D14.5D8 - GroupNames

G = D₁₄.₅D₈ order 448 = 2⁶·7

2^nd non-split extension by D₁₄ of D₈ acting via D₈/D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₄.₅D₈, C₂.D₈⋊₂D₇, C₂.₁₃(D₇×D₈), D₁₄⋊C₈⋊₂₅C₂, C₄⋊C₄.₄₇D₁₄, (C₂×C₈).₂₇D₁₄, C₁₄.₂₉(C₂×D₈), C₄⋊D₂₈.₇C₂, C₂.D₅₆⋊₂₆C₂, C₁₄.D₈⋊₂₀C₂, C₄.₈₀(C₄○D₂₈), C₂₈.₃₇(C₄○D₄), C₂₈.Q₈⋊₂₁C₂, (C₂×Dic₇).₄₈D₄, (C₂²×D₇).₈₅D₄, C₂².₂₂₈(D₄×D₇), C₇⋊₃(C₂².D₈), (C₂×C₂₈).₂₉₈C₂³, (C₂×C₅₆).₂₄₁C₂², C₄.₂₈(Q₈⋊₂D₇), (C₂×D₂₈).₈₁C₂², C₂.₂₂(Q₁₆⋊D₇), C₁₄.₇₀(C₈.C₂²), C₄⋊Dic₇.₁₂₄C₂², C₂.₁₅(D₁₄.₅D₄), C₁₄.₄₅(C₂².D₄), (D₇×C₄⋊C₄)⋊₇C₂, (C₇×C₂.D₈)⋊₁₁C₂, (C₂×C₇⋊C₈).₆₉C₂², (C₂×C₄×D₇).₃₈C₂², (C₂×C₁₄).₃₀₃(C₂×D₄), (C₇×C₄⋊C₄).₉₁C₂², (C₂×C₄).₄₀₁(C₂²×D₇), SmallGroup(448,416)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — D₁₄.₅D₈

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — C₂×C₄×D₇ — D₇×C₄⋊C₄ — D₁₄.₅D₈

Lower central

C₇ — C₁₄ — C₂×C₂₈ — D₁₄.₅D₈

Upper central

C₁ — C₂² — C₂×C₄ — C₂.D₈

Generators and relations for D₁₄.₅D₈
G = < a,b,c,d | a¹⁴=b²=c⁸=1, d²=a⁷, bab=a^-1, ac=ca, ad=da, cbc^-1=dbd^-1=a⁷b, dcd^-1=c^-1 >

Subgroups: 748 in 114 conjugacy classes, 39 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, D₇, C₁₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂×D₄, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂²⋊C₈, D₄⋊C₄, C₂.D₈, C₂.D₈, C₂×C₄⋊C₄, C₄⋊D₄, C₇⋊C₈, C₅₆, C₄×D₇, D₂₈, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×D₇, C₂²×D₇, C₂².D₈, C₂×C₇⋊C₈, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₇×C₄⋊C₄, C₂×C₅₆, C₂×C₄×D₇, C₂×C₄×D₇, C₂×D₂₈, C₂×D₂₈, C₂₈.Q₈, C₁₄.D₈, D₁₄⋊C₈, C₂.D₅₆, C₇×C₂.D₈, D₇×C₄⋊C₄, C₄⋊D₂₈, D₁₄.₅D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, D₈, C₂×D₄, C₄○D₄, D₁₄, C₂².D₄, C₂×D₈, C₈.C₂², C₂²×D₇, C₂².D₈, C₄○D₂₈, D₄×D₇, Q₈⋊₂D₇, D₁₄.₅D₄, D₇×D₈, Q₁₆⋊D₇, D₁₄.₅D₈

Smallest permutation representation of D₁₄.₅D₈
►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 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 42)(43 52)(44 51)(45 50)(46 49)(47 48)(53 56)(54 55)(57 64)(58 63)(59 62)(60 61)(65 70)(66 69)(67 68)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(113 123)(114 122)(115 121)(116 120)(117 119)(124 126)(127 130)(128 129)(131 140)(132 139)(133 138)(134 137)(135 136)(141 148)(142 147)(143 146)(144 145)(149 154)(150 153)(151 152)(155 162)(156 161)(157 160)(158 159)(163 168)(164 167)(165 166)(169 181)(170 180)(171 179)(172 178)(173 177)(174 176)(183 189)(184 188)(185 187)(190 196)(191 195)(192 194)(197 207)(198 206)(199 205)(200 204)(201 203)(208 210)(211 213)(214 224)(215 223)(216 222)(217 221)(218 220)

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

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

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

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

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

61 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	28A	···	28F	28G	···	28R	56A	···	56L
order	1	2	2	2	2	2	2	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	56	2	2	4	4	8	28	28	28	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	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+		+	+	+		-	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₇	C₄○D₄	D₈	D₁₄	D₁₄	C₄○D₂₈	C₈.C₂²	Q₈⋊₂D₇	D₄×D₇	D₇×D₈	Q₁₆⋊D₇
kernel	D₁₄.₅D₈	C₂₈.Q₈	C₁₄.D₈	D₁₄⋊C₈	C₂.D₅₆	C₇×C₂.D₈	D₇×C₄⋊C₄	C₄⋊D₂₈	C₂×Dic₇	C₂²×D₇	C₂.D₈	C₂₈	D₁₄	C₄⋊C₄	C₂×C₈	C₄	C₁₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	1	1	3	4	4	6	3	12	1	3	3	6	6

Matrix representation of D₁₄.₅D₈ ►in GL₆(𝔽₁₁₃)

34	88	0	0	0	0
25	88	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	112	0	0	0
0	0	5	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

112	0	0	0	0	0
0	112	0	0	0	0
0	0	110	44	0	0
0	0	82	3	0	0
0	0	0	0	0	15
0	0	0	0	15	62

1	0	0	0	0	0
0	1	0	0	0	0
0	0	45	18	0	0
0	0	13	68	0	0
0	0	0	0	44	89
0	0	0	0	10	69

G:=sub<GL(6,GF(113))| [34,25,0,0,0,0,88,88,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,112,5,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,110,82,0,0,0,0,44,3,0,0,0,0,0,0,0,15,0,0,0,0,15,62],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,45,13,0,0,0,0,18,68,0,0,0,0,0,0,44,10,0,0,0,0,89,69] >;

D₁₄.₅D₈ in GAP, Magma, Sage, TeX

D_{14}._5D_8

% in TeX

G:=Group("D14.5D8");

// GroupNames label

G:=SmallGroup(448,416);

// by ID

G=gap.SmallGroup(448,416);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,254,219,268,851,102,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^8=1,d^2=a^7,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=c^-1>;

// generators/relations

G = D14.5D8 order 448 = 26·7

2nd non-split extension by D14 of D8 acting via D8/D4=C2

G = D₁₄.₅D₈ order 448 = 2⁶·7

2^nd non-split extension by D₁₄ of D₈ acting via D₈/D₄=C₂