metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊3Q16, C56.27D4, (C2×Q16)⋊5D7, (C14×Q16)⋊5C2, C56⋊1C4⋊24C2, C2.19(D7×Q16), (C2×C8).243D14, C28.186(C2×D4), C8.28(C7⋊D4), C7⋊5(C8.18D4), (C2×Q8).64D14, C14.30(C2×Q16), C14.81(C4○D8), Q8⋊Dic7⋊35C2, (C2×C56).95C22, D14⋊3Q8.8C2, (C22×D7).60D4, C22.279(D4×D7), C28.107(C4○D4), C4.36(D4⋊2D7), C2.22(C28⋊2D4), (C2×C28).462C23, (C2×Dic7).117D4, (Q8×C14).91C22, C2.18(Q8.D14), C14.121(C4⋊D4), C4⋊Dic7.185C22, (D7×C2×C8).5C2, C4.85(C2×C7⋊D4), (C2×C14).373(C2×D4), (C2×C7⋊C8).278C22, (C2×C4×D7).241C22, (C2×C4).550(C22×D7), SmallGroup(448,722)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14⋊3Q16
G = < a,b,c,d | a14=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a7b, dcd-1=c-1 >
Subgroups: 516 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C23, D7, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, Dic7, C28, C28, D14, D14, C2×C14, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C8.18D4, C8×D7, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C2×C56, C7×Q16, C2×C4×D7, Q8×C14, C56⋊1C4, Q8⋊Dic7, D7×C2×C8, D14⋊3Q8, C14×Q16, D14⋊3Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, C4○D4, D14, C4⋊D4, C2×Q16, C4○D8, C7⋊D4, C22×D7, C8.18D4, D4×D7, D4⋊2D7, C2×C7⋊D4, D7×Q16, Q8.D14, C28⋊2D4, D14⋊3Q16
►On 224 points
Generators in S224
(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)
(1 176)(2 175)(3 174)(4 173)(5 172)(6 171)(7 170)(8 169)(9 182)(10 181)(11 180)(12 179)(13 178)(14 177)(15 131)(16 130)(17 129)(18 128)(19 127)(20 140)(21 139)(22 138)(23 137)(24 136)(25 135)(26 134)(27 133)(28 132)(29 197)(30 210)(31 209)(32 208)(33 207)(34 206)(35 205)(36 204)(37 203)(38 202)(39 201)(40 200)(41 199)(42 198)(43 156)(44 155)(45 168)(46 167)(47 166)(48 165)(49 164)(50 163)(51 162)(52 161)(53 160)(54 159)(55 158)(56 157)(57 99)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 88)(72 87)(73 86)(74 85)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(113 212)(114 211)(115 224)(116 223)(117 222)(118 221)(119 220)(120 219)(121 218)(122 217)(123 216)(124 215)(125 214)(126 213)(141 194)(142 193)(143 192)(144 191)(145 190)(146 189)(147 188)(148 187)(149 186)(150 185)(151 184)(152 183)(153 196)(154 195)
(1 137 71 121 177 24 89 219)(2 138 72 122 178 25 90 220)(3 139 73 123 179 26 91 221)(4 140 74 124 180 27 92 222)(5 127 75 125 181 28 93 223)(6 128 76 126 182 15 94 224)(7 129 77 113 169 16 95 211)(8 130 78 114 170 17 96 212)(9 131 79 115 171 18 97 213)(10 132 80 116 172 19 98 214)(11 133 81 117 173 20 85 215)(12 134 82 118 174 21 86 216)(13 135 83 119 175 22 87 217)(14 136 84 120 176 23 88 218)(29 158 68 196 197 55 102 153)(30 159 69 183 198 56 103 154)(31 160 70 184 199 43 104 141)(32 161 57 185 200 44 105 142)(33 162 58 186 201 45 106 143)(34 163 59 187 202 46 107 144)(35 164 60 188 203 47 108 145)(36 165 61 189 204 48 109 146)(37 166 62 190 205 49 110 147)(38 167 63 191 206 50 111 148)(39 168 64 192 207 51 112 149)(40 155 65 193 208 52 99 150)(41 156 66 194 209 53 100 151)(42 157 67 195 210 54 101 152)
