Copied to
clipboard

## G = D7×C8⋊C4order 448 = 26·7

### Direct product of D7 and C8⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D7×C8⋊C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — D7×C42 — D7×C8⋊C4
 Lower central C7 — C14 — D7×C8⋊C4
 Upper central C1 — C2×C4 — C8⋊C4

Generators and relations for D7×C8⋊C4
G = < a,b,c,d | a7=b2=c8=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 516 in 146 conjugacy classes, 83 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, D7, C14, C14, C42, C42, C2×C8, C2×C8, C22×C4, Dic7, Dic7, C28, C28, D14, C2×C14, C8⋊C4, C8⋊C4, C2×C42, C22×C8, C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C2×C8⋊C4, C8×D7, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4×C28, C2×C56, C2×C4×D7, C2×C4×D7, C42.D7, C56⋊C4, C7×C8⋊C4, D7×C42, D7×C2×C8, D7×C8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, M4(2), C22×C4, D14, C8⋊C4, C2×C42, C2×M4(2), C4×D7, C22×D7, C2×C8⋊C4, C2×C4×D7, D7×C42, D7×M4(2), D7×C8⋊C4

Smallest permutation representation of D7×C8⋊C4
On 224 points
Generators in S224
(1 146 73 42 53 198 187)(2 147 74 43 54 199 188)(3 148 75 44 55 200 189)(4 149 76 45 56 193 190)(5 150 77 46 49 194 191)(6 151 78 47 50 195 192)(7 152 79 48 51 196 185)(8 145 80 41 52 197 186)(9 119 106 62 174 24 209)(10 120 107 63 175 17 210)(11 113 108 64 176 18 211)(12 114 109 57 169 19 212)(13 115 110 58 170 20 213)(14 116 111 59 171 21 214)(15 117 112 60 172 22 215)(16 118 105 61 173 23 216)(25 121 101 154 65 163 36)(26 122 102 155 66 164 37)(27 123 103 156 67 165 38)(28 124 104 157 68 166 39)(29 125 97 158 69 167 40)(30 126 98 159 70 168 33)(31 127 99 160 71 161 34)(32 128 100 153 72 162 35)(81 224 202 177 90 136 141)(82 217 203 178 91 129 142)(83 218 204 179 92 130 143)(84 219 205 180 93 131 144)(85 220 206 181 94 132 137)(86 221 207 182 95 133 138)(87 222 208 183 96 134 139)(88 223 201 184 89 135 140)
(1 191)(2 192)(3 185)(4 186)(5 187)(6 188)(7 189)(8 190)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 113)(16 114)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 153)(40 154)(41 45)(42 46)(43 47)(44 48)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(65 167)(66 168)(67 161)(68 162)(69 163)(70 164)(71 165)(72 166)(81 220)(82 221)(83 222)(84 223)(85 224)(86 217)(87 218)(88 219)(89 93)(90 94)(91 95)(92 96)(105 212)(106 213)(107 214)(108 215)(109 216)(110 209)(111 210)(112 211)(121 125)(122 126)(123 127)(124 128)(129 182)(130 183)(131 184)(132 177)(133 178)(134 179)(135 180)(136 181)(137 202)(138 203)(139 204)(140 205)(141 206)(142 207)(143 208)(144 201)(145 193)(146 194)(147 195)(148 196)(149 197)(150 198)(151 199)(152 200)(169 173)(170 174)(171 175)(172 176)
(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)
(1 114 217 164)(2 119 218 161)(3 116 219 166)(4 113 220 163)(5 118 221 168)(6 115 222 165)(7 120 223 162)(8 117 224 167)(9 83 71 188)(10 88 72 185)(11 85 65 190)(12 82 66 187)(13 87 67 192)(14 84 68 189)(15 81 69 186)(16 86 70 191)(17 135 100 51)(18 132 101 56)(19 129 102 53)(20 134 103 50)(21 131 104 55)(22 136 97 52)(23 133 98 49)(24 130 99 54)(25 76 64 181)(26 73 57 178)(27 78 58 183)(28 75 59 180)(29 80 60 177)(30 77 61 182)(31 74 62 179)(32 79 63 184)(33 150 105 207)(34 147 106 204)(35 152 107 201)(36 149 108 206)(37 146 109 203)(38 151 110 208)(39 148 111 205)(40 145 112 202)(41 172 90 125)(42 169 91 122)(43 174 92 127)(44 171 93 124)(45 176 94 121)(46 173 95 126)(47 170 96 123)(48 175 89 128)(137 154 193 211)(138 159 194 216)(139 156 195 213)(140 153 196 210)(141 158 197 215)(142 155 198 212)(143 160 199 209)(144 157 200 214)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 7A 7B 7C 8A ··· 8H 8I ··· 8P 14A ··· 14I 28A ··· 28L 28M ··· 28X 56A ··· 56X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 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 7 7 7 7 1 1 1 1 2 2 2 2 7 7 7 7 14 14 14 14 2 2 2 2 ··· 2 14 ··· 14 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D7 M4(2) D14 D14 C4×D7 C4×D7 D7×M4(2) kernel D7×C8⋊C4 C42.D7 C56⋊C4 C7×C8⋊C4 D7×C42 D7×C2×C8 C8×D7 C4×Dic7 C2×C4×D7 C8⋊C4 D14 C42 C2×C8 C8 C2×C4 C2 # reps 1 1 2 1 1 2 16 4 4 3 8 3 6 24 12 12

Matrix representation of D7×C8⋊C4 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 79 1 0 0 112 0
,
 112 0 0 0 0 112 0 0 0 0 25 88 0 0 34 88
,
 29 72 0 0 76 84 0 0 0 0 15 0 0 0 0 15
,
 82 104 0 0 69 31 0 0 0 0 15 0 0 0 0 15
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,79,112,0,0,1,0],[112,0,0,0,0,112,0,0,0,0,25,34,0,0,88,88],[29,76,0,0,72,84,0,0,0,0,15,0,0,0,0,15],[82,69,0,0,104,31,0,0,0,0,15,0,0,0,0,15] >;

D7×C8⋊C4 in GAP, Magma, Sage, TeX

D_7\times C_8\rtimes C_4
% in TeX

G:=Group("D7xC8:C4");
// GroupNames label

G:=SmallGroup(448,238);
// by ID

G=gap.SmallGroup(448,238);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,387,58,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^8=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

׿
×
𝔽