metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.80D14, C4⋊Q8⋊4D7, C7⋊C8.8D4, C4.18(D4×D7), C28.38(C2×D4), C7⋊3(C8.2D4), (C2×C28).295D4, (C2×Q8).46D14, C4.D28.8C2, C14.24(C4⋊1D4), C2.15(C28⋊D4), (C2×C28).404C23, (C4×C28).133C22, C42.D7⋊14C2, (Q8×C14).64C22, (C2×D28).108C22, C14.96(C8.C22), C2.17(C28.C23), (C2×Dic14).113C22, (C7×C4⋊Q8)⋊4C2, (C2×Q8⋊D7).7C2, (C2×C7⋊Q16)⋊15C2, (C2×C14).535(C2×D4), (C2×C4).73(C7⋊D4), (C2×C7⋊C8).137C22, (C2×C4).501(C22×D7), C22.207(C2×C7⋊D4), SmallGroup(448,620)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.80D14
G = < a,b,c,d | a4=b4=1, c14=b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c13 >
Subgroups: 652 in 124 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic7, C28, C28, D14, C2×C14, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C7⋊C8, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C8.2D4, C2×C7⋊C8, D14⋊C4, Q8⋊D7, C7⋊Q16, C4×C28, C7×C4⋊C4, C2×Dic14, C2×D28, Q8×C14, C42.D7, C4.D28, C2×Q8⋊D7, C2×C7⋊Q16, C7×C4⋊Q8, C42.80D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C8.C22, C7⋊D4, C22×D7, C8.2D4, D4×D7, C2×C7⋊D4, C28⋊D4, C28.C23, C42.80D14
►On 224 points
Generators in S224
(1 49 179 64)(2 65 180 50)(3 51 181 66)(4 67 182 52)(5 53 183 68)(6 69 184 54)(7 55 185 70)(8 71 186 56)(9 29 187 72)(10 73 188 30)(11 31 189 74)(12 75 190 32)(13 33 191 76)(14 77 192 34)(15 35 193 78)(16 79 194 36)(17 37 195 80)(18 81 196 38)(19 39 169 82)(20 83 170 40)(21 41 171 84)(22 57 172 42)(23 43 173 58)(24 59 174 44)(25 45 175 60)(26 61 176 46)(27 47 177 62)(28 63 178 48)(85 147 203 119)(86 120 204 148)(87 149 205 121)(88 122 206 150)(89 151 207 123)(90 124 208 152)(91 153 209 125)(92 126 210 154)(93 155 211 127)(94 128 212 156)(95 157 213 129)(96 130 214 158)(97 159 215 131)(98 132 216 160)(99 161 217 133)(100 134 218 162)(101 163 219 135)(102 136 220 164)(103 165 221 137)(104 138 222 166)(105 167 223 139)(106 140 224 168)(107 141 197 113)(108 114 198 142)(109 143 199 115)(110 116 200 144)(111 145 201 117)(112 118 202 146)
(1 105 15 91)(2 92 16 106)(3 107 17 93)(4 94 18 108)(5 109 19 95)(6 96 20 110)(7 111 21 97)(8 98 22 112)(9 85 23 99)(10 100 24 86)(11 87 25 101)(12 102 26 88)(13 89 27 103)(14 104 28 90)(29 147 43 161)(30 162 44 148)(31 149 45 163)(32 164 46 150)(33 151 47 165)(34 166 48 152)(35 153 49 167)(36 168 50 154)(37 155 51 141)(38 142 52 156)(39 157 53 143)(40 144 54 158)(41 159 55 145)(42 146 56 160)(57 118 71 132)(58 133 72 119)(59 120 73 134)(60 135 74 121)(61 122 75 136)(62 137 76 123)(63 124 77 138)(64 139 78 125)(65 126 79 140)(66 113 80 127)(67 128 81 114)(68 115 82 129)(69 130 83 116)(70 117 84 131)(169 213 183 199)(170 200 184 214)(171 215 185 201)(172 202 186 216)(173 217 187 203)(174 204 188 218)(175 219 189 205)(176 206 190 220)(177 221 191 207)(178 208 192 222)(179 223 193 209)(180 210 194 224)(181 197 195 211)(182 212 196 198)
(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 14 105 104 15 28 91 90)(2 89 92 27 16 103 106 13)(3 12 107 102 17 26 93 88)(4 87 94 25 18 101 108 11)(5 10 109 100 19 24 95 86)(6 85 96 23 20 99 110 9)(7 8 111 98 21 22 97 112)(29 83 147 116 43 69 161 130)(30 129 162 68 44 115 148 82)(31 81 149 114 45 67 163 128)(32 127 164 66 46 113 150 80)(33 79 151 140 47 65 165 126)(34 125 166 64 48 139 152 78)(35 77 153 138 49 63 167 124)(36 123 168 62 50 137 154 76)(37 75 155 136 51 61 141 122)(38 121 142 60 52 135 156 74)(39 73 157 134 53 59 143 120)(40 119 144 58 54 133 158 72)(41 71 159 132 55 57 145 118)(42 117 146 84 56 131 160 70)(169 174 213 204 183 188 199 218)(170 217 200 187 184 203 214 173)(171 172 215 202 185 186 201 216)(175 196 219 198 189 182 205 212)(176 211 206 181 190 197 220 195)(177 194 221 224 191 180 207 210)(178 209 208 179 192 223 222 193)
G:=sub<Sym(224)| (1,49,179,64)(2,65,180,50)(3,51,181,66)(4,67,182,52)(5,53,183,68)(6,69,184,54)(7,55,185,70)(8,71,186,56)(9,29,187,72)(10,73,188,30)(11,31,189,74)(12,75,190,32)(13,33,191,76)(14,77,192,34)(15,35,193,78)(16,79,194,36)(17,37,195,80)(18,81,196,38)(19,39,169,82)(20,83,170,40)(21,41,171,84)(22,57,172,42)(23,43,173,58)(24,59,174,44)(25,45,175,60)(26,61,176,46)(27,47,177,62)(28,63,178,48)(85,147,203,119)(86,120,204,148)(87,149,205,121)(88,122,206,150)(89,151,207,123)(90,124,208,152)(91,153,209,125)(92,126,210,154)(93,155,211,127)(94,128,212,156)(95,157,213,129)(96,130,214,158)(97,159,215,131)(98,132,216,160)(99,161,217,133)(100,134,218,162)(101,163,219,135)(102,136,220,164)(103,165,221,137)(104,138,222,166)(105,167,223,139)(106,140,224,168)(107,141,197,113)(108,114,198,142)(109,143,199,115)(110,116,200,144)(111,145,201,117)(112,118,202,146), (1,105,15,91)(2,92,16,106)(3,107,17,93)(4,94,18,108)(5,109,19,95)(6,96,20,110)(7,111,21,97)(8,98,22,112)(9,85,23,99)(10,100,24,86)(11,87,25,101)(12,102,26,88)(13,89,27,103)(14,104,28,90)(29,147,43,161)(30,162,44,148)(31,149,45,163)(32,164,46,150)(33,151,47,165)(34,166,48,152)(35,153,49,167)(36,168,50,154)(37,155,51,141)(38,142,52,156)(39,157,53,143)(40,144,54,158)(41,159,55,145)(42,146,56,160)(57,118,71,132)(58,133,72,119)(59,120,73,134)(60,135,74,121)(61,122,75,136)(62,137,76,123)(63,124,77,138)(64,139,78,125)(65,126,79,140)(66,113,80,127)(67,128,81,114)(68,115,82,129)(69,130,83,116)(70,117,84,131)(169,213,183,199)(170,200,184,214)(171,215,185,201)(172,202,186,216)(173,217,187,203)(174,204,188,218)(175,219,189,205)(176,206,190,220)(177,221,191,207)(178,208,192,222)(179,223,193,209)(180,210,194,224)(181,197,195,211)(182,212,196,198), (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), (1,14,105,104,15,28,91,90)(2,89,92,27,16,103,106,13)(3,12,107,102,17,26,93,88)(4,87,94,25,18,101,108,11)(5,10,109,100,19,24,95,86)(6,85,96,23,20,99,110,9)(7,8,111,98,21,22,97,112)(29,83,147,116,43,69,161,130)(30,129,162,68,44,115,148,82)(31,81,149,114,45,67,163,128)(32,127,164,66,46,113,150,80)(33,79,151,140,47,65,165,126)(34,125,166,64,48,139,152,78)(35,77,153,138,49,63,167,124)(36,123,168,62,50,137,154,76)(37,75,155,136,51,61,141,122)(38,121,142,60,52,135,156,74)(39,73,157,134,53,59,143,120)(40,119,144,58,54,133,158,72)(41,71,159,132,55,57,145,118)(42,117,146,84,56,131,160,70)(169,174,213,204,183,188,199,218)(170,217,200,187,184,203,214,173)(171,172,215,202,185,186,201,216)(175,196,219,198,189,182,205,212)(176,211,206,181,190,197,220,195)(177,194,221,224,191,180,207,210)(178,209,208,179,192,223,222,193)>;
