 0 | 0 | 0 | 0 |
| 94 | 16 | 57 | 107 | 0 | 0 | 0 | 0 |
| 72 | 11 | 84 | 47 | 0 | 0 | 0 | 0 |
| 50 | 51 | 6 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 40 | 90 | 105 |
| 0 | 0 | 0 | 0 | 73 | 111 | 8 | 23 |
| 0 | 0 | 0 | 0 | 90 | 105 | 111 | 73 |
| 0 | 0 | 0 | 0 | 8 | 23 | 40 | 2 |
| 103 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 104 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 79 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 79 | 112 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 102 | 103 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 25 | 0 | 33 | 103 | 0 | 0 | 0 | 0 |
| 55 | 88 | 41 | 80 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 88 | 79 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 88 | 79 |
G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[44,6,4,57,0,0,0,0,110,63,30,70,0,0,0,0,27,95,66,10,0,0,0,0,14,103,3,53,0,0,0,0,0,0,0,0,4,33,4,33,0,0,0,0,80,109,80,109,0,0,0,0,109,80,4,33,0,0,0,0,33,4,80,109],[95,94,72,50,0,0,0,0,66,16,11,51,0,0,0,0,84,57,84,6,0,0,0,0,31,107,47,31,0,0,0,0,0,0,0,0,2,73,90,8,0,0,0,0,40,111,105,23,0,0,0,0,90,8,111,40,0,0,0,0,105,23,73,2],[103,2,0,0,0,0,0,0,112,34,0,0,0,0,0,0,0,0,104,72,0,0,0,0,0,0,112,33,0,0,0,0,0,0,0,0,79,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,79,1,0,0,0,0,0,0,112,0],[102,12,25,55,0,0,0,0,103,11,0,88,0,0,0,0,0,0,33,41,0,0,0,0,0,0,103,80,0,0,0,0,0,0,0,0,34,88,0,0,0,0,0,0,1,79,0,0,0,0,0,0,0,0,34,88,0,0,0,0,0,0,1,79] >;
C2×Q16⋊D7 in GAP, Magma, Sage, TeX
C_2\times Q_{16}\rtimes D_7 % in TeX
G:=Group("C2xQ16:D7"); // GroupNames label
G:=SmallGroup(448,1217);
// by ID
G=gap.SmallGroup(448,1217);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,1123,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=d^7=e^2=1,c^2=b^4,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,e*b*e=b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations