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## G = (C22×C8)⋊D7order 448 = 26·7

### 3rd semidirect product of C22×C8 and D7 acting via D7/C7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (C22×C8)⋊D7
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — C2×C4○D28 — (C22×C8)⋊D7
 Lower central C7 — C2×C14 — (C22×C8)⋊D7
 Upper central C1 — C2×C4 — C22×C8

Generators and relations for (C22×C8)⋊D7
G = < a,b,c,d,e | a2=b2=c8=d7=e2=1, ab=ba, ece=ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc4, cd=dc, ede=d-1 >

Subgroups: 740 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, (C22×C8)⋊C2, C2×C7⋊C8, C4.Dic7, C2×C56, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, D14⋊C8, C2×C4.Dic7, C22×C56, C2×C4○D28, (C22×C8)⋊D7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C8○D4, C4×D7, D28, C7⋊D4, C22×D7, (C22×C8)⋊C2, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, D28.2C4, C2×D14⋊C4, (C22×C8)⋊D7

Smallest permutation representation of (C22×C8)⋊D7
On 224 points
Generators in S224
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 165)(10 166)(11 167)(12 168)(13 161)(14 162)(15 163)(16 164)(17 110)(18 111)(19 112)(20 105)(21 106)(22 107)(23 108)(24 109)(25 150)(26 151)(27 152)(28 145)(29 146)(30 147)(31 148)(32 149)(33 185)(34 186)(35 187)(36 188)(37 189)(38 190)(39 191)(40 192)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 177)(56 178)(57 196)(58 197)(59 198)(60 199)(61 200)(62 193)(63 194)(64 195)(65 173)(66 174)(67 175)(68 176)(69 169)(70 170)(71 171)(72 172)(73 144)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 113)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 121)(97 134)(98 135)(99 136)(100 129)(101 130)(102 131)(103 132)(104 133)(153 206)(154 207)(155 208)(156 201)(157 202)(158 203)(159 204)(160 205)(209 218)(210 219)(211 220)(212 221)(213 222)(214 223)(215 224)(216 217)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 165)(10 166)(11 167)(12 168)(13 161)(14 162)(15 163)(16 164)(17 110)(18 111)(19 112)(20 105)(21 106)(22 107)(23 108)(24 109)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 145)(33 189)(34 190)(35 191)(36 192)(37 185)(38 186)(39 187)(40 188)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 177)(56 178)(57 200)(58 193)(59 194)(60 195)(61 196)(62 197)(63 198)(64 199)(65 173)(66 174)(67 175)(68 176)(69 169)(70 170)(71 171)(72 172)(73 144)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 118)(82 119)(83 120)(84 113)(85 114)(86 115)(87 116)(88 117)(89 126)(90 127)(91 128)(92 121)(93 122)(94 123)(95 124)(96 125)(97 134)(98 135)(99 136)(100 129)(101 130)(102 131)(103 132)(104 133)(153 202)(154 203)(155 204)(156 205)(157 206)(158 207)(159 208)(160 201)(209 222)(210 223)(211 224)(212 217)(213 218)(214 219)(215 220)(216 221)
(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 9 78 179 176 99 20)(2 10 79 180 169 100 21)(3 11 80 181 170 101 22)(4 12 73 182 171 102 23)(5 13 74 183 172 103 24)(6 14 75 184 173 104 17)(7 15 76 177 174 97 18)(8 16 77 178 175 98 19)(25 36 199 125 160 209 116)(26 37 200 126 153 210 117)(27 38 193 127 154 211 118)(28 39 194 128 155 212 119)(29 40 195 121 156 213 120)(30 33 196 122 157 214 113)(31 34 197 123 158 215 114)(32 35 198 124 159 216 115)(41 163 139 55 66 134 111)(42 164 140 56 67 135 112)(43 165 141 49 68 136 105)(44 166 142 50 69 129 106)(45 167 143 51 70 130 107)(46 168 144 52 71 131 108)(47 161 137 53 72 132 109)(48 162 138 54 65 133 110)(57 89 202 223 88 147 185)(58 90 203 224 81 148 186)(59 91 204 217 82 149 187)(60 92 205 218 83 150 188)(61 93 206 219 84 151 189)(62 94 207 220 85 152 190)(63 95 208 221 86 145 191)(64 96 201 222 87 146 192)
(1 91)(2 125)(3 93)(4 127)(5 95)(6 121)(7 89)(8 123)(9 59)(10 199)(11 61)(12 193)(13 63)(14 195)(15 57)(16 197)(17 156)(18 202)(19 158)(20 204)(21 160)(22 206)(23 154)(24 208)(25 180)(26 51)(27 182)(28 53)(29 184)(30 55)(31 178)(32 49)(33 139)(34 77)(35 141)(36 79)(37 143)(38 73)(39 137)(40 75)(41 122)(42 90)(43 124)(44 92)(45 126)(46 94)(47 128)(48 96)(50 150)(52 152)(54 146)(56 148)(58 164)(60 166)(62 168)(64 162)(65 87)(66 113)(67 81)(68 115)(69 83)(70 117)(71 85)(72 119)(74 191)(76 185)(78 187)(80 189)(82 176)(84 170)(86 172)(88 174)(97 223)(98 215)(99 217)(100 209)(101 219)(102 211)(103 221)(104 213)(105 159)(106 205)(107 153)(108 207)(109 155)(110 201)(111 157)(112 203)(114 175)(116 169)(118 171)(120 173)(129 218)(130 210)(131 220)(132 212)(133 222)(134 214)(135 224)(136 216)(138 192)(140 186)(142 188)(144 190)(145 183)(147 177)(149 179)(151 181)(161 194)(163 196)(165 198)(167 200)

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

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

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

124 conjugacy classes

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

124 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D7 D14 D14 C8○D4 C4×D7 D28 C7⋊D4 C4×D7 D28.2C4 kernel (C22×C8)⋊D7 D14⋊C8 C2×C4.Dic7 C22×C56 C2×C4○D28 C2×Dic14 C2×D28 C2×C7⋊D4 C2×C28 C22×C8 C2×C8 C22×C4 C14 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 4 3 6 3 8 6 12 12 6 48

Matrix representation of (C22×C8)⋊D7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 1 0 0 0 0 1 0 0 0 54 112
,
 95 0 0 0 0 18 0 0 0 0 95 0 0 0 0 95
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 73 109
,
 0 1 0 0 1 0 0 0 0 0 91 5 0 0 39 22
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,1,0,0,0,0,1,54,0,0,0,112],[95,0,0,0,0,18,0,0,0,0,95,0,0,0,0,95],[1,0,0,0,0,1,0,0,0,0,28,73,0,0,0,109],[0,1,0,0,1,0,0,0,0,0,91,39,0,0,5,22] >;

(C22×C8)⋊D7 in GAP, Magma, Sage, TeX

(C_2^2\times C_8)\rtimes D_7
% in TeX

G:=Group("(C2^2xC8):D7");
// GroupNames label

G:=SmallGroup(448,644);
// by ID

G=gap.SmallGroup(448,644);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,422,58,136,18822]);
// Polycyclic

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

׿
×
𝔽