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## G = C7×C8⋊6D4order 448 = 26·7

### Direct product of C7 and C8⋊6D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C8⋊6D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C2×C56 — C7×C22⋊C8 — C7×C8⋊6D4
 Lower central C1 — C22 — C7×C8⋊6D4
 Upper central C1 — C2×C28 — C7×C8⋊6D4

Generators and relations for C7×C86D4
G = < a,b,c,d | a7=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 178 in 122 conjugacy classes, 74 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C28, C28, C28, C2×C14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C56, C56, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C86D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C7×M4(2), C22×C28, D4×C14, C4×C56, C7×C22⋊C8, C7×C4⋊C8, D4×C28, C14×M4(2), C7×C86D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, M4(2), C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C2×M4(2), C8○D4, C2×C28, C7×D4, C22×C14, C86D4, C7×M4(2), C22×C28, D4×C14, C7×C4○D4, D4×C28, C14×M4(2), C7×C8○D4, C7×C86D4

Smallest permutation representation of C7×C86D4
On 224 points
Generators in S224
(1 67 12 87 31 79 23)(2 68 13 88 32 80 24)(3 69 14 81 25 73 17)(4 70 15 82 26 74 18)(5 71 16 83 27 75 19)(6 72 9 84 28 76 20)(7 65 10 85 29 77 21)(8 66 11 86 30 78 22)(33 120 105 49 97 41 89)(34 113 106 50 98 42 90)(35 114 107 51 99 43 91)(36 115 108 52 100 44 92)(37 116 109 53 101 45 93)(38 117 110 54 102 46 94)(39 118 111 55 103 47 95)(40 119 112 56 104 48 96)(57 221 164 213 156 205 148)(58 222 165 214 157 206 149)(59 223 166 215 158 207 150)(60 224 167 216 159 208 151)(61 217 168 209 160 201 152)(62 218 161 210 153 202 145)(63 219 162 211 154 203 146)(64 220 163 212 155 204 147)(121 173 194 137 186 129 178)(122 174 195 138 187 130 179)(123 175 196 139 188 131 180)(124 176 197 140 189 132 181)(125 169 198 141 190 133 182)(126 170 199 142 191 134 183)(127 171 200 143 192 135 184)(128 172 193 144 185 136 177)
(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)
(1 128 35 150)(2 121 36 151)(3 122 37 152)(4 123 38 145)(5 124 39 146)(6 125 40 147)(7 126 33 148)(8 127 34 149)(9 198 112 220)(10 199 105 221)(11 200 106 222)(12 193 107 223)(13 194 108 224)(14 195 109 217)(15 196 110 218)(16 197 111 219)(17 179 93 201)(18 180 94 202)(19 181 95 203)(20 182 96 204)(21 183 89 205)(22 184 90 206)(23 177 91 207)(24 178 92 208)(25 187 101 209)(26 188 102 210)(27 189 103 211)(28 190 104 212)(29 191 97 213)(30 192 98 214)(31 185 99 215)(32 186 100 216)(41 156 77 134)(42 157 78 135)(43 158 79 136)(44 159 80 129)(45 160 73 130)(46 153 74 131)(47 154 75 132)(48 155 76 133)(49 164 85 142)(50 165 86 143)(51 166 87 144)(52 167 88 137)(53 168 81 138)(54 161 82 139)(55 162 83 140)(56 163 84 141)(57 65 170 120)(58 66 171 113)(59 67 172 114)(60 68 173 115)(61 69 174 116)(62 70 175 117)(63 71 176 118)(64 72 169 119)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(57 170)(58 175)(59 172)(60 169)(61 174)(62 171)(63 176)(64 173)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(90 94)(92 96)(98 102)(100 104)(106 110)(108 112)(113 117)(115 119)(121 147)(122 152)(123 149)(124 146)(125 151)(126 148)(127 145)(128 150)(129 155)(130 160)(131 157)(132 154)(133 159)(134 156)(135 153)(136 158)(137 163)(138 168)(139 165)(140 162)(141 167)(142 164)(143 161)(144 166)(177 207)(178 204)(179 201)(180 206)(181 203)(182 208)(183 205)(184 202)(185 215)(186 212)(187 209)(188 214)(189 211)(190 216)(191 213)(192 210)(193 223)(194 220)(195 217)(196 222)(197 219)(198 224)(199 221)(200 218)

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

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

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

196 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A ··· 7F 8A ··· 8H 8I 8J 8K 8L 14A ··· 14R 14S ··· 14AD 28A ··· 28X 28Y ··· 28AV 28AW ··· 28BH 56A ··· 56AV 56AW ··· 56BT order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 7 ··· 7 8 ··· 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 1 1 4 4 1 1 1 1 2 2 2 2 4 4 1 ··· 1 2 ··· 2 4 4 4 4 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C7 C14 C14 C14 C14 C14 C28 C28 C28 D4 M4(2) C4○D4 C8○D4 C7×D4 C7×M4(2) C7×C4○D4 C7×C8○D4 kernel C7×C8⋊6D4 C4×C56 C7×C22⋊C8 C7×C4⋊C8 D4×C28 C14×M4(2) C7×C22⋊C4 C7×C4⋊C4 D4×C14 C8⋊6D4 C4×C8 C22⋊C8 C4⋊C8 C4×D4 C2×M4(2) C22⋊C4 C4⋊C4 C2×D4 C56 C28 C28 C14 C8 C4 C4 C2 # reps 1 1 2 1 1 2 4 2 2 6 6 12 6 6 12 24 12 12 2 4 2 4 12 24 12 24

Matrix representation of C7×C86D4 in GL4(𝔽113) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 89 82 0 0 30 24 0 0 0 0 15 0 0 0 0 15
,
 93 106 0 0 25 20 0 0 0 0 0 112 0 0 1 0
,
 1 92 0 0 0 112 0 0 0 0 1 0 0 0 0 112
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[89,30,0,0,82,24,0,0,0,0,15,0,0,0,0,15],[93,25,0,0,106,20,0,0,0,0,0,1,0,0,112,0],[1,0,0,0,92,112,0,0,0,0,1,0,0,0,0,112] >;

C7×C86D4 in GAP, Magma, Sage, TeX

C_7\times C_8\rtimes_6D_4
% in TeX

G:=Group("C7xC8:6D4");
// GroupNames label

G:=SmallGroup(448,844);
// by ID

G=gap.SmallGroup(448,844);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,4790,2403,604,124]);
// Polycyclic

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

׿
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𝔽