metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊C8⋊2D4, C7⋊1(C8⋊2D4), C28⋊2D4⋊2C2, C4⋊D28⋊4C2, C4⋊C4.13D14, C4.161(D4×D7), D4⋊C4⋊19D7, (C2×D4).29D14, C28.111(C2×D4), (C2×C8).170D14, C4.Dic14⋊6C2, C2.D56⋊20C2, C4.27(C4○D28), C28.10(C4○D4), (C2×Dic7).23D4, (C22×D7).14D4, C22.180(D4×D7), C2.18(D8⋊D7), C14.18(C4⋊D4), C2.11(D56⋊C2), C14.36(C8⋊C22), (C2×C28).222C23, (C2×C56).187C22, (C2×D28).52C22, (D4×C14).43C22, C4⋊Dic7.75C22, C2.21(D14⋊D4), (C2×D4⋊D7)⋊5C2, (C2×C8⋊D7)⋊18C2, (C2×C7⋊C8).20C22, (C7×D4⋊C4)⋊25C2, (C2×C4×D7).14C22, (C2×C14).235(C2×D4), (C7×C4⋊C4).23C22, (C2×C4).329(C22×D7), SmallGroup(448,316)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊C8⋊D4
G = < a,b,c,d | a7=b8=c4=d2=1, bab-1=cac-1=dad=a-1, cbc-1=b3, dbd=b5, dcd=c-1 >
Subgroups: 852 in 130 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C7⋊C8, C56, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C8⋊2D4, C8⋊D7, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, D4⋊D7, C23.D7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C2×D28, C2×C7⋊D4, D4×C14, C4.Dic14, C2.D56, C7×D4⋊C4, C4⋊D28, C2×C8⋊D7, C2×D4⋊D7, C28⋊2D4, C7⋊C8⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, C22×D7, C8⋊2D4, C4○D28, D4×D7, D14⋊D4, D8⋊D7, D56⋊C2, C7⋊C8⋊D4
►On 224 points
Generators in S224
(1 141 76 178 203 44 187)(2 188 45 204 179 77 142)(3 143 78 180 205 46 189)(4 190 47 206 181 79 144)(5 137 80 182 207 48 191)(6 192 41 208 183 73 138)(7 139 74 184 201 42 185)(8 186 43 202 177 75 140)(9 68 158 119 111 167 27)(10 28 168 112 120 159 69)(11 70 160 113 105 161 29)(12 30 162 106 114 153 71)(13 72 154 115 107 163 31)(14 32 164 108 116 155 65)(15 66 156 117 109 165 25)(16 26 166 110 118 157 67)(17 171 37 62 103 121 214)(18 215 122 104 63 38 172)(19 173 39 64 97 123 216)(20 209 124 98 57 40 174)(21 175 33 58 99 125 210)(22 211 126 100 59 34 176)(23 169 35 60 101 127 212)(24 213 128 102 61 36 170)(49 149 129 81 96 217 200)(50 193 218 89 82 130 150)(51 151 131 83 90 219 194)(52 195 220 91 84 132 152)(53 145 133 85 92 221 196)(54 197 222 93 86 134 146)(55 147 135 87 94 223 198)(56 199 224 95 88 136 148)
(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)
(1 124 221 161)(2 127 222 164)(3 122 223 167)(4 125 224 162)(5 128 217 165)(6 123 218 168)(7 126 219 163)(8 121 220 166)(9 46 63 87)(10 41 64 82)(11 44 57 85)(12 47 58 88)(13 42 59 83)(14 45 60 86)(15 48 61 81)(16 43 62 84)(17 52 118 75)(18 55 119 78)(19 50 120 73)(20 53 113 76)(21 56 114 79)(22 51 115 74)(23 54 116 77)(24 49 117 80)(25 191 102 96)(26 186 103 91)(27 189 104 94)(28 192 97 89)(29 187 98 92)(30 190 99 95)(31 185 100 90)(32 188 101 93)(33 136 71 206)(34 131 72 201)(35 134 65 204)(36 129 66 207)(37 132 67 202)(38 135 68 205)(39 130 69 208)(40 133 70 203)(105 141 209 196)(106 144 210 199)(107 139 211 194)(108 142 212 197)(109 137 213 200)(110 140 214 195)(111 143 215 198)(112 138 216 193)(145 160 178 174)(146 155 179 169)(147 158 180 172)(148 153 181 175)(149 156 182 170)(150 159 183 173)(151 154 184 176)(152 157 177 171)