G:=Group( (1,49,179,64)(2,65,180,50)(3,51,181,66)(4,67,182,52)(5,53,183,68)(6,69,184,54)(7,55,185,70)(8,71,186,56)(9,29,187,72)(10,73,188,30)(11,31,189,74)(12,75,190,32)(13,33,191,76)(14,77,192,34)(15,35,193,78)(16,79,194,36)(17,37,195,80)(18,81,196,38)(19,39,169,82)(20,83,170,40)(21,41,171,84)(22,57,172,42)(23,43,173,58)(24,59,174,44)(25,45,175,60)(26,61,176,46)(27,47,177,62)(28,63,178,48)(85,147,203,119)(86,120,204,148)(87,149,205,121)(88,122,206,150)(89,151,207,123)(90,124,208,152)(91,153,209,125)(92,126,210,154)(93,155,211,127)(94,128,212,156)(95,157,213,129)(96,130,214,158)(97,159,215,131)(98,132,216,160)(99,161,217,133)(100,134,218,162)(101,163,219,135)(102,136,220,164)(103,165,221,137)(104,138,222,166)(105,167,223,139)(106,140,224,168)(107,141,197,113)(108,114,198,142)(109,143,199,115)(110,116,200,144)(111,145,201,117)(112,118,202,146), (1,105,15,91)(2,92,16,106)(3,107,17,93)(4,94,18,108)(5,109,19,95)(6,96,20,110)(7,111,21,97)(8,98,22,112)(9,85,23,99)(10,100,24,86)(11,87,25,101)(12,102,26,88)(13,89,27,103)(14,104,28,90)(29,147,43,161)(30,162,44,148)(31,149,45,163)(32,164,46,150)(33,151,47,165)(34,166,48,152)(35,153,49,167)(36,168,50,154)(37,155,51,141)(38,142,52,156)(39,157,53,143)(40,144,54,158)(41,159,55,145)(42,146,56,160)(57,118,71,132)(58,133,72,119)(59,120,73,134)(60,135,74,121)(61,122,75,136)(62,137,76,123)(63,124,77,138)(64,139,78,125)(65,126,79,140)(66,113,80,127)(67,128,81,114)(68,115,82,129)(69,130,83,116)(70,117,84,131)(169,213,183,199)(170,200,184,214)(171,215,185,201)(172,202,186,216)(173,217,187,203)(174,204,188,218)(175,219,189,205)(176,206,190,220)(177,221,191,207)(178,208,192,222)(179,223,193,209)(180,210,194,224)(181,197,195,211)(182,212,196,198), (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), (1,14,105,104,15,28,91,90)(2,89,92,27,16,103,106,13)(3,12,107,102,17,26,93,88)(4,87,94,25,18,101,108,11)(5,10,109,100,19,24,95,86)(6,85,96,23,20,99,110,9)(7,8,111,98,21,22,97,112)(29,83,147,116,43,69,161,130)(30,129,162,68,44,115,148,82)(31,81,149,114,45,67,163,128)(32,127,164,66,46,113,150,80)(33,79,151,140,47,65,165,126)(34,125,166,64,48,139,152,78)(35,77,153,138,49,63,167,124)(36,123,168,62,50,137,154,76)(37,75,155,136,51,61,141,122)(38,121,142,60,52,135,156,74)(39,73,157,134,53,59,143,120)(40,119,144,58,54,133,158,72)(41,71,159,132,55,57,145,118)(42,117,146,84,56,131,160,70)(169,174,213,204,183,188,199,218)(170,217,200,187,184,203,214,173)(171,172,215,202,185,186,201,216)(175,196,219,198,189,182,205,212)(176,211,206,181,190,197,220,195)(177,194,221,224,191,180,207,210)(178,209,208,179,192,223,222,193) );
G=PermutationGroup([[(1,49,179,64),(2,65,180,50),(3,51,181,66),(4,67,182,52),(5,53,183,68),(6,69,184,54),(7,55,185,70),(8,71,186,56),(9,29,187,72),(10,73,188,30),(11,31,189,74),(12,75,190,32),(13,33,191,76),(14,77,192,34),(15,35,193,78),(16,79,194,36),(17,37,195,80),(18,81,196,38),(19,39,169,82),(20,83,170,40),(21,41,171,84),(22,57,172,42),(23,43,173,58),(24,59,174,44),(25,45,175,60),(26,61,176,46),(27,47,177,62),(28,63,178,48),(85,147,203,119),(86,120,204,148),(87,149,205,121),(88,122,206,150),(89,151,207,123),(90,124,208,152),(91,153,209,125),(92,126,210,154),(93,155,211,127),(94,128,212,156),(95,157,213,129),(96,130,214,158),(97,159,215,131),(98,132,216,160),(99,161,217,133),(100,134,218,162),(101,163,219,135),(102,136,220,164),(103,165,221,137),(104,138,222,166),(105,167,223,139),(106,140,224,168),(107,141,197,113),(108,114,198,142),(109,143,199,115),(110,116,200,144),(111,145,201,117),(112,118,202,146)], [(1,105,15,91),(2,92,16,106),(3,107,17,93),(4,94,18,108