direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q16⋊D7, Q16⋊8D14, C56.41C23, C28.11C24, 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, C4.11(C23×D7), C8.13(C22×D7), Q8.5(C22×D7), (C7×Q8).5C23, 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
Subgroups: 1252 in 258 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×8], C22, C22 [×8], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×16], D4 [×7], Q8 [×4], Q8 [×9], C23 [×2], D7 [×4], C14, C14 [×2], C2×C8, C2×C8, M4(2) [×4], SD16 [×8], Q16 [×4], Q16 [×4], C22×C4 [×3], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×8], C4○D4 [×6], Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×4], D14 [×2], D14 [×6], C2×C14, C2×M4(2), C2×SD16 [×2], C2×Q16, C2×Q16, C8.C22 [×8], C22×Q8, C2×C4○D4, C7⋊C8 [×2], C56 [×2], Dic14 [×2], Dic14 [×5], C4×D7 [×4], C4×D7 [×8], D28 [×2], D28 [×5], C2×Dic7, C2×Dic7, C2×C28, C2×C28 [×2], C7×Q8 [×4], C7×Q8 [×2], C22×D7, C22×D7, C2×C8.C22, C8⋊D7 [×4], C56⋊C2 [×4], C2×C7⋊C8, Q8⋊D7 [×4], C7⋊Q16 [×4], C2×C56, C7×Q16 [×4], C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7 [×2], C2×D28, C2×D28, Q8×D7 [×4], Q8×D7 [×2], Q8⋊2D7 [×4], Q8⋊2D7 [×2], Q8×C14 [×2], C2×C8⋊D7, C2×C56⋊C2, Q16⋊D7 [×8], C2×Q8⋊D7, C2×C7⋊Q16, C14×Q16, C2×Q8×D7, C2×Q8⋊2D7, C2×Q16⋊D7
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8.C22 [×2], C22×D4, C22×D7 [×7], C2×C8.C22, D4×D7 [×2], C23×D7, Q16⋊D7 [×2], C2×D4×D7, C2×Q16⋊D7
Generators and relations
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 >
►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)(41 144)(42 137)(43 138)(44 139)(45 140)(46 141)(47 142)(48 143)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 136)(56 129)(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)(73 127)(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)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 145)(96 146)(97 155)(98 156)(99 157)(100 158)(101 159)(102 160)(103 153)(104 154)(105 163)(106 164)(107 165)(108 166)(109 167)(110 168)(111 161)(112 162)
(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 59 21 63)(18 58 22 62)(19 57 23 61)(20 64 24 60)(25 198 29 194)(26 197 30 193)(27 196 31 200)(28 195 32 199)(33 96 37 92)(34 95 38 91)(35 94 39 90)(36 93 40 89)(41 102 45 98)(42 101 46 97)(43 100 47 104)(44 99 48 103)(49 108 53 112)(50 107 54 111)(51 106 55 110)(52 105 56 109)(73 83 77 87)(74 82 78 86)(75 81 79 85)(76 88 80 84)(113 147 117 151)(114 146 118 150)(115 145 119 149)(116 152 120 148)(121 188 125 192)(122 187 126 191)(123 186 127 190)(124 185 128 189)(129 167 133 163)(130 166 134 162)(131 165 135 161)(132 164 136 168)(137 159 141 155)(138 158 142 154)(139 157 143 153)(140 156 144 160)(169 215 173 211)(170 214 174 210)(171 213 175 209)(172 212 176 216)(177 207 181 203)(178 206 182 202)(179 205 183 201)(180 204 184 208)
(1 47 50 177 174 151 189)(2 48 51 178 175 152 190)(3 41 52 179 176 145 191)(4 42 53 180 169 146 192)(5 43 54 181 170 147 185)(6 44 55 182 171 148 186)(7 45 56 183 172 149 187)(8 46 49 184 173 150 188)(9 20 96 85 199 137 134)(10 21 89 86 200 138 135)(11 22 90 87 193 139 136)(12 23 91 88 194 140 129)(13 24 92 81 195 141 130)(14 17 93 82 196 142 131)(15 18 94 83 197 143 132)(16 19 95 84 198 144 133)(25 156 167 72 61 34 80)(26 157 168 65 62 35 73)(27 158 161 66 63 36 74)(28 159 162 67 64 37 75)(29 160 163 68 57 38 76)(30 153 164 69 58 39 77)(31 154 165 70 59 40 78)(32 155 166 71 60 33 79)(97 108 208 211 114 125 220)(98 109 201 212 115 126 221)(99 110 202 213 116 127 222)(100 111 203 214 117 128 223)(101 112 204 215 118 121 224)(102 105 205 216 119 122 217)(103 106 206 209 120 123 218)(104 107 207 210 113 124 219)