(1 65 177 99)(2 66 178 100)(3 67 179 101)(4 68 180 102)(5 69 181 103)(6 70 182 104)(7 57 169 105)(8 58 170 106)(9 59 171 107)(10 60 172 108)(11 61 173 109)(12 62 174 110)(13 63 175 111)(14 64 176 112)(15 43 128 160)(16 44 129 161)(17 45 130 162)(18 46 131 163)(19 47 132 164)(20 48 133 165)(21 49 134 166)(22 50 135 167)(23 51 136 168)(24 52 137 155)(25 53 138 156)(26 54 139 157)(27 55 140 158)(28 56 127 159)(29 92 197 74)(30 93 198 75)(31 94 199 76)(32 95 200 77)(33 96 201 78)(34 97 202 79)(35 98 203 80)(36 85 204 81)(37 86 205 82)(38 87 206 83)(39 88 207 84)(40 89 208 71)(41 90 209 72)(42 91 210 73)(113 142 211 185)(114 143 212 186)(115 144 213 187)(116 145 214 188)(117 146 215 189)(118 147 216 190)(119 148 217 191)(120 149 218 192)(121 150 219 193)(122 151 220 194)(123 152 221 195)(124 153 222 196)(125 154 223 183)(126 141 224 184)
G:=sub<Sym(224)| (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), (1,176)(2,175)(3,174)(4,173)(5,172)(6,171)(7,170)(8,169)(9,182)(10,181)(11,180)(12,179)(13,178)(14,177)(15,131)(16,130)(17,129)(18,128)(19,127)(20,140)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,197)(30,210)(31,209)(32,208)(33,207)(34,206)(35,205)(36,204)(37,203)(38,202)(39,201)(40,200)(41,199)(42,198)(43,156)(44,155)(45,168)(46,167)(47,166)(48,165)(49,164)(50,163)(51,162)(52,161)(53,160)(54,159)(55,158)(56,157)(57,99)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,88)(72,87)(73,86)(74,85)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(113,212)(114,211)(115,224)(116,223)(117,222)(118,221)(119,220)(120,219)(121,218)(122,217)(123,216)(124,215)(125,214)(126,213)(141,194)(142,193)(143,192)(144,191)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,196)(154,195), (1,137,71,121,177,24,89,219)(2,138,72,122,178,25,90,220)(3,139,73,123,179,26,91,221)(4,140,74,124,180,27,92,222)(5,127,75,125,181,28,93,223)(6,128,76,126,182,15,94,224)(7,129,77,113,169,16,95,211)(8,130,78,114,170,17,96,212)(9,131,79,115,171,18,97,213)(10,132,80,116,172,19,98,214)(11,133,81,117,173,20,85,215)(12,134,82,118,174,21,86,216)(13,135,83,119,175,22,87,217)(14,136,84,120,176,23,88,218)(29,158,68,196,197,55,102,153)(30,159,69,183,198,56,103,154)(31,160,70,184,199,43,104,141)(32,161,57,185,200,44,105,142)(33,162,58,186,201,45,106,143)(34,163,59,187,202,46,107,144)(35,164,60,188,203,47,108,145)(36,165,61,189,204,48,109,146)(37,166,62,190,205,49,110,147)(38,167,63,191,206,50,111,148)(39,168,64,192,207,51,112,149)(40,155,65,193,208,52,99,150)(41,156,66,194,209,53,100,151)(42,157,67,195,210,54,101,152), (1,65,177,99)(2,66,178,100)(3,67,179,101)(4,68,180,102)(5,69,181,103)(6,70,182,104)(7,57,169,105)(8,58,170,106)(9,59,171,107)(10,60,172,108)(11,61,173,109)(12,62,174,110)(13,63,175,111)(14,64,176,112)(15,43,128,160)(16,44,129,161)(17,45,130,162)(18,46,131,163)(19,47,132,164)(20,48,133,165)(21,49,134,166)(22,50,135,167)(23,51,136,168)(24,52,137,155)(25,53,138,156)(26,54,139,157)(27,55,140,158)(28,56,127,159)(29,92,197,74)(30,93,198,75)(31,94,199,76)(32,95,200,77)(33,96,201,78)(34,97,202,79)(35,98,203,80)(36,85,204,81)(37,86,205,82)(38,87,206,83)(39,88,207,84)(40,89,208,71)(41,90,209,72)(42,91,210,73)(113,142,211,185)(114,143,212,186)(115,144,213,187)(116,145,214,188)(117,146,215,189)(118,147,216,190)(119,148,217,191)(120,149,218,192)(121,150,219,193)(122,151,220,194)(123,152,221,195)(124,153,222,196)(125,154,223,183)(126,141,224,184)>;