),(5,109,19,95),(6,96,20,110),(7,111,21,97),(8,98,22,112),(9,85,23,99),(10,100,24,86),(11,87,25,101),(12,102,26,88),(13,89,27,103),(14,104,28,90),(29,147,43,161),(30,162,44,148),(31,149,45,163),(32,164,46,150),(33,151,47,165),(34,166,48,152),(35,153,49,167),(36,168,50,154),(37,155,51,141),(38,142,52,156),(39,157,53,143),(40,144,54,158),(41,159,55,145),(42,146,56,160),(57,118,71,132),(58,133,72,119),(59,120,73,134),(60,135,74,121),(61,122,75,136),(62,137,76,123),(63,124,77,138),(64,139,78,125),(65,126,79,140),(66,113,80,127),(67,128,81,114),(68,115,82,129),(69,130,83,116),(70,117,84,131),(169,213,183,199),(170,200,184,214),(171,215,185,201),(172,202,186,216),(173,217,187,203),(174,204,188,218),(175,219,189,205),(176,206,190,220),(177,221,191,207),(178,208,192,222),(179,223,193,209),(180,210,194,224),(181,197,195,211),(182,212,196,198)], [(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)], [(1,14,105,104,15,28,91,90),(2,89,92,27,16,103,106,13),(3,12,107,102,17,26,93,88),(4,87,94,25,18,101,108,11),(5,10,109,100,19,24,95,86),(6,85,96,23,20,99,110,9),(7,8,111,98,21,22,97,112),(29,83,147,116,43,69,161,130),(30,129,162,68,44,115,148,82),(31,81,149,114,45,67,163,128),(32,127,164,66,46,113,150,80),(33,79,151,140,47,65,165,126),(34,125,166,64,48,139,152,78),(35,77,153,138,49,63,167,124),(36,123,168,62,50,137,154,76),(37,75,155,136,51,61,141,122),(38,121,142,60,52,135,156,74),(39,73,157,134,53,59,143,120),(40,119,144,58,54,133,158,72),(41,71,159,132,55,57,145,118),(42,117,146,84,56,131,160,70),(169,174,213,204,183,188,199,218),(170,217,200,187,184,203,214,173),(171,172,215,202,185,186,201,216),(175,196,219,198,189,182,205,212),(176,211,206,181,190,197,220,195),(177,194,221,224,191,180,207,210),(178,209,208,179,192,223,222,193)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 56 | 2 | 2 | 4 | 4 | 8 | 8 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C7⋊D4 | C8.C22 | D4×D7 | C28.C23 |
| kernel | C42.80D14 | C42.D7 | C4.D28 | C2×Q8⋊D7 | C2×C7⋊Q16 | C7×C4⋊Q8 | C7⋊C8 | C2×C28 | C4⋊Q8 | C42 | C2×Q8 | C2×C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 3 | 3 | 6 | 12 | 2 | 6 | 12 |
Matrix representation of C42.80D14 ►in GL8(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 77 | 0 | 0 | 0 | 0 |
| 0 | 0 | 82 | 76 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 15 | 98 |
| 0 | 0 | 0 | 0 | 112 | 0 | 98 | 98 |
| 0 | 0 | 0 | 0 | 98 | 15 | 0 | 1 |
| 0 | 0 | 0 | 0 | 15 | 15 | 112 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 25 | 33 | 0 | 0 | 0 | 0 | 0 | 0 |
| 54 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 | 0 | 0 |
| 79 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,37,82,0,0,0,0,0,0,77,76,0,0,0,0,0,0,0,0,0,112,98,15,0,0,0,0,1,0,15,15,0,0,0,0,15,98,0,112,0,0,0,0,98,98,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0],[25,54,0,0,0,0,0,0,33,112,0,0,0,0,0,0,0,0,1,46,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[79,60,0,0,0,0,0,0,9,34,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
C42.80D14 in GAP, Magma, Sage, TeX
C_4^2._{80}D_{14} % in TeX
G:=Group("C4^2.80D14"); // GroupNames label
G:=SmallGroup(448,620);
// by ID
G=gap.SmallGroup(448,620);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,555,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^13>;
// generators/relations