D14.Q16 - GroupNames

G = D₁₄.Q₁₆ order 448 = 2⁶·7

1^st non-split extension by D₁₄ of Q₁₆ acting via Q₁₆/Q₈=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₄.₁Q₁₆, D₁₄⋊C₈.₃C₂, Q₈⋊C₄⋊₅D₇, C₅₆⋊₁C₄⋊₁₀C₂, (C₂×C₈).₁₇D₁₄, C₂.₁₀(D₇×Q₁₆), C₄⋊C₄.₁₅₂D₁₄, Q₈⋊Dic₇⋊₈C₂, C₂₈.Q₈⋊₉C₂, (C₂×Q₈).₁₇D₁₄, C₁₄.₁₈(C₂×Q₁₆), C₄.₅₆(C₄○D₂₈), (C₂×C₅₆).₁₇C₂², D₁₄⋊₃Q₈.₂C₂, (C₂×Dic₇).₃₂D₄, (C₂²×D₇).₇₉D₄, C₂².₁₉₉(D₄×D₇), C₂₈.₁₆₂(C₄○D₄), C₄.₈₇(D₄⋊₂D₇), C₂.₁₆(D₅₆⋊C₂), C₁₄.₆₂(C₈⋊C₂²), (C₂×C₂₈).₂₄₉C₂³, C₇⋊₂(C₂³.₄₈D₄), C₄⋊Dic₇.₉₅C₂², (Q₈×C₁₄).₃₂C₂², C₂.₁₇(D₁₄.D₄), C₁₄.₂₅(C₂².D₄), (D₇×C₄⋊C₄).₃C₂, (C₇×Q₈⋊C₄)⋊₅C₂, (C₂×C₇⋊C₈).₄₀C₂², (C₂×C₄×D₇).₂₂C₂², (C₂×C₁₄).₂₆₂(C₂×D₄), (C₇×C₄⋊C₄).₅₀C₂², (C₂×C₄).₃₅₆(C₂²×D₇), SmallGroup(448,343)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — D₁₄.Q₁₆

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — C₂×C₄×D₇ — D₇×C₄⋊C₄ — D₁₄.Q₁₆

Lower central

C₇ — C₁₄ — C₂×C₂₈ — D₁₄.Q₁₆

Upper central

C₁ — C₂² — C₂×C₄ — Q₈⋊C₄

Generators and relations for D₁₄.Q₁₆
G = < a,b,c,d | a¹⁴=b²=c⁸=1, d²=c⁴, bab=a^-1, ac=ca, ad=da, cbc^-1=dbd^-1=a⁷b, dcd^-1=a⁷c^-1 >

Subgroups: 532 in 104 conjugacy classes, 39 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, D₇, C₁₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂×Q₈, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂²⋊C₈, Q₈⋊C₄, Q₈⋊C₄, C₂.D₈, C₂×C₄⋊C₄, C₂²⋊Q₈, C₇⋊C₈, C₅₆, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂³.₄₈D₄, C₂×C₇⋊C₈, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₇×C₄⋊C₄, C₂×C₅₆, C₂×C₄×D₇, C₂×C₄×D₇, Q₈×C₁₄, C₂₈.Q₈, C₅₆⋊₁C₄, D₁₄⋊C₈, Q₈⋊Dic₇, C₇×Q₈⋊C₄, D₇×C₄⋊C₄, D₁₄⋊₃Q₈, D₁₄.Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, Q₁₆, C₂×D₄, C₄○D₄, D₁₄, C₂².D₄, C₂×Q₁₆, C₈⋊C₂², C₂²×D₇, C₂³.₄₈D₄, C₄○D₂₈, D₄×D₇, D₄⋊₂D₇, D₁₄.D₄, D₅₆⋊C₂, D₇×Q₁₆, D₁₄.Q₁₆

Smallest permutation representation of D₁₄.Q₁₆
►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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 16)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(29 35)(30 34)(31 33)(36 42)(37 41)(38 40)(43 49)(44 48)(45 47)(50 56)(51 55)(52 54)(57 62)(58 61)(59 60)(63 70)(64 69)(65 68)(66 67)(71 75)(72 74)(76 84)(77 83)(78 82)(79 81)(85 93)(86 92)(87 91)(88 90)(94 98)(95 97)(99 109)(100 108)(101 107)(102 106)(103 105)(110 112)(113 114)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)(127 137)(128 136)(129 135)(130 134)(131 133)(138 140)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(153 154)(156 168)(157 167)(158 166)(159 165)(160 164)(161 163)(169 172)(170 171)(173 182)(174 181)(175 180)(176 179)(177 178)(183 193)(184 192)(185 191)(186 190)(187 189)(194 196)(197 200)(198 199)(201 210)(202 209)(203 208)(204 207)(205 206)(211 220)(212 219)(213 218)(214 217)(215 216)(221 224)(222 223)

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

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

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

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

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

61 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	28A	···	28F	28G	···	28R	56A	···	56L
order	1	2	2	2	2	2	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	14	14	2	2	4	4	8	28	28	28	56	2	2	2	4	4	28	28	2	···	2	4	···	4	8	···	8	4	···	4

61 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	D₇	C₄○D₄	Q₁₆	D₁₄	D₁₄	D₁₄	C₄○D₂₈	C₈⋊C₂²	D₄⋊₂D₇	D₄×D₇	D₅₆⋊C₂	D₇×Q₁₆
kernel	D₁₄.Q₁₆	C₂₈.Q₈	C₅₆⋊₁C₄	D₁₄⋊C₈	Q₈⋊Dic₇	C₇×Q₈⋊C₄	D₇×C₄⋊C₄	D₁₄⋊₃Q₈	C₂×Dic₇	C₂²×D₇	Q₈⋊C₄	C₂₈	D₁₄	C₄⋊C₄	C₂×C₈	C₂×Q₈	C₄	C₁₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	1	1	3	4	4	3	3	3	12	1	3	3	6	6

Matrix representation of D₁₄.Q₁₆ ►in GL₆(𝔽₁₁₃)

112	0	0	0	0	0
0	112	0	0	0	0
0	0	103	103	0	0
0	0	10	89	0	0
0	0	0	0	1	0
0	0	0	0	0	1

112	112	0	0	0	0
0	1	0	0	0	0
0	0	103	103	0	0
0	0	89	10	0	0
0	0	0	0	1	0
0	0	0	0	0	1

15	0	0	0	0	0
83	98	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	31	82
0	0	0	0	31	31

1	0	0	0	0	0
111	112	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	76	72
0	0	0	0	72	37

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,103,10,0,0,0,0,103,89,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,112,1,0,0,0,0,0,0,103,89,0,0,0,0,103,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,83,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,31,0,0,0,0,82,31],[1,111,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,76,72,0,0,0,0,72,37] >;

D₁₄.Q₁₆ in GAP, Magma, Sage, TeX

D_{14}.Q_{16}

% in TeX

G:=Group("D14.Q16");

// GroupNames label

G:=SmallGroup(448,343);

// by ID

G=gap.SmallGroup(448,343);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,926,219,184,851,438,102,18822]);

// 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,c*b*c^-1=d*b*d^-1=a^7*b,d*c*d^-1=a^7*c^-1>;

// generators/relations

G = D14.Q16 order 448 = 26·7

1st non-split extension by D14 of Q16 acting via Q16/Q8=C2

G = D₁₄.Q₁₆ order 448 = 2⁶·7

1^st non-split extension by D₁₄ of Q₁₆ acting via Q₁₆/Q₈=C₂