G:=Group( (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), (1,176)(2,175)(3,174)(4,173)(5,172)(6,171)(7,170)(8,169)(9,182)(10,181)(11,180)(12,179)(13,178)(14,177)(15,131)(16,130)(17,129)(18,128)(19,127)(20,140)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,197)(30,210)(31,209)(32,208)(33,207)(34,206)(35,205)(36,204)(37,203)(38,202)(39,201)(40,200)(41,199)(42,198)(43,156)(44,155)(45,168)(46,167)(47,166)(48,165)(49,164)(50,163)(51,162)(52,161)(53,160)(54,159)(55,158)(56,157)(57,99)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,88)(72,87)(73,86)(74,85)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(113,212)(114,211)(115,224)(116,223)(117,222)(118,221)(119,220)(120,219)(121,218)(122,217)(123,216)(124,215)(125,214)(126,213)(141,194)(142,193)(143,192)(144,191)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,196)(154,195), (1,137,71,121,177,24,89,219)(2,138,72,122,178,25,90,220)(3,139,73,123,179,26,91,221)(4,140,74,124,180,27,92,222)(5,127,75,125,181,28,93,223)(6,128,76,126,182,15,94,224)(7,129,77,113,169,16,95,211)(8,130,78,114,170,17,96,212)(9,131,79,115,171,18,97,213)(10,132,80,116,172,19,98,214)(11,133,81,117,173,20,85,215)(12,134,82,118,174,21,86,216)(13,135,83,119,175,22,87,217)(14,136,84,120,176,23,88,218)(29,158,68,196,197,55,102,153)(30,159,69,183,198,56,103,154)(31,160,70,184,199,43,104,141)(32,161,57,185,200,44,105,142)(33,162,58,186,201,45,106,143)(34,163,59,187,202,46,107,144)(35,164,60,188,203,47,108,145)(36,165,61,189,204,48,109,146)(37,166,62,190,205,49,110,147)(38,167,63,191,206,50,111,148)(39,168,64,192,207,51,112,149)(40,155,65,193,208,52,99,150)(41,156,66,194,209,53,100,151)(42,157,67,195,210,54,101,152), (1,65,177,99)(2,66,178,100)(3,67,179,101)(4,68,180,102)(5,69,181,103)(6,70,182,104)(7,57,169,105)(8,58,170,106)(9,59,171,107)(10,60,172,108)(11,61,173,109)(12,62,174,110)(13,63,175,111)(14,64,176,112)(15,43,128,160)(16,44,129,161)(17,45,130,162)(18,46,131,163)(19,47,132,164)(20,48,133,165)(21,49,134,166)(22,50,135,167)(23,51,136,168)(24,52,137,155)(25,53,138,156)(26,54,139,157)(27,55,140,158)(28,56,127,159)(29,92,197,74)(30,93,198,75)(31,94,199,76)(32,95,200,77)(33,96,201,78)(34,97,202,79)(35,98,203,80)(36,85,204,81)(37,86,205,82)(38,87,206,83)(39,88,207,84)(40,89,208,71)(41,90,209,72)(42,91,210,73)(113,142,211,185)(114,143,212,186)(115,144,213,187)(116,145,214,188)(117,146,215,189)(118,147,216,190)(119,148,217,191)(120,149,218,192)(121,150,219,193)(122,151,220,194)(123,152,221,195)(124,153,222,196)(125,154,223,183)(126,141,224,184) );
G=PermutationGroup([[(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)], [(1,176),(2,175),(3,174),(4,173),(5,172),(6,171),(7,170),(8,169),(9,182),(10,181),(11,180),(12,179),(13,178),(14,177),(15,131),(16,130),(17,129),(18,128),(19,127),(20,140),(21,139),(22,138),(23,137),(24,136),(25,135),(26,134),(27,133),(28,132),(29,197),(30,210),(31,209),(32,208),(33,207),(34,206),(35,205),(36,204),(37,203),(38,202),(39,201),(40,200),(41,199),(42,198),(43,156),(44,155),(45,168),(46,167),(47,166),(48,165),(49,164),(50,163),(51,162),(52,161),(53,160),(54,159),(55,158),(56,157),(57,99),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,88),(72,87),(73,86),(74,85),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(113,212),(114,211),(115,224),(116,223),(