metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×Q16)⋊2D7, (C7×Q8).8D4, (Q8×Dic7)⋊7C2, (C2×C8).37D14, Dic7⋊C8⋊29C2, (C14×Q16)⋊12C2, C28.185(C2×D4), (C2×Q8).61D14, Q8.3(C7⋊D4), C7⋊7(Q8.D4), C14.79(C4○D8), Q8⋊Dic7⋊32C2, (C2×Dic7).75D4, C22.276(D4×D7), C2.D56.12C2, C28.106(C4○D4), C4.14(D4⋊2D7), (C2×C56).251C22, (C2×C28).459C23, C28.23D4.5C2, (Q8×C14).88C22, C2.16(Q8.D14), C14.120(C4⋊D4), C2.27(Q16⋊D7), (C2×D28).124C22, C14.77(C8.C22), C4⋊Dic7.182C22, (C4×Dic7).55C22, C2.29(Dic7⋊D4), C4.47(C2×C7⋊D4), (C2×Q8⋊D7).8C2, (C2×C14).370(C2×D4), (C2×C7⋊C8).164C22, (C2×C4).547(C22×D7), SmallGroup(448,719)
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, ece=ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=ab-1, cd=dc, ede=d-1 >
Subgroups: 580 in 112 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C7⋊C8, C56, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, Q8.D4, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4⋊Dic7, C4⋊Dic7, D14⋊C4, Q8⋊D7, C2×C56, C7×Q16, C2×D28, Q8×C14, Dic7⋊C8, C2.D56, Q8⋊Dic7, C2×Q8⋊D7, Q8×Dic7, C28.23D4, C14×Q16, (C2×Q16)⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C4○D8, C8.C22, C7⋊D4, C22×D7, Q8.D4, D4×D7, D4⋊2D7, C2×C7⋊D4, Q16⋊D7, Q8.D14, Dic7⋊D4, (C2×Q16)⋊D7
►On 224 points
Generators in S224
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 113)(9 104)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 77)(18 78)(19 79)(20 80)(21 73)(22 74)(23 75)(24 76)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 127)(34 128)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 87)(42 88)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 205)(50 206)(51 207)(52 208)(53 201)(54 202)(55 203)(56 204)(57 193)(58 194)(59 195)(60 196)(61 197)(62 198)(63 199)(64 200)(65 222)(66 223)(67 224)(68 217)(69 218)(70 219)(71 220)(72 221)(105 166)(106 167)(107 168)(108 161)(109 162)(110 163)(111 164)(112 165)(129 214)(130 215)(131 216)(132 209)(133 210)(134 211)(135 212)(136 213)(137 186)(138 187)(139 188)(140 189)(141 190)(142 191)(143 192)(144 185)(145 169)(146 170)(147 171)(148 172)(149 173)(150 174)(151 175)(152 176)(153 181)(154 182)(155 183)(156 184)(157 177)(158 178)(159 179)(160 180)
(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)
(1 108 5 112)(2 107 6 111)(3 106 7 110)(4 105 8 109)(9 72 13 68)(10 71 14 67)(11 70 15 66)(12 69 16 65)(17 91 21 95)(18 90 22 94)(19 89 23 93)(20 96 24 92)(25 75 29 79)(26 74 30 78)(27 73 31 77)(28 80 32 76)(33 153 37 157)(34 160 38 156)(35 159 39 155)(36 158 40 154)(41 49 45 53)(42 56 46 52)(43 55 47 51)(44 54 48 50)(57 191 61 187)(58 190 62 186)(59 189 63 185)(60 188 64 192)(81 203 85 207)(82 202 86 206)(83 201 87 205)(84 208 88 204)(97 220 101 224)(98 219 102 223)(99 218 103 222)(100 217 104 221)(113 162 117 166)(114 161 118 165)(115 168 119 164)(116 167 120 163)(121 179 125 183)(122 178 126 182)(123 177 127 181)(124 184 128 180)(129 172 133 176)(130 171 134 175)(131 170 135 174)(132 169 136 173)(137 194 141 198)(138 193 142 197)(139 200 143 196)(140 199 144 195)(145 213 149 209)(146 212 150 216)(147 211 151 215)(148 210 152 214)
