Dic7:5SD16 - GroupNames

G = Dic₇⋊₅SD₁₆ order 448 = 2⁶·7

2^nd semidirect product of Dic₇ and SD₁₆ acting via SD₁₆/Q₈=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₇⋊₅SD₁₆, (C₇×Q₈)⋊₄D₄, (C₂×SD₁₆)⋊₇D₇, (Q₈×Dic₇)⋊₄C₂, Q₈⋊₁(C₇⋊D₄), C₇⋊₅(C₄⋊SD₁₆), Dic₇⋊C₈⋊₃₃C₂, (C₂×D₄).₆₈D₁₄, C₂₈.₁₇₁(C₂×D₄), (C₂×C₈).₁₄₄D₁₄, C₂₈⋊D₄.₅C₂, C₂.D₅₆⋊₃₃C₂, C₂.₂₇(D₇×SD₁₆), (C₁₄×SD₁₆)⋊₁₈C₂, C₂₈.₉₇(C₄○D₄), D₄⋊Dic₇⋊₃₂C₂, (C₂×Q₈).₁₁₃D₁₄, (C₂×Dic₇).₆₇D₄, C₁₄.₄₄(C₂×SD₁₆), C₂².₂₆₁(D₄×D₇), C₄.₁₀(D₄⋊₂D₇), C₂.₂₆(D₅₆⋊C₂), C₁₄.₇₅(C₈⋊C₂²), (C₂×C₅₆).₂₉₁C₂², (C₂×C₂₈).₄₄₀C₂³, (D₄×C₁₄).₈₉C₂², (Q₈×C₁₄).₇₀C₂², C₁₄.₁₁₃(C₄⋊D₄), (C₂×D₂₈).₁₁₇C₂², C₄⋊Dic₇.₁₇₀C₂², (C₄×Dic₇).₄₇C₂², C₂.₂₅(Dic₇⋊D₄), (C₂×Q₈⋊D₇)⋊₁₆C₂, C₄.₃₉(C₂×C₇⋊D₄), (C₂×C₁₄).₃₅₂(C₂×D₄), (C₂×C₇⋊C₈).₁₅₂C₂², (C₂×C₄).₅₂₉(C₂²×D₇), SmallGroup(448,697)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — Dic₇⋊₅SD₁₆

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₄×Dic₇ — C₂₈⋊D₄ — Dic₇⋊₅SD₁₆

Lower central

C₇ — C₁₄ — C₂×C₂₈ — Dic₇⋊₅SD₁₆

Upper central

C₁ — C₂² — C₂×C₄ — C₂×SD₁₆

Generators and relations for Dic₇⋊₅SD₁₆
G = < a,b,c,d | a¹⁴=c⁸=d²=1, b²=a⁷, bab^-1=a^-1, ac=ca, ad=da, cbc^-1=dbd=a⁷b, dcd=c³ >

Subgroups: 740 in 128 conjugacy classes, 43 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, D₇, C₁₄, C₁₄, C₄², C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂×D₄, C₂×D₄, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, D₄⋊C₄, C₄⋊C₈, C₄×Q₈, C₄⋊₁D₄, C₂×SD₁₆, C₂×SD₁₆, C₇⋊C₈, C₅₆, D₂₈, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, C₄⋊SD₁₆, C₂×C₇⋊C₈, C₄×Dic₇, C₄×Dic₇, C₄⋊Dic₇, C₄⋊Dic₇, Q₈⋊D₇, C₂×C₅₆, C₇×SD₁₆, C₂×D₂₈, C₂×C₇⋊D₄, D₄×C₁₄, Q₈×C₁₄, Dic₇⋊C₈, C₂.D₅₆, D₄⋊Dic₇, C₂₈⋊D₄, C₂×Q₈⋊D₇, Q₈×Dic₇, C₁₄×SD₁₆, Dic₇⋊₅SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, SD₁₆, C₂×D₄, C₄○D₄, D₁₄, C₄⋊D₄, C₂×SD₁₆, C₈⋊C₂², C₇⋊D₄, C₂²×D₇, C₄⋊SD₁₆, D₄×D₇, D₄⋊₂D₇, C₂×C₇⋊D₄, D₇×SD₁₆, D₅₆⋊C₂, Dic₇⋊D₄, Dic₇⋊₅SD₁₆

Smallest permutation representation of Dic₇⋊₅SD₁₆
►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)

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

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

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

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

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

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

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	14J	···	14O	28A	···	28F	28G	···	28L	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	14	···	14	28	···	28	28	···	28	56	···	56
size	1	1	1	1	8	56	2	2	4	4	14	14	28	28	28	2	2	2	4	4	28	28	2	···	2	8	···	8	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	SD₁₆	C₄○D₄	D₁₄	D₁₄	D₁₄	C₇⋊D₄	C₈⋊C₂²	D₄⋊₂D₇	D₄×D₇	D₇×SD₁₆	D₅₆⋊C₂
kernel	Dic₇⋊₅SD₁₆	Dic₇⋊C₈	C₂.D₅₆	D₄⋊Dic₇	C₂₈⋊D₄	C₂×Q₈⋊D₇	Q₈×Dic₇	C₁₄×SD₁₆	C₂×Dic₇	C₇×Q₈	C₂×SD₁₆	Dic₇	C₂₈	C₂×C₈	C₂×D₄	C₂×Q₈	Q₈	C₁₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	2	2	3	4	2	3	3	3	12	1	3	3	6	6

Matrix representation of Dic₇⋊₅SD₁₆ ►in GL₆(𝔽₁₁₃)

104	1	0	0	0	0
111	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

112	105	0	0	0	0
0	1	0	0	0	0
0	0	6	46	0	0
0	0	95	107	0	0
0	0	0	0	112	0
0	0	0	0	0	112

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	15	1	0	0
0	0	0	0	87	59
0	0	0	0	23	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	98	112	0	0
0	0	0	0	1	0
0	0	0	0	33	112

G:=sub<GL(6,GF(113))| [104,111,0,0,0,0,1,88,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,105,1,0,0,0,0,0,0,6,95,0,0,0,0,46,107,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,15,0,0,0,0,0,1,0,0,0,0,0,0,87,23,0,0,0,0,59,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,98,0,0,0,0,0,112,0,0,0,0,0,0,1,33,0,0,0,0,0,112] >;

Dic₇⋊₅SD₁₆ in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes_5{\rm SD}_{16}

% in TeX

G:=Group("Dic7:5SD16");

// GroupNames label

G:=SmallGroup(448,697);

// by ID

G=gap.SmallGroup(448,697);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,135,184,570,297,136,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^14=c^8=d^2=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^7*b,d*c*d=c^3>;

// generators/relations

G = Dic7⋊5SD16 order 448 = 26·7

2nd semidirect product of Dic7 and SD16 acting via SD16/Q8=C2

G = Dic₇⋊₅SD₁₆ order 448 = 2⁶·7

2^nd semidirect product of Dic₇ and SD₁₆ acting via SD₁₆/Q₈=C₂