(1 189)(2 186)(3 191)(4 188)(5 185)(6 190)(7 187)(8 192)(9 13)(11 15)(17 131)(18 136)(19 133)(20 130)(21 135)(22 132)(23 129)(24 134)(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)(49 169)(50 174)(51 171)(52 176)(53 173)(54 170)(55 175)(56 172)(57 167)(58 164)(59 161)(60 166)(61 163)(62 168)(63 165)(64 162)(66 70)(68 72)(81 199)(82 196)(83 193)(84 198)(85 195)(86 200)(87 197)(88 194)(89 138)(90 143)(91 140)(92 137)(93 142)(94 139)(95 144)(96 141)(97 114)(98 119)(99 116)(100 113)(101 118)(102 115)(103 120)(104 117)(105 212)(106 209)(107 214)(108 211)(109 216)(110 213)(111 210)(112 215)(121 224)(122 221)(123 218)(124 223)(125 220)(126 217)(127 222)(128 219)(178 182)(180 184)(201 205)(203 207)
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)(41,144)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,129)(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)(73,127)(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)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,145)(96,146)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,153)(104,154)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,161)(112,162), (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)(17,59,21,63)(18,58,22,62)(19,57,23,61)(20,64,24,60)(25,198,29,194)(26,197,30,193)(27,196,31,200)(28,195,32,199)(33,96,37,92)(34,95,38,91)(35,94,39,90)(36,93,40,89)(41,102,45,98)(42,101,46,97)(43,100,47,104)(44,99,48,103)(49,108,53,112)(50,107,54,111)(51,106,55,110)(52,105,56,109)(73,83,77,87)(74,82,78,86)(75,81,79,85)(76,88,80,84)(113,147,117,151)(114,146,118,150)(115,145,119,149)(116,152,120,148)(121,188,125,192)(122,187,126,191)(123,186,127,190)(124,185,128,189)(129,167,133,163)(130,166,134,162)(131,165,135,161)(132,164,136,168)(137,159,141,155)(138,158,142,154)(139,157,143,153)(140,156,144,160)(169,215,173,211)(170,214,174,210)(171,213,175,209)(172,212,176,216)(177,207,181,203)(178,206,182,202)(179,205,183,201)(180,204,184,208), (1,47,50,177,174,151,189)(2,48,51,178,175,152,190)(3,41,52,179,176,145,191)(4,42,53,180,169,146,192)(5,43,54,181,170,147,185)(6,44,55,182,171,148,186)(7,45,56,183,172,149,187)(8,46,49,184,173,150,188)(9,20,96,85,199,137,134)(10,21,89,86,200,138,135)(11,22,90,87,193,139,136)(12,23,91,88,194,140,129)(13,24,92,81,195,141,130)(14,17,93,82,196,142,131)(15,18,94,83,197,143,132)(16,19,95,84,198,144,133)(25,156,167,72,61,34,80)(26,157,168,65,62,35,73)(27,158,161,66,63,36,74)(28,159,162,67,64,37,75)(29,160,163,68,57,38,76)(30,153,164,69,58,39,77)(31,154,165,70,59,40,78)(32,155,166,71,60,33,79)(97,108,208,211,114,125,220)(98,109,201,212,115,126,221)(99,110,202,213,116,127,222)(100,111,203,214,117,128,223)(101,112,204,215,118,121,224)(102,105,205,216,119,122,217)(103,106,206,209,120,123,218)(104,107,207,210,113,124,219), (1,189)(2,186)(3,191)(4,188)(5,185)(6,190)(7,187)(8,192)(9,13)(11,15)(17,131)(18,136)(19,133)(20,130)(21,135)(22,132)(23,129)(24,134)(