(1 217)(2 222)(3 219)(4 224)(5 221)(6 218)(7 223)(8 220)(9 115)(10 120)(11 117)(12 114)(13 119)(14 116)(15 113)(16 118)(17 62)(18 59)(19 64)(20 61)(21 58)(22 63)(23 60)(24 57)(25 105)(26 110)(27 107)(28 112)(29 109)(30 106)(31 111)(32 108)(33 175)(34 172)(35 169)(36 174)(37 171)(38 176)(39 173)(40 170)(41 50)(42 55)(43 52)(44 49)(45 54)(46 51)(47 56)(48 53)(65 155)(66 160)(67 157)(68 154)(69 159)(70 156)(71 153)(72 158)(73 82)(74 87)(75 84)(76 81)(77 86)(78 83)(79 88)(80 85)(89 138)(90 143)(91 140)(92 137)(93 142)(94 139)(95 144)(96 141)(97 216)(98 213)(99 210)(100 215)(101 212)(102 209)(103 214)(104 211)(122 126)(124 128)(129 178)(130 183)(131 180)(132 177)(133 182)(134 179)(135 184)(136 181)(145 207)(146 204)(147 201)(148 206)(149 203)(150 208)(151 205)(152 202)(161 165)(163 167)(185 198)(186 195)(187 200)(188 197)(189 194)(190 199)(191 196)(192 193)
G:=sub<Sym(224)| (1,141,76,178,203,44,187)(2,188,45,204,179,77,142)(3,143,78,180,205,46,189)(4,190,47,206,181,79,144)(5,137,80,182,207,48,191)(6,192,41,208,183,73,138)(7,139,74,184,201,42,185)(8,186,43,202,177,75,140)(9,68,158,119,111,167,27)(10,28,168,112,120,159,69)(11,70,160,113,105,161,29)(12,30,162,106,114,153,71)(13,72,154,115,107,163,31)(14,32,164,108,116,155,65)(15,66,156,117,109,165,25)(16,26,166,110,118,157,67)(17,171,37,62,103,121,214)(18,215,122,104,63,38,172)(19,173,39,64,97,123,216)(20,209,124,98,57,40,174)(21,175,33,58,99,125,210)(22,211,126,100,59,34,176)(23,169,35,60,101,127,212)(24,213,128,102,61,36,170)(49,149,129,81,96,217,200)(50,193,218,89,82,130,150)(51,151,131,83,90,219,194)(52,195,220,91,84,132,152)(53,145,133,85,92,221,196)(54,197,222,93,86,134,146)(55,147,135,87,94,223,198)(56,199,224,95,88,136,148), (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), (1,124,221,161)(2,127,222,164)(3,122,223,167)(4,125,224,162)(5,128,217,165)(6,123,218,168)(7,126,219,163)(8,121,220,166)(9,46,63,87)(10,41,64,82)(11,44,57,85)(12,47,58,88)(13,42,59,83)(14,45,60,86)(15,48,61,81)(16,43,62,84)(17,52,118,75)(18,55,119,78)(19,50,120,73)(20,53,113,76)(21,56,114,79)(22,51,115,74)(23,54,116,77)(24,49,117,80)(25,191,102,96)(26,186,103,91)(27,189,104,94)(28,192,97,89)(29,187,98,92)(30,190,99,95)(31,185,100,90)(32,188,101,93)(33,136,71,206)(34,131,72,201)(35,134,65,204)(36,129,66,207)(37,132,67,202)(38,135,68,205)(39,130,69,208)(40,133,70,203)(105,141,209,196)(106,144,210,199)(107,139,211,194)(108,142,212,197)(109,137,213,200)(110,140,214,195)(111,143,215,198)(112,138,216,193)(145,160,178,174