117,222),(118,221),(119,220),(120,219),(121,218),(122,217),(123,216),(124,215),(125,214),(126,213),(141,194),(142,193),(143,192),(144,191),(145,190),(146,189),(147,188),(148,187),(149,186),(150,185),(151,184),(152,183),(153,196),(154,195)], [(1,137,71,121,177,24,89,219),(2,138,72,122,178,25,90,220),(3,139,73,123,179,26,91,221),(4,140,74,124,180,27,92,222),(5,127,75,125,181,28,93,223),(6,128,76,126,182,15,94,224),(7,129,77,113,169,16,95,211),(8,130,78,114,170,17,96,212),(9,131,79,115,171,18,97,213),(10,132,80,116,172,19,98,214),(11,133,81,117,173,20,85,215),(12,134,82,118,174,21,86,216),(13,135,83,119,175,22,87,217),(14,136,84,120,176,23,88,218),(29,158,68,196,197,55,102,153),(30,159,69,183,198,56,103,154),(31,160,70,184,199,43,104,141),(32,161,57,185,200,44,105,142),(33,162,58,186,201,45,106,143),(34,163,59,187,202,46,107,144),(35,164,60,188,203,47,108,145),(36,165,61,189,204,48,109,146),(37,166,62,190,205,49,110,147),(38,167,63,191,206,50,111,148),(39,168,64,192,207,51,112,149),(40,155,65,193,208,52,99,150),(41,156,66,194,209,53,100,151),(42,157,67,195,210,54,101,152)], [(1,65,177,99),(2,66,178,100),(3,67,179,101),(4,68,180,102),(5,69,181,103),(6,70,182,104),(7,57,169,105),(8,58,170,106),(9,59,171,107),(10,60,172,108),(11,61,173,109),(12,62,174,110),(13,63,175,111),(14,64,176,112),(15,43,128,160),(16,44,129,161),(17,45,130,162),(18,46,131,163),(19,47,132,164),(20,48,133,165),(21,49,134,166),(22,50,135,167),(23,51,136,168),(24,52,137,155),(25,53,138,156),(26,54,139,157),(27,55,140,158),(28,56,127,159),(29,92,197,74),(30,93,198,75),(31,94,199,76),(32,95,200,77),(33,96,201,78),(34,97,202,79),(35,98,203,80),(36,85,204,81),(37,86,205,82),(38,87,206,83),(39,88,207,84),(40,89,208,71),(41,90,209,72),(42,91,210,73),(113,142,211,185),(114,143,212,186),(115,144,213,187),(116,145,214,188),(117,146,215,189),(118,147,216,190),(119,148,217,191),(120,149,218,192),(121,150,219,193),(122,151,220,194),(123,152,221,195),(124,153,222,196),(125,154,223,183),(126,141,224,184)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 2 | 2 | 8 | 8 | 14 | 14 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | Q16 | D14 | D14 | C4○D8 | C7⋊D4 | D4⋊2D7 | D4×D7 | D7×Q16 | Q8.D14 |
| kernel | D14⋊3Q16 | C56⋊1C4 | Q8⋊Dic7 | D7×C2×C8 | D14⋊3Q8 | C14×Q16 | C56 | C2×Dic7 | C22×D7 | C2×Q16 | C28 | D14 | C2×C8 | C2×Q8 | C14 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 3 | 6 | 4 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of D14⋊3Q16 ►in GL4(𝔽113) generated by
| 80 | 80 | 0 | 0 |
| 33 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 80 | 80 | 0 | 0 |
| 9 | 33 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 62 | 51 |
| 0 | 0 | 31 | 0 |
| 79 | 108 | 0 | 0 |
| 5 | 34 | 0 | 0 |
| 0 | 0 | 95 | 94 |
| 0 | 0 | 29 | 18 |
G:=sub<GL(4,GF(113))| [80,33,0,0,80,9,0,0,0,0,1,0,0,0,0,1],[80,9,0,0,80,33,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,62,31,0,0,51,0],[79,5,0,0,108,34,0,0,0,0,95,29,0,0,94,18] >;
D14⋊3Q16 in GAP, Magma, Sage, TeX
D_{14}\rtimes_3Q_{16} % in TeX
G:=Group("D14:3Q16"); // GroupNames label
G:=SmallGroup(448,722);
// by ID
G=gap.SmallGroup(448,722);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,184,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations