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## G = C2×Q16⋊D7order 448 = 26·7

### Direct product of C2 and Q16⋊D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×Q16⋊D7
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — C2×Q8×D7 — C2×Q16⋊D7
 Lower central C7 — C14 — C28 — C2×Q16⋊D7
 Upper central C1 — C22 — C2×C4 — C2×Q16

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, Q82D7, Q82D7, Q8×C14, C2×C8⋊D7, C2×C56⋊C2, Q16⋊D7, C2×Q8⋊D7, C2×C7⋊Q16, C14×Q16, C2×Q8×D7, C2×Q82D7, 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

Smallest permutation representation of 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

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