)(146,155,179,169)(147,158,180,172)(148,153,181,175)(149,156,182,170)(150,159,183,173)(151,154,184,176)(152,157,177,171), (1,217)(2,222)(3,219)(4,224)(5,221)(6,218)(7,223)(8,220)(9,115)(10,120)(11,117)(12,114)(13,119)(14,116)(15,113)(16,118)(17,62)(18,59)(19,64)(20,61)(21,58)(22,63)(23,60)(24,57)(25,105)(26,110)(27,107)(28,112)(29,109)(30,106)(31,111)(32,108)(33,175)(34,172)(35,169)(36,174)(37,171)(38,176)(39,173)(40,170)(41,50)(42,55)(43,52)(44,49)(45,54)(46,51)(47,56)(48,53)(65,155)(66,160)(67,157)(68,154)(69,159)(70,156)(71,153)(72,158)(73,82)(74,87)(75,84)(76,81)(77,86)(78,83)(79,88)(80,85)(89,138)(90,143)(91,140)(92,137)(93,142)(94,139)(95,144)(96,141)(97,216)(98,213)(99,210)(100,215)(101,212)(102,209)(103,214)(104,211)(122,126)(124,128)(129,178)(130,183)(131,180)(132,177)(133,182)(134,179)(135,184)(136,181)(145,207)(146,204)(147,201)(148,206)(149,203)(150,208)(151,205)(152,202)(161,165)(163,167)(185,198)(186,195)(187,200)(188,197)(189,194)(190,199)(191,196)(192,193)>;
G:=Group( (1,141,76,178,203,44,187)(2,188,45,204,179,77,142)(3,143,78,180,205,46,189)(4,190,47,206,181,79,144)(5,137,80,182,207,48,191)(6,192,41,208,183,73,138)(7,139,74,184,201,42,185)(8,186,43,202,177,75,140)(9,68,158,119,111,167,27)(10,28,168,112,120,159,69)(11,70,160,113,105,161,29)(12,30,162,106,114,153,71)(13,72,154,115,107,163,31)(14,32,164,108,116,155,65)(15,66,156,117,109,165,25)(16,26,166,110,118,157,67)(17,171,37,62,103,121,214)(18,215,122,104,63,38,172)(19,173,39,64,97,123,216)(20,209,124,98,57,40,174)(21,175,33,58,99,125,210)(22,211,126,100,59,34,176)(23,169,35,60,101,127,212)(24,213,128,102,61,36,170)(49,149,129,81,96,217,200)(50,193,218,89,82,130,150)(51,151,131,83,90,219,194)(52,195,220,91,84,132,152)(53,145,133,85,92,221,196)(54,197,222,93,86,134,146)(55,147,135,87,94,223,198)(56,199,224,95,88,136,148), (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), (1,124,221,161)(2,127,222,164)(3,122,223,167)(4,125,224,162)(5,128,217,165)(6,123,218,168)(7,126,219,163)(8,121,220,166)(9,46,63,87)(10,41,64,82)(11,44,57,85)(12,47,58,88)(13,42,59,83)(14,45,60,86)(15,48,61,81)(16,43,62,84)(17,52,118,75)(18,55,119,78)(19,50,120,73)(20,53,113,76)(21,56,114,79)(22,51,115,74)(23,54,116,77)(24,49,117,80)(25,191,102,96)(26,186,103,91)(27,189,104,94)(28,192,97,89)(29,187,98,92)(30,190,99,95)(31,185,100,90)(32,188,101,93)(33,136,71,206)(34,131,72,201)(35,134,65,204)(36,129,66,207)(37,132,67,202)(38,135,68,205)(39,130,69,208)(40,133,70,203)(105,141,209,196)(106,144,210,199)(107,139,211,194)(108,142,212,197)(109,137,213,200)(110,140,214,195)(111,143,215,198)(112,138,216,193)(145,160,178,174)(146,155,179,169)(147,158,180,172)(148,153,181,175)(149,156,182,170)(150,159,183,173)(151,154,184,176)(152,157,177,171), (1,217)(2,222)(3,219)(4,224)(5,221)(6,218)(7,223)(8,220)(9,115)(10,120)(11,117)(12,114)(13,119)(14,116)(15,113)(16,118)(17,62)(18,59)(19,64)(20,61)(21,58)(22,63)(23,60)(24,57)(25,105)(26,110)(27,107)(28,112)(29,109)(30,106)(31,111)(32,108)(33,175)(34,172)(35,169)(36,174)(37,171)(38,176)(39,173)(40,170)(41,50)(42,55)(43,52)(44,49)(45,54)(46,51)(47,56)(48,53)(65,155)(66,160)(67,157)(68,154)(69,159)(70,156)(71,153)(72,158)(73,82)(74,87)(75,84)(76,81)(77,86)(78,83)(79,88)(80,85)(89,138)(90,143)(91,140)(92,137)(93,142)(94,139)(95,144)(96,141)(97,216)(98,213)(99,210)(100,215)(101,212)(102,209)(103,214)(104,211)(122,126)(124,128)(129,178)(130,183)(131,180)(132,177)(133,182)(134,179)(135,184)(136,181)(145,207)(146,204)(147,201)(148,206)(149,203)(150,208)(151,205)(152,202)(161,165)(163,167)(185,198)(186,195)(187,200)(188,197)(189,194)(190,199)(191,196)(192,193) );
G=PermutationGroup([[(1,141,76,178,203,44,187),(2,188,45,204,179,77,142),(3,143,78,180,205,46,189),(4,190,47,206,181,79,144),(5,137,80,182,207,48,191),(6,192,41,208,183,73,138),(7,139,74,184,201,42,185),(8,186,43,202,177,75,140),(9,68,158,119,111,167,27),(10,28,168,112,120,159,69),(11,70,160,113,105,161,29),(12,30,162,106,114,153,71),(13,72,154,115,107,163,31),(14,32,164,108,116,155,65),(15,66,156,117,109,165,25),(16,26,166,110,118,157,67),(17,171,37,62,103,121,214),(18,215,122,104,63,38,172),(19,173,39,64,97,123,216),(20,209,124,98,57,40,174),(21,175,33,58,99,125,210),(22,211,126,100,59,34,176),(23,169,35,60,101,127,212),(24,213,128,102,61,36,170),(49,149,129,81,96,217,200),(50,193,218,89,82,130,150),(51,151,131,83,90,219,194),(52,195,220,91,84,132,152),(53,145,133,85,92,221,196),(54,197,222,93,86,134,146),(55,147,135,87,94,223,198),(56,199,224,95,88,136,148)], [(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)], [(1,124,221,161),(2,127,222,164),(3,122,223,167),(4,125,224,162),(5,128,217,165),(6,123,218,168),(7,126,219,163),(8,121,220,166),(9,46,63,87),(10,41,64,82),(11,44,57,85),(12,47,58,88),(13,42,59,83),(14,45,60,86),(15,48,61,81),(16,43,62,84),(17,52,118,75),(18,55,119,78),(19,50,120,73),(20,53,113,76),(21,56,114,79),(22,51,115,74),(23,54,116,77),(24,49,117,80),(25,191,102,96),(26,186,103,91),(27,189,104,94),(28,192,97,89),(29,187,98,92),(30,190,99,95),(31,185,100,90),(32,188,101,93),(33,136,71,206),(34,131,72,201),(35,134,65,204),(36,129,66,207),(37,132,67,202),(38,135,68,205),(39,130,69,208),(40,133,70,203),(105,141,209,196),(106,144,210,199),(107,139,211,194),(108,142,212,197),(109,137,213,200),(110,140,214,195),(111,143,215,198),(112,138,216,193),(145,160,178,174),(146,155,179,169),(147,158,180,172),(148,153,181,175),(149,156,182,170),(150,159,183,173),(151,154,184,176),(152,157,177,171)], [(1,217),(2,222),(3,219),(4,224),(5,221),(6,218),(7,223