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)(49,169)(50,174)(51,171)(52,176)(53,173)(54,170)(55,175)(56,172)(57,167)(58,164)(59,161)(60,166)(61,163)(62,168)(63,165)(64,162)(66,70)(68,72)(81,199)(82,196)(83,193)(84,198)(85,195)(86,200)(87,197)(88,194)(89,138)(90,143)(91,140)(92,137)(93,142)(94,139)(95,144)(96,141)(97,114)(98,119)(99,116)(100,113)(101,118)(102,115)(103,120)(104,117)(105,212)(106,209)(107,214)(108,211)(109,216)(110,213)(111,210)(112,215)(121,224)(122,221)(123,218)(124,223)(125,220)(126,217)(127,222)(128,219)(178,182)(180,184)(201,205)(203,207)>;
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)(41,144)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,129)(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)(73,127)(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)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,145)(96,146)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,153)(104,154)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,161)(112,162), (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)(17,59,21,63)(18,58,22,62)(19,57,23,61)(20,64,24,60)(25,198,29,194)(26,197,30,193)(27,196,31,200)(28,195,32,199)(33,96,37,92)(34,95,38,91)(35,94,39,90)(36,93,40,89)(41,102,45,98)(42,101,46,97)(43,100,47,104)(44,99,48,103)(49,108,53,112)(50,107,54,111)(51,106,55,110)(52,105,56,109)(73,83,77,87)(74,82,78,86)(75,81,79,85)(76,88,80,84)(113,147,117,151)(114,146,118,150)(115,145,119,149)(116,152,120,148)(121,188,125,192)(122,187,126,191)(123,186,127,190)(124,185,128,189)(129,167,133,163)(130,166,134,162)(131,165,135,161)(132,164,136,168)(137,159,141,155)(138,158,142,154)(139,157,143,153)(140,156,144,160)(169,215,173,211)(170,214,174,210)(171,213,175,209)(172,212,176,216)(177,207,181,203)(178,206,182,202)(179,205,183,201)(180,204,184,208), (1,47,50,177,174,151,189)(2,48,51,178,175,152,190)(3,41,52,179,176,145,191)(4,42,53,180,169,146,192)(5,43,54,181,170,147,185)(6,44,55,182,171,148,186)(7,45,56,183,172,149,187)(8,46,49,184,173,150,188)(9,20,96,85,199,137,134)(10,21,89,86,200,138,135)(11,22,90,87,193,139,136)(12,23,91,88,194,140,129)(13,24,92,81,195,141,130)(14,17,93,82,196,142,131)(15,18,94,83,197,143,132)(16,19,95,84,198,144,133)(25,156,167,72,61,34,80)(26,157,168,65,62,35,73)(27,158,161,66,63,36,74)(28,159,162,67,64,37,75)(29,160,163,68,57,38,76)(30,153,164,69,58,39,77)(31,154,165,70,59,40,78)(32,155,166,71,60,33,79)(97,108,208,211,114,125,220)(98,109,201,212,115,126,221)(99,110,202,213,116,127,222)(100,111,203,214,117,128,223)(101,112,204,215,118,121,224)(102,105,205,216,119,122,217)(103,106,206,209,120,123,218)(104,107,207,210,113,124,219), (1,189)(2,186)(3,191)(4,188)(5,185)(6,190)(7,187)(8,192)(9,13)(11,15)(17,131)(18,136)(19,133)(20,130)(21,135)(22,132)(23,129)(24,134)(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)(49,169)(50,174)(51,171)(52,176)(53,173)(54,170)(55,175)(56,172)(57,167)(58,164)(59,161)(60,166)(61,163)(62,168)(63,165)(64,162)(66,70)(68,72)(81,199)(82,196)(83,193)(84,198)(85,195)(86,200)(87,197)(88,194)(89,138)(90,143)(91,140)(92,137)(93,142)(94,139)(95,144)(96,141)(97,114)(98,119)(99,116)(100,113)(101,118)(102,115)(103,120)(104,117)(105,212)(106,209)(107,214)(108,211)(109,216)(110,213)(111,210)(112,215)(121,224)(122,221)(123,218)(124,223)(125,220)(126,217)(127,222)(128,219)(178,182)(180,184)(201,205)(203,207) );
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),(41,144),(42,137),(43,138),(44,139),(45,140),(46,141),(47,142),(48,143),(49,130),(50,131),(51,132),(52,133),(53,134),(54,135),(55,136),(56,129),(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),(73,127),(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),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,145),(96,146),(97,155),(98,156),(99,157),(100,158),(101,159),(102,160),(103,153),(104,154),(105,163),(106,164),(107,165),(108,166),(109,167),(110,168),(111,161),(112,162)], [(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),(17,59,21,63),(18,58,22,62),(19,57,23,61),(20,64,24,60),(25,198,29,194),(26,197,30,193),(27,196,31,200),(28,195,32,199),(33,96,37,92),(34,95,38,91),(35,94,39,90),(36,93,40,89),(41,102,45,98),(42,101,46,97),(43,100,47,104),(44,99,48,103),(49,108,53,112),(50,107,54,111),(51,106,55,110),(52,105,56,109),(73,83,77,87),(74,82,78,86),(75,81,79,85),(76,88,80,84),(113,147,117,151),(114,146,118,150),(115,145,119,149),(116,152,120,148),(121,188,125,192),(122,187,126,191),(123,186,127,190),(124,185,128,189),(129,167,133,163),(130,166,134,162),(131,165,135,161),(132,164,136,168),(137,159,141,155),(138,158,142,154),(139,157,143,153),(140,156,144,160),(169,215,173,211),(170,214,174,210),(171,213,175,209),(172,212,176,216),(177,207,181,203),(178,206,182,202),(179,205,183,201),(180,204,184,208)], [(1,47,50,177,174,151,189),(2,48,51,178,175,152,190),(3,41,52,179,176,145,191),(4,42,53,180,169,146,192),(5,43,54,181,170,147,185),(6,44,55,182,171,148,186),(7,45,56,183,172,149,187),(8,46,49,184,173,150,188),(9,20,96,85,199,137,134),(10,21,89,86,200,138,135),(11,22,90,87,193,139,136),(12,23,91,88,194,140,129),(13,24,92,81,195,141,130),(14,17,93,82,196,142,131),(15,18,94,83,197,143,132),(16,19,95,84,198,144,133),(25,156,167,72,61,34,80),(26,157,168,65,62,35,73),(27,158,161,66,63,36,74),(28,159,162,67,64,37,75),(29,160,163,68,57,38,76),(30,153,164,69,58,39,77),(31,154,165,70,59,40,78),(32,155,166,71,60,33,79),(97,108,208,211,114,125,220),(98,109,201,212,115,126,221),(99,110,202,213,116,127,222),(100,111,203,214,117,128,223),(101,112,204,215,118,121,224),(102,105,205,216,119,122,217),(103,106,206,209,120,123,218),(104,107,207,210,113,124,219)], [(1,189),(2,186),(3,191),(4,188),(5,185),(6,190),(7,187),(8,192),(9,13),(11,15),(17,131),(18,136),(19,133),(20,130),(21,135),(22,132),(23,129),(24,134),(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),(49,169),(50,174),(51,171),(52,176),(53,173),(54,170),(55,175),(56,172),(57,167),(58,164),(59,161),(60,166),(61,163),(62,168),(63,165),(64,162),(66,70),(68,72),(81,199),(82,196),(83,193),(84,198),(85,195),(86,200),(87,197),(88,194),(89,138),(90,143),(91,140),(92,137),(93,142),(94,139),(95,144),(96,141),(97,114),(98,119),(99,116),(100,113),(101,118),(102,115),(103,120),(104,117),(105,212),(106,209),(107,214),(108,211),(109,216),(110,213),(111,210),(112,215),(121,224),(122,221),(123,218),(124,223),(125,220),(126,217),(127,222),(128,219),(178,182),(180,184),(201,205),(203,207)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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ℤ
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