(1 60 96 171 203 123 72)(2 61 89 172 204 124 65)(3 62 90 173 205 125 66)(4 63 91 174 206 126 67)(5 64 92 175 207 127 68)(6 57 93 176 208 128 69)(7 58 94 169 201 121 70)(8 59 95 170 202 122 71)(9 112 192 20 130 81 181)(10 105 185 21 131 82 182)(11 106 186 22 132 83 183)(12 107 187 23 133 84 184)(13 108 188 24 134 85 177)(14 109 189 17 135 86 178)(15 110 190 18 136 87 179)(16 111 191 19 129 88 180)(25 148 56 38 222 115 197)(26 149 49 39 223 116 198)(27 150 50 40 224 117 199)(28 151 51 33 217 118 200)(29 152 52 34 218 119 193)(30 145 53 35 219 120 194)(31 146 54 36 220 113 195)(32 147 55 37 221 114 196)(41 159 102 163 141 78 213)(42 160 103 164 142 79 214)(43 153 104 165 143 80 215)(44 154 97 166 144 73 216)(45 155 98 167 137 74 209)(46 156 99 168 138 75 210)(47 157 100 161 139 76 211)(48 158 101 162 140 77 212)
(1 72)(2 220)(3 70)(4 218)(5 68)(6 224)(7 66)(8 222)(9 165)(10 111)(11 163)(12 109)(13 161)(14 107)(15 167)(16 105)(17 84)(18 45)(19 82)(20 43)(21 88)(22 41)(23 86)(24 47)(25 202)(26 53)(27 208)(28 51)(29 206)(30 49)(31 204)(32 55)(33 200)(34 63)(35 198)(36 61)(37 196)(38 59)(39 194)(40 57)(42 73)(44 79)(46 77)(48 75)(50 93)(52 91)(54 89)(56 95)(58 125)(60 123)(62 121)(64 127)(65 113)(67 119)(69 117)(71 115)(74 87)(76 85)(78 83)(80 81)(90 201)(92 207)(94 205)(96 203)(97 164)(98 110)(99 162)(100 108)(101 168)(102 106)(103 166)(104 112)(114 221)(116 219)(118 217)(120 223)(122 197)(124 195)(126 193)(128 199)(129 131)(130 215)(132 213)(133 135)(134 211)(136 209)(137 179)(138 158)(139 177)(140 156)(141 183)(142 154)(143 181)(144 160)(145 149)(146 172)(148 170)(150 176)(152 174)(153 192)(155 190)(157 188)(159 186)(169 173)(178 187)(180 185)(182 191)(184 189)(210 212)(214 216)
G:=sub<Sym(224)| (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,113)(9,104)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,77)(18,78)(19,79)(20,80)(21,73)(22,74)(23,75)(24,76)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,127)(34,128)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,87)(42,88)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,205)(50,206)(51,207)(52,208)(53,201)(54,202)(55,203)(56,204)(57,193)(58,194)(59,195)(60,196)(61,197)(62,198)(63,199)(64,200)(65,222)(66,223)(67,224)(68,217)(69,218)(70,219)(71,220)(72,221)(105,166)(106,167)(107,168)(108,161)(109,162)(110,163)(111,164)(112,165)(129,214)(130,215)(131,216)(132,209)(133,210)(134,211)(135,212)(136,213)(137,186)(138,187)(139,188)(140,189)(141,190)(142,191)(143,192)(144,185)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,181)(154,182)(155,183)(156,184)(157,177)(158,178)(159,179)(160,180), (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), (1,108,5,112)(2,107,6,111)(3,106,7,110)(4,105,8,109)(9,72,13,68)(10,71,14,67)(11,70,15,66)(12,69,16,65)(17,91,21,95)(18,90,22,94)(19,89,23,93)(20,96,24,92)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(33,153,37,157)(34,160,38,156)(35,159,39,155)(36,158,40,154)(41,49,45,53)(42,56,46,52)(43,55,47,51)(44,54,48,50)(57,191,61,187)(58,190,62,186)(59,189,63,185)(60,188,64,192)(81,203,85,207)(82,202,86,206)(83,201,87,205)(84,208,88,204)(97,220,101,224)(98,219,102,223)(99,218,103,222)(100,217,104,221)(113,162,117,166)(114,161,118,165)(115,168,119,164)(116,167,120,163)(121,179,125,183)(122,178,126,182)(123,177,127,181)(124,184,128,180)(129,172,133,176)(130,171,134,175)(131,170,135,174)(132,169,136,173)(137,194,141,198)(138,193,142,197)(139,200,143,196)(140,199,144,195)(145,213,149,209)(146,212,150,216)(147,211,151,215)(148,210,152,214), (1,60,96,171,203,123,72)(2,61,89,172,204,124,65)(3,62,90,173,205,125,66)(4,63,91,174,206,126,67)(5,64,92,175,207,127,68)(6,57,93,176,208,128,69)(7,58,94,169,201,121,70)(8,59,95,170,202,122,71)(9,112,192,20,130,81,181)(10,105,185,21,131,82,182)(11,106,186,22,132,83,183)(12,107,187,23,133,84,184)(13,108,188,24,134,85,177)(14,109,189,17,135,86,178)(15,110,190,18,136,87,179)(16,111,191,19,129,88,180)(25,148,56,38,222,115,197)(26,149,49,39,223,116,198)(27,150,50,40,224,117,199)(28,151,51,33,217,118,200)(29,152,52,34,218,119,193)(30,145,53,35,219,120,194)(31,146,54,36,220,113,195)(32,147,55,37,221,114,196)(41,159,102,163,141,78,213)(42,160,103,164,142,79,214)(43,153,104,165,143,80,215)(44,154,97,166,144,73,216)(45,155,98,167,137,74,209)(46,156,99,168,138,75,210)(47,157,100,161,139,76,211)(48,158,101,162,140,77,212), (1,72)(2,220)(3,70)(4,218)(5,68)(6,224)(7,66)(8,222)(9,165)(10,111)(11,163)(12,109)(13,161)(14,107)(15,167)(16,105)(17,84)(18,45)(19,82)(20,43)(21,88)(22,41)(23,86)(24,47)(25,202)(26,53)(27,208)(28,51)(29,206)(30,49)(31,204)(32,55)(33,200)(34,63)(35,198)(36,61)(37,196)(38,59)(39,194)(40,57)(42,73)(44,79)(46,77)(48,75)(50,93)(52,91)(54,89)(56,95)(58,125)(60,123)(62,121)(64,127)(65,113)(67,119)(69,117)(71,115)(74,87)(76,85)(78,83)(80,81)(90,201)(92,207)(94,205)(96,203)(97,164)(98,110)(99,162)(100,108)(101,168)(102,106)(103,166)(104,112)(114,221)(116,219)(118,217)(120,223)(122,197)(124,195)(126,193)(128,199)(129,131)(130,215)(132,213)(133,135)(134,211)(136,209)(137,179)(138,158)(139,177)(140,156)(141,183)(142,154)(143,181)(144,160)(145,149)(146,172)(148,170)(150,176)(152,174)(153,192)(155,190)(157,188)(159,186)(169,173)(178,187)(180,185)(182,191)(184,189)(210,212)(214,216)>;
G:=Group( (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,113)(9,104)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,77)(18,78)(19,79)(20,80)(21,73)(22,74)(23,75)(24,76)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,127)(34,128)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,87)(42,88)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,205)(50,206)(51,207)(52,208)(53,201)(54,202)(55,203)(56,204)(57,193)(58,194)(59,195)(60,196)(61,197)(62,198)(63,199)(64,200)(65,222)(66,223)(67,224)(68,217)(69,218)(70,219)(71,220)(72,221)(105,166)(106,167)(107,168)(108,161)(109,162)(110,163)(111,164)(112,165)(129,214)(130,215)(131,216)(132,209)(133,210)(134,211)(135,212)(136,213)(137,186)(138,187)(139,188)(140,189)(141,190)(142,191)(143,192)(144,185)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,181)(154,182)(155,183)(156,184)(157,177)(158,178)(159,179)(160,180), (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), (1,108,5,112)(2,107,6,111)(3,106,7,110)(4,105,8,109)(9,72,13,68)(10,71,14,67)(11,70,15,66)(12,69,16,65)(17,91,21,95)(18,90,22,94)(19,89,23,93)(20,96,24,92)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(33,153,37,157)(34,160,38,156)(35,159,39,155)(36,158,40,154)(41,49,45,53)(42,56,46,52)(43,55,47,51)(44,54,48,50)(57,191,61,187)(58,190,62,186)(59,189,63,185)(60,188,64,192)(81,203,85,207)(82,202,86,206)(83,201,87,205)(84,208,88,204)(97,220,101,224)(98,219,102,223)(99,218,103,222)(100,217,104,221)(113,162,117,166)(114,161,118,165)(115,168,119,164)(116,167,120,163)(121,179,125,183)(122,178,126,182)(123,177,127,181)(124,184,128,180)(129,172,133,176)(130,171,134,175)(131,170,135,174)(132,169,136,173)(137,194,141,198)(138,193,142,197)(139,200,143,196)(140,199,144,195)(145,213,149,209)(146,212,150,216)(147,211,151,215)(148,210,152,214), (1,60,96,171,203,123,72)(2,61,89,172,204,124,65)(3,62,90,173,205,125,66)(4,63,91,174,206,126,67)(5,64,92,175,207,127,68)(6,57,93,176,208,128,69)(7,58,94,169,201,121,70)(8,59,95,170,202,122,71)(9,112,192,20,130,81,181)(10,105,185,21,131,82,182)(11,106,186,22,132,83,183)(12,107,187,23,133,84,184)(13,108,188,24,134,85,177)(14,109,189,17,135,86,178)(15,110,190,18,136,87,179)(16,111,191,19,129,88,180)(25,148,56,38,222,115,197)(26,149,49,39,223,116,198)(27,150,50,40,224,117,199)(28,151,51,33,217,118,200)(29,152,52,34,218,119,193)(30,145,53,35,219,120,194)(31,146,54,36,220,113,195)(32,147,55,37,221,114,196)(41,159,102,163,141,78,213)(42,160,103,164,142,79,214)(43,153,104,165,143,80,215)(44,154,97,166,144,73,216)(45,155,98,167,137,74,209)(46,156,99,168,138,75,210)(47,157,100,161,139,76,211)(48,158,101,162,140,77,212), (1,72)(2,220)(3,70)(4,218)(5,68)(6,224)(7,66)(8,222)(9,165)(10,111)(11,163)(12,109)(13,161)(14,107)(15,167)(16,105)(17,84)(18,45)(19,82)(20,43)(21,88)(22,41)(23,86)(24,47)(25,202)(26,53)(27,208)(28,51)(29,206)(30,49)(31,204)(32,55)(33,200)(34,63)(35,198)(36,61)(37,196)(38,59)(39,194)(40,57)(42,73)(44,79)(46,77)(48,75)(50,93)(52,91)(54,89)(56,95)(58,125)(60,123)(62,121)(64,127)(65,113)(67,119)(69,117)(71,115)(74,87)(76,85)(78,83)(80,81)(90,201)(92,207)(94,205)(96,203)(97,164)(98,110)(99,162)(100,108)(101,168)(102,106)(103,166)(104,112)(114,221)(116,219)(118,217)(120,223)(122,197)(124,195)(126,193)(128,199)(129,131)(130,215)(132,213)(133,135)(134,211)(136,209)(137,179)(138,158)(139,177)(140,156)(141,183)(142,154)(143,181)(144,160)(145,149)(146,172)(148,170)(150,176)(152,174)(153,192)(155,190)(157,188)(159,186)(169,173)(178,187)(180,185)(182,191)(184,189)(210,212)(214,216) );
G=PermutationGroup([[(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,113),(9,104),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,77),(18,78),(19,79),(20,80),(21,73),(22,74),(23,75),(24,76),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,127),(34,128),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,87),(42,88),(43,81),(44,82),(45,83),(46,84),(47,85),(48,86),(49,205),(50,206),(51,207),(52,208),(53,201),(54,202),(55,203),(56,204),(57,193),(58,194),(59,195),(60,196),(61,197),(62,198),(63,199),(64,200),(65,222),(66,223),(67,224),(68,217),(69,218),(70,219),(71,220),(72,221),(105,166),(106,167),(107,168),(108,161),(109,162),(110,163),(111,164),(112,165),(129,214),(130,215),(131,216),(132,209),(133,210),(134,211),(135,212),(136,213),(137,186),(138,187),(139,188),(140,189),(141,190),(142,191),(143,192),(144,185),(145,169),(146,170),(147,171),(148,172),(149,173),(150,174),(151,175),(152,176),(153,181),(154,182),(155,183),(156,184),(157,177),(158,178),(159,179),(160,180)], [(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)], [(1,108,5,112),(2,107,6,111),(3,106,7,110),(4,105,8,109),(9,72,13,68),(10,71,14,67),(11,70,15,66),(12,69,16,65),(17,91,21,95),(18,90,22,94),(19,89,23,93),(20,96,24,92),(25,75,29,79),(26,74,30,78),(27,73,31,77),(28,80,32,76),(33,153,37,157),(34,160,38,156),(35,159,39,155),(36,158,40,154),(41,49,45,53),(42,56,46,52),(43,55,47,51),(44,54,48,50),(57,191,61,187),(58,190,62,186),(59,189,63,185),(60,188,64,192),(81,203,85,207),(82,202,86,206),(83,201,87,205),(84,208,88,204),(97,220,101,224),(98,219,102,223),(99,218,103,222),(100,217,104,221),(113,162,117,166),(114,161,118,165),(115,168,119,164),(116,167,120,163),(121,179,125,183),(122,178,126,182),(123,177,127,181),(124,184,128,180),(129,172,133,176),(130,171,134,