),(8,220),(9,115),(10,120),(11,117),(12,114),(13,119),(14,116),(15,113),(16,118),(17,62),(18,59),(19,64),(20,61),(21,58),(22,63),(23,60),(24,57),(25,105),(26,110),(27,107),(28,112),(29,109),(30,106),(31,111),(32,108),(33,175),(34,172),(35,169),(36,174),(37,171),(38,176),(39,173),(40,170),(41,50),(42,55),(43,52),(44,49),(45,54),(46,51),(47,56),(48,53),(65,155),(66,160),(67,157),(68,154),(69,159),(70,156),(71,153),(72,158),(73,82),(74,87),(75,84),(76,81),(77,86),(78,83),(79,88),(80,85),(89,138),(90,143),(91,140),(92,137),(93,142),(94,139),(95,144),(96,141),(97,216),(98,213),(99,210),(100,215),(101,212),(102,209),(103,214),(104,211),(122,126),(124,128),(129,178),(130,183),(131,180),(132,177),(133,182),(134,179),(135,184),(136,181),(145,207),(146,204),(147,201),(148,206),(149,203),(150,208),(151,205),(152,202),(161,165),(163,167),(185,198),(186,195),(187,200),(188,197),(189,194),(190,199),(191,196),(192,193)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 28 | 56 | 2 | 2 | 8 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | C8⋊C22 | D4×D7 | D4×D7 | D8⋊D7 | D56⋊C2 |
| kernel | C7⋊C8⋊D4 | C4.Dic14 | C2.D56 | C7×D4⋊C4 | C4⋊D28 | C2×C8⋊D7 | C2×D4⋊D7 | C28⋊2D4 | C7⋊C8 | C2×Dic7 | C22×D7 | D4⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×D4 | C4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 12 | 2 | 3 | 3 | 6 | 6 |
Matrix representation of C7⋊C8⋊D4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 79 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 80 | 112 |
| 0 | 0 | 0 | 0 | 81 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 43 |
| 0 | 0 | 0 | 0 | 111 | 50 |
| 0 | 0 | 109 | 17 | 91 | 7 |
| 0 | 0 | 36 | 21 | 28 | 22 |
| 80 | 2 | 0 | 0 | 0 | 0 |
| 20 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 34 | 1 |
| 0 | 0 | 89 | 99 | 89 | 78 |
| 0 | 0 | 27 | 4 | 103 | 99 |
| 0 | 0 | 17 | 90 | 90 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 33 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 80 | 112 |
| 0 | 0 | 0 | 0 | 71 | 33 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,79,1,0,0,0,0,112,0,0,0,0,0,0,0,80,81,0,0,0,0,112,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,109,36,0,0,0,0,17,21,0,0,20,111,91,28,0,0,43,50,7,22],[80,20,0,0,0,0,2,33,0,0,0,0,0,0,14,89,27,17,0,0,0,99,4,90,0,0,34,89,103,90,0,0,1,78,99,10],[1,33,0,0,0,0,0,112,0,0,0,0,0,0,112,34,0,0,0,0,0,1,0,0,0,0,0,0,80,71,0,0,0,0,112,33] >;
C7⋊C8⋊D4 in GAP, Magma, Sage, TeX
C_7\rtimes C_8\rtimes D_4
% in TeX
G:=Group("C7:C8:D4"); // GroupNames label
G:=SmallGroup(448,316);
// by ID
G=gap.SmallGroup(448,316);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,1094,135,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=b^3,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations