Dic7:3SD16 - GroupNames

G = Dic₇⋊₃SD₁₆ order 448 = 2⁶·7

2^nd semidirect product of Dic₇ and SD₁₆ acting via SD₁₆/D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₇⋊₃SD₁₆, (C₇×D₄).₇D₄, Dic₇⋊C₈⋊₃₂C₂, (C₂×C₈).₁₄₃D₁₄, C₂₈.₁₇₀(C₂×D₄), C₇⋊₅(D₄.D₄), D₄.₂(C₇⋊D₄), (C₂×Q₈).₅₀D₁₄, (D₄×Dic₇).₈C₂, (C₂×SD₁₆).₃D₇, C₂.₂₆(D₇×SD₁₆), Dic₇⋊Q₈⋊₃C₂, (C₂×D₄).₁₄₃D₁₄, C₂₈.₉₆(C₄○D₄), Q₈⋊Dic₇⋊₂₅C₂, C₄.₉(D₄⋊₂D₇), (C₁₄×SD₁₆).₆C₂, C₁₄.₄₃(C₂×SD₁₆), C₂².₂₆₀(D₄×D₇), C₂₈.₄₄D₄⋊₃₃C₂, (C₂×C₂₈).₄₃₉C₂³, (C₂×C₅₆).₂₉₀C₂², (C₂×Dic₇).₁₈₂D₄, (D₄×C₁₄).₈₈C₂², (Q₈×C₁₄).₆₉C₂², C₁₄.₁₁₂(C₄⋊D₄), C₂.₂₆(SD₁₆⋊D₇), C₁₄.₄₅(C₈.C₂²), C₄⋊Dic₇.₁₆₉C₂², (C₄×Dic₇).₄₆C₂², C₂.₂₄(Dic₇⋊D₄), (C₂×Dic₁₄).₁₂₃C₂², C₄.₃₈(C₂×C₇⋊D₄), (C₂×D₄.D₇).₈C₂, (C₂×C₁₄).₃₅₁(C₂×D₄), (C₂×C₇⋊C₈).₁₅₁C₂², (C₂×C₄).₅₂₈(C₂²×D₇), SmallGroup(448,696)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — Dic₇⋊₃SD₁₆

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₄×Dic₇ — D₄×Dic₇ — Dic₇⋊₃SD₁₆

Lower central

C₇ — C₁₄ — C₂×C₂₈ — Dic₇⋊₃SD₁₆

Upper central

C₁ — C₂² — C₂×C₄ — C₂×SD₁₆

Generators and relations for Dic₇⋊₃SD₁₆
G = < a,b,c,d | a¹⁴=c⁸=d²=1, b²=a⁷, bab^-1=a^-1, ac=ca, ad=da, cbc^-1=a⁷b, bd=db, dcd=c³ >

Subgroups: 548 in 120 conjugacy classes, 43 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, C₂×C₁₄, C₂×C₁₄, Q₈⋊C₄, C₄⋊C₈, C₄×D₄, C₄⋊Q₈, C₂×SD₁₆, C₂×SD₁₆, C₇⋊C₈, C₅₆, Dic₁₄, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×D₄, C₇×Q₈, C₂²×C₁₄, D₄.D₄, C₂×C₇⋊C₈, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, D₄.D₇, C₂³.D₇, C₂×C₅₆, C₇×SD₁₆, C₂×Dic₁₄, C₂²×Dic₇, D₄×C₁₄, Q₈×C₁₄, Dic₇⋊C₈, C₂₈.₄₄D₄, Q₈⋊Dic₇, C₂×D₄.D₇, D₄×Dic₇, Dic₇⋊Q₈, C₁₄×SD₁₆, Dic₇⋊₃SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, SD₁₆, C₂×D₄, C₄○D₄, D₁₄, C₄⋊D₄, C₂×SD₁₆, C₈.C₂², C₇⋊D₄, C₂²×D₇, D₄.D₄, D₄×D₇, D₄⋊₂D₇, C₂×C₇⋊D₄, D₇×SD₁₆, SD₁₆⋊D₇, Dic₇⋊D₄, Dic₇⋊₃SD₁₆

Smallest permutation representation of Dic₇⋊₃SD₁₆
►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 43 8 50)(2 56 9 49)(3 55 10 48)(4 54 11 47)(5 53 12 46)(6 52 13 45)(7 51 14 44)(15 71 22 78)(16 84 23 77)(17 83 24 76)(18 82 25 75)(19 81 26 74)(20 80 27 73)(21 79 28 72)(29 212 36 219)(30 211 37 218)(31 224 38 217)(32 223 39 216)(33 222 40 215)(34 221 41 214)(35 220 42 213)(57 179 64 172)(58 178 65 171)(59 177 66 170)(60 176 67 169)(61 175 68 182)(62 174 69 181)(63 173 70 180)(85 202 92 209)(86 201 93 208)(87 200 94 207)(88 199 95 206)(89 198 96 205)(90 197 97 204)(91 210 98 203)(99 196 106 189)(100 195 107 188)(101 194 108 187)(102 193 109 186)(103 192 110 185)(104 191 111 184)(105 190 112 183)(113 165 120 158)(114 164 121 157)(115 163 122 156)(116 162 123 155)(117 161 124 168)(118 160 125 167)(119 159 126 166)(127 143 134 150)(128 142 135 149)(129 141 136 148)(130 154 137 147)(131 153 138 146)(132 152 139 145)(133 151 140 144)

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

(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 136)(22 137)(23 138)(24 139)(25 140)(26 127)(27 128)(28 129)(29 170)(30 171)(31 172)(32 173)(33 174)(34 175)(35 176)(36 177)(37 178)(38 179)(39 180)(40 181)(41 182)(42 169)(57 224)(58 211)(59 212)(60 213)(61 214)(62 215)(63 216)(64 217)(65 218)(66 219)(67 220)(68 221)(69 222)(70 223)(71 154)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(81 150)(82 151)(83 152)(84 153)(99 157)(100 158)(101 159)(102 160)(103 161)(104 162)(105 163)(106 164)(107 165)(108 166)(109 167)(110 168)(111 155)(112 156)(113 195)(114 196)(115 183)(116 184)(117 185)(118 186)(119 187)(120 188)(121 189)(122 190)(123 191)(124 192)(125 193)(126 194)

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

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

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

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	4	4	2	2	8	14	14	28	28	28	56	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	SD₁₆⋊D₇
kernel	Dic₇⋊₃SD₁₆	Dic₇⋊C₈	C₂₈.₄₄D₄	Q₈⋊Dic₇	C₂×D₄.D₇	D₄×Dic₇	Dic₇⋊Q₈	C₁₄×SD₁₆	C₂×Dic₇	C₇×D₄	C₂×SD₁₆	Dic₇	C₂₈	C₂×C₈	C₂×D₄	C₂×Q₈	D₄	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⋊₃SD₁₆ ►in GL₆(𝔽₁₁₃)

112	0	0	0	0	0
0	112	0	0	0	0
0	0	0	1	0	0
0	0	112	9	0	0
0	0	0	0	1	0
0	0	0	0	0	1

59	14	0	0	0	0
58	54	0	0	0	0
0	0	48	41	0	0
0	0	21	65	0	0
0	0	0	0	112	0
0	0	0	0	0	112

112	41	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	87	60
0	0	0	0	81	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	72	112

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[59,58,0,0,0,0,14,54,0,0,0,0,0,0,48,21,0,0,0,0,41,65,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,41,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,87,81,0,0,0,0,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,112] >;

Dic₇⋊₃SD₁₆ in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes_3{\rm SD}_{16}

% in TeX

G:=Group("Dic7:3SD16");

// GroupNames label

G:=SmallGroup(448,696);

// by ID

G=gap.SmallGroup(448,696);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,422,135,184,570,297,136,18822]);

// 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=a^7*b,b*d=d*b,d*c*d=c^3>;

// generators/relations

G = Dic7⋊3SD16 order 448 = 26·7

2nd semidirect product of Dic7 and SD16 acting via SD16/D4=C2

G = Dic₇⋊₃SD₁₆ order 448 = 2⁶·7

2^nd semidirect product of Dic₇ and SD₁₆ acting via SD₁₆/D₄=C₂