175),(131,170,135,174),(132,169,136,173),(137,194,141,198),(138,193,142,197),(139,200,143,196),(140,199,144,195),(145,213,149,209),(146,212,150,216),(147,211,151,215),(148,210,152,214)], [(1,60,96,171,203,123,72),(2,61,89,172,204,124,65),(3,62,90,173,205,125,66),(4,63,91,174,206,126,67),(5,64,92,175,207,127,68),(6,57,93,176,208,128,69),(7,58,94,169,201,121,70),(8,59,95,170,202,122,71),(9,112,192,20,130,81,181),(10,105,185,21,131,82,182),(11,106,186,22,132,83,183),(12,107,187,23,133,84,184),(13,108,188,24,134,85,177),(14,109,189,17,135,86,178),(15,110,190,18,136,87,179),(16,111,191,19,129,88,180),(25,148,56,38,222,115,197),(26,149,49,39,223,116,198),(27,150,50,40,224,117,199),(28,151,51,33,217,118,200),(29,152,52,34,218,119,193),(30,145,53,35,219,120,194),(31,146,54,36,220,113,195),(32,147,55,37,221,114,196),(41,159,102,163,141,78,213),(42,160,103,164,142,79,214),(43,153,104,165,143,80,215),(44,154,97,166,144,73,216),(45,155,98,167,137,74,209),(46,156,99,168,138,75,210),(47,157,100,161,139,76,211),(48,158,101,162,140,77,212)], [(1,72),(2,220),(3,70),(4,218),(5,68),(6,224),(7,66),(8,222),(9,165),(10,111),(11,163),(12,109),(13,161),(14,107),(15,167),(16,105),(17,84),(18,45),(19,82),(20,43),(21,88),(22,41),(23,86),(24,47),(25,202),(26,53),(27,208),(28,51),(29,206),(30,49),(31,204),(32,55),(33,200),(34,63),(35,198),(36,61),(37,196),(38,59),(39,194),(40,57),(42,73),(44,79),(46,77),(48,75),(50,93),(52,91),(54,89),(56,95),(58,125),(60,123),(62,121),(64,127),(65,113),(67,119),(69,117),(71,115),(74,87),(76,85),(78,83),(80,81),(90,201),(92,207),(94,205),(96,203),(97,164),(98,110),(99,162),(100,108),(101,168),(102,106),(103,166),(104,112),(114,221),(116,219),(118,217),(120,223),(122,197),(124,195),(126,193),(128,199),(129,131),(130,215),(132,213),(133,135),(134,211),(136,209),(137,179),(138,158),(139,177),(140,156),(141,183),(142,154),(143,181),(144,160),(145,149),(146,172),(148,170),(150,176),(152,174),(153,192),(155,190),(157,188),(159,186),(169,173),(178,187),(180,185),(182,191),(184,189),(210,212),(214,216)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 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 | 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 | 56 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 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 | Q16⋊D7 | Q8.D14 |
| kernel | (C2×Q16)⋊D7 | Dic7⋊C8 | C2.D56 | Q8⋊Dic7 | C2×Q8⋊D7 | Q8×Dic7 | C28.23D4 | C14×Q16 | C2×Dic7 | C7×Q8 | C2×Q16 | C28 | C2×C8 | C2×Q8 | C14 | Q8 | 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×Q16)⋊D7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 91 | 12 | 0 | 0 |
| 101 | 22 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 31 | 62 |
| 91 | 12 | 0 | 0 |
| 101 | 22 | 0 | 0 |
| 0 | 0 | 98 | 83 |
| 0 | 0 | 0 | 15 |
| 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 | 2 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[91,101,0,0,12,22,0,0,0,0,0,31,0,0,51,62],[91,101,0,0,12,22,0,0,0,0,98,0,0,0,83,15],[0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,112] >;
(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(448,719);
// by ID
G=gap.SmallGroup(448,719);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,1094,135,184,570,297,136,18822]);
// 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,e*c*e=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=a*b^-1,c*d=d*c,e*d*e=d^-1>;
// generators/relations