Copied to
clipboard

## G = C2×D7×Q16order 448 = 26·7

### Direct product of C2, D7 and Q16

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×D7×Q16
G = < a,b,c,d,e | a2=b7=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 1124 in 258 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, Q16, Q16, C22×C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×C14, C22×C8, C2×Q16, C2×Q16, C22×Q8, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×Q16, C8×D7, Dic28, C2×C7⋊C8, C7⋊Q16, C2×C56, C7×Q16, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, Q8×D7, Q8×D7, Q8×C14, D7×C2×C8, C2×Dic28, D7×Q16, C2×C7⋊Q16, C14×Q16, C2×Q8×D7, C2×D7×Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, C24, D14, C2×Q16, C22×D4, C22×D7, C22×Q16, D4×D7, C23×D7, D7×Q16, C2×D4×D7, C2×D7×Q16

Smallest permutation representation of C2×D7×Q16
On 224 points
Generators in S224
(1 164)(2 165)(3 166)(4 167)(5 168)(6 161)(7 162)(8 163)(9 156)(10 157)(11 158)(12 159)(13 160)(14 153)(15 154)(16 155)(17 150)(18 151)(19 152)(20 145)(21 146)(22 147)(23 148)(24 149)(25 141)(26 142)(27 143)(28 144)(29 137)(30 138)(31 139)(32 140)(33 111)(34 112)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 89)(56 90)(57 213)(58 214)(59 215)(60 216)(61 209)(62 210)(63 211)(64 212)(65 221)(66 222)(67 223)(68 224)(69 217)(70 218)(71 219)(72 220)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(113 184)(114 177)(115 178)(116 179)(117 180)(118 181)(119 182)(120 183)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 169)(129 185)(130 186)(131 187)(132 188)(133 189)(134 190)(135 191)(136 192)
(1 194 219 17 14 113 103)(2 195 220 18 15 114 104)(3 196 221 19 16 115 97)(4 197 222 20 9 116 98)(5 198 223 21 10 117 99)(6 199 224 22 11 118 100)(7 200 217 23 12 119 101)(8 193 218 24 13 120 102)(25 136 112 96 128 207 210)(26 129 105 89 121 208 211)(27 130 106 90 122 201 212)(28 131 107 91 123 202 213)(29 132 108 92 124 203 214)(30 133 109 93 125 204 215)(31 134 110 94 126 205 216)(32 135 111 95 127 206 209)(33 53 176 86 61 140 191)(34 54 169 87 62 141 192)(35 55 170 88 63 142 185)(36 56 171 81 64 143 186)(37 49 172 82 57 144 187)(38 50 173 83 58 137 188)(39 51 174 84 59 138 189)(40 52 175 85 60 139 190)(41 166 76 65 152 155 178)(42 167 77 66 145 156 179)(43 168 78 67 146 157 180)(44 161 79 68 147 158 181)(45 162 80 69 148 159 182)(46 163 73 70 149 160 183)(47 164 74 71 150 153 184)(48 165 75 72 151 154 177)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 224)(16 217)(17 21)(18 22)(19 23)(20 24)(25 214)(26 215)(27 216)(28 209)(29 210)(30 211)(31 212)(32 213)(33 172)(34 173)(35 174)(36 175)(37 176)(38 169)(39 170)(40 171)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 161)(49 53)(50 54)(51 55)(52 56)(57 140)(58 141)(59 142)(60 143)(61 144)(62 137)(63 138)(64 139)(65 159)(66 160)(67 153)(68 154)(69 155)(70 156)(71 157)(72 158)(73 179)(74 180)(75 181)(76 182)(77 183)(78 184)(79 177)(80 178)(81 190)(82 191)(83 192)(84 185)(85 186)(86 187)(87 188)(88 189)(89 93)(90 94)(91 95)(92 96)(105 125)(106 126)(107 127)(108 128)(109 121)(110 122)(111 123)(112 124)(113 198)(114 199)(115 200)(116 193)(117 194)(118 195)(119 196)(120 197)(129 204)(130 205)(131 206)(132 207)(133 208)(134 201)(135 202)(136 203)(145 149)(146 150)(147 151)(148 152)
(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)
(1 144 5 140)(2 143 6 139)(3 142 7 138)(4 141 8 137)(9 169 13 173)(10 176 14 172)(11 175 15 171)(12 174 16 170)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 163 29 167)(26 162 30 166)(27 161 31 165)(28 168 32 164)(33 219 37 223)(34 218 38 222)(35 217 39 221)(36 224 40 220)(41 211 45 215)(42 210 46 214)(43 209 47 213)(44 216 48 212)(57 99 61 103)(58 98 62 102)(59 97 63 101)(60 104 64 100)(65 105 69 109)(66 112 70 108)(67 111 71 107)(68 110 72 106)(73 132 77 136)(74 131 78 135)(75 130 79 134)(76 129 80 133)(81 118 85 114)(82 117 86 113)(83 116 87 120)(84 115 88 119)(89 148 93 152)(90 147 94 151)(91 146 95 150)(92 145 96 149)(121 159 125 155)(122 158 126 154)(123 157 127 153)(124 156 128 160)(177 201 181 205)(178 208 182 204)(179 207 183 203)(180 206 184 202)(185 200 189 196)(186 199 190 195)(187 198 191 194)(188 197 192 193)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 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 4 4 7 7 7 8 8 8 8 8 8 8 8 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 7 7 7 7 2 2 4 4 4 4 14 14 28 28 28 28 2 2 2 2 2 2 2 14 14 14 14 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 Q16 D14 D14 D14 D4×D7 D4×D7 D7×Q16 kernel C2×D7×Q16 D7×C2×C8 C2×Dic28 D7×Q16 C2×C7⋊Q16 C14×Q16 C2×Q8×D7 C4×D7 C2×Dic7 C22×D7 C2×Q16 D14 C2×C8 Q16 C2×Q8 C4 C22 C2 # reps 1 1 1 8 2 1 2 2 1 1 3 8 3 12 6 3 3 12

Matrix representation of C2×D7×Q16 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 104 1 0 0 41 33 0 0 0 0 1 0 0 0 0 1
,
 33 112 0 0 71 80 0 0 0 0 112 0 0 0 0 112
,
 1 0 0 0 0 1 0 0 0 0 51 88 0 0 104 0
,
 1 0 0 0 0 1 0 0 0 0 30 110 0 0 112 83
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[104,41,0,0,1,33,0,0,0,0,1,0,0,0,0,1],[33,71,0,0,112,80,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,51,104,0,0,88,0],[1,0,0,0,0,1,0,0,0,0,30,112,0,0,110,83] >;

C2×D7×Q16 in GAP, Magma, Sage, TeX

C_2\times D_7\times Q_{16}
% in TeX

G:=Group("C2xD7xQ16");
// GroupNames label

G:=SmallGroup(448,1216);
// by ID

G=gap.SmallGroup(448,1216);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,185,136,438,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^7=c^2=d^8=1,e^2=d^4,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,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