metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7.Q8, C4⋊C4.5D7, C2.5(Q8×D7), (C2×C4).10D14, C4⋊Dic7.6C2, C14.11(C2×Q8), C7⋊2(C42.C2), Dic7⋊C4.5C2, (C4×Dic7).9C2, C14.11(C4○D4), C2.13(C4○D28), (C2×C14).30C23, (C2×C28).55C22, C2.11(D4⋊2D7), (C2×Dic7).9C22, C22.47(C22×D7), (C7×C4⋊C4).6C2, SmallGroup(224,84)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7.Q8
G = < a,b,c,d | a14=c4=1, b2=a7, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a7b, bd=db, dcd-1=a7c-1 >
Subgroups: 182 in 56 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C14, C42, C4⋊C4, C4⋊C4, Dic7, Dic7, C28, C2×C14, C42.C2, C2×Dic7, C2×C28, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, Dic7.Q8
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, D14, C42.C2, C22×D7, C4○D28, D4⋊2D7, Q8×D7, Dic7.Q8
►Regular action 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)
(1 184 8 191)(2 183 9 190)(3 196 10 189)(4 195 11 188)(5 194 12 187)(6 193 13 186)(7 192 14 185)(15 116 22 123)(16 115 23 122)(17 114 24 121)(18 113 25 120)(19 126 26 119)(20 125 27 118)(21 124 28 117)(29 80 36 73)(30 79 37 72)(31 78 38 71)(32 77 39 84)(33 76 40 83)(34 75 41 82)(35 74 42 81)(43 224 50 217)(44 223 51 216)(45 222 52 215)(46 221 53 214)(47 220 54 213)(48 219 55 212)(49 218 56 211)(57 201 64 208)(58 200 65 207)(59 199 66 206)(60 198 67 205)(61 197 68 204)(62 210 69 203)(63 209 70 202)(85 112 92 105)(86 111 93 104)(87 110 94 103)(88 109 95 102)(89 108 96 101)(90 107 97 100)(91 106 98 99)(127 153 134 146)(128 152 135 145)(129 151 136 144)(130 150 137 143)(131 149 138 142)(132 148 139 141)(133 147 140 154)(155 175 162 182)(156 174 163 181)(157 173 164 180)(158 172 165 179)(159 171 166 178)(160 170 167 177)(161 169 168 176)
(1 15 43 84)(2 16 44 71)(3 17 45 72)(4 18 46 73)(5 19 47 74)(6 20 48 75)(7 21 49 76)(8 22 50 77)(9 23 51 78)(10 24 52 79)(11 25 53 80)(12 26 54 81)(13 27 55 82)(14 28 56 83)(29 188 113 214)(30 189 114 215)(31 190 115 216)(32 191 116 217)(33 192 117 218)(34 193 118 219)(35 194 119 220)(36 195 120 221)(37 196 121 222)(38 183 122 223)(39 184 123 224)(40 185 124 211)(41 186 125 212)(42 187 126 213)(57 132 164 108)(58 133 165 109)(59 134 166 110)(60 135 167 111)(61 136 168 112)(62 137 155 99)(63 138 156 100)(64 139 157 101)(65 140 158 102)(66 127 159 103)(67 128 160 104)(68 129 161 105)(69 130 162 106)(70 131 163 107)(85 197 151 176)(86 198 152 177)(87 199 153 178)(88 200 154 179)(89 201 141 180)(90 202 142 181)(91 203 143 182)(92 204 144 169)(93 205 145 170)(94 206 146 171)(95 207 147 172)(96 208 148 173)(97 209 149 174)(98 210 150 175)
(1 93 43 145)(2 92 44 144)(3 91 45 143)(4 90 46 142)(5 89 47 141)(6 88 48 154)(7 87 49 153)(8 86 50 152)(9 85 51 151)(10 98 52 150)(11 97 53 149)(12 96 54 148)(13 95 55 147)(14 94 56 146)(15 177 84 198)(16 176 71 197)(17 175 72 210)(18 174 73 209)(19 173 74 208)(20 172 75 207)(21 171 76 206)(22 170 77 205)(23 169 78 204)(24 182 79 203)(25 181 80 202)(26 180 81 201)(27 179 82 200)(28 178 83 199)(29 70 113 163)(30 69 114 162)(31 68 115 161)(32 67 116 160)(33 66 117 159)(34 65 118 158)(35 64 119 157)(36 63 120 156)(37 62 121 155)(38 61 122 168)(39 60 123 167)(40 59 124 166)(41 58 125 165)(42 57 126 164)(99 215 137 189)(100 214 138 188)(101 213 139 187)(102 212 140 186)(103 211 127 185)(104 224 128 184)(105 223 129 183)(106 222 130 196)(107 221 131 195)(108 220 132 194)(109 219 133 193)(110 218 134 192)(111 217 135 191)(112 216 136 190)
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), (1,184,8,191)(2,183,9,190)(3,196,10,189)(4,195,11,188)(5,194,12,187)(6,193,13,186)(7,192,14,185)(15,116,22,123)(16,115,23,122)(17,114,24,121)(18,113,25,120)(19,126,26,119)(20,125,27,118)(21,124,28,117)(29,80,36,73)(30,79,37,72)(31,78,38,71)(32,77,39,84)(33,76,40,83)(34,75,41,82)(35,74,42,81)(43,224,50,217)(44,223,51,216)(45,222,52,215)(46,221,53,214)(47,220,54,213)(48,219,55,212)(49,218,56,211)(57,201,64,208)(58,200,65,207)(59,199,66,206)(60,198,67,205)(61,197,68,204)(62,210,69,203)(63,209,70,202)(85,112,92,105)(86,111,93,104)(87,110,94,103)(88,109,95,102)(89,108,96,101)(90,107,97,100)(91,106,98,99)(127,153,134,146)(128,152,135,145)(129,151,136,144)(130,150,137,143)(131,149,138,142)(132,148,139,141)(133,147,140,154)(155,175,162,182)(156,174,163,181)(157,173,164,180)(158,172,165,179)(159,171,166,178)(160,170,167,177)(161,169,168,176), (1,15,43,84)(2,16,44,71)(3,17,45,72)(4,18,46,73)(5,19,47,74)(6,20,48,75)(7,21,49,76)(8,22,50,77)(9,23,51,78)(10,24,52,79)(11,25,53,80)(12,26,54,81)(13,27,55,82)(14,28,56,83)(29,188,113,214)(30,189,114,215)(31,190,115,216)(32,191,116,217)(33,192,117,218)(34,193,118,219)(35,194,119,220)(36,195,120,221)(37,196,121,222)(38,183,122,223)(39,184,123,224)(40,185,124,211)(41,186,125,212)(42,187,126,213)(57,132,164,108)(58,133,165,109)(59,134,166,110)(60,135,167,111)(61,136,168,112)(62,137,155,99)(63,138,156,100)(64,139,157,101)(65,140,158,102)(66,127,159,103)(67,128,160,104)(68,129,161,105)(69,130,162,106)(70,131,163,107)(85,197,151,176)(86,198,152,177)(87,199,153,178)(88,200,154,179)(89,201,141,180)(90,202,142,181)(91,203,143,182)(92,204,144,169)(93,205,145,170)(94,206,146,171)(95,207,147,172)(96,208,148,173)(97,209,149,174)(98,210,150,175), (1,93,43,145)(2,92,44,144)(3,91,45,143)(4,90,46,142)(5,89,47,141)(6,88,48,154)(7,87,49,153)(8,86,50,152)(9,85,51,151)(10,98,52,150)(11,97,53,149)(12,96,54,148)(13,95,55,147)(14,94,56,146)(15,177,84,198)(16,176,71,197)(17,175,72,210)(18,174,73,209)(19,173,74,208)(20,172,75,207)(21,171,76,206)(22,170,77,205)(23,169,78,204)(24,182,79,203)(25,181,80,202)(26,180,81,201)(27,179,82,200)(28,178,83,199)(29,70,113,163)(30,69,114,162)(31,68,115,161)(32,67,116,160)(33,66,117,159)(34,65,118,158)(35,64,119,157)(36,63,120,156)(37,62,121,155)(38,61,122,168)(39,60,123,167)(40,59,124,166)(41,58,125,165)(42,57,126,164)(99,215,137,189)(100,214,138,188)(101,213,139,187)(102,212,140,186)(103,211,127,185)(104,224,128,184)(105,223,129,183)(106,222,130,196)(107,221,131,195)(108,220,132,194)(109,219,133,193)(110,218,134,192)(111,217,135,191)(112,216,136,190)>;
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), (1,184,8,191)(2,183,9,190)(3,196,10,189)(4,195,11,188)(5,194,12,187)(6,193,13,186)(7,192,14,185)(15,116,22,123)(16,115,23,122)(17,114,24,121)(18,113,25,120)(19,126,26,119)(20,125,27,118)(21,124,28,117)(29,80,36,73)(30,79,37,72)(31,78,38,71)(32,77,39,84)(33,76,40,83)(34,75,41,82)(35,74,42,81)(43,224,50,217)(44,223,51,216)(45,222,52,215)(46,221,53,214)(47,220,54,213)(48,219,55,212)(49,218,56,211)(57,201,64,208)(58,200,65,207)(59,199,66,206)(60,198,67,205)(61,197,68,204)(62,210,69,203)(63,209,70,202)(85,112,92,105)(86,111,93,104)(87,110,94,103)(88,109,95,102)(89,108,96,101)(90,107,97,100)(91,106,98,99)(127,153,134,146)(128,152,135,145)(129,151,136,144)(130,150,137,143)(131,149,138,142)(132,148,139,141)(133,147,140,154)(155,175,162,182)(156,174,163,181)(157,173,164,180)(158,172,165,179)(159,171,166,178)(160,170,167,177)(161,169,168,176), (1,15,43,84)(2,16,44,71)(3,17,45,72)(4,18,46,73)(5,19,47,74)(6,20,48,75)(7,21,49,76)(8,22,50,77)(9,23,51,78)(10,24,52,79)(11,25,53,80)(12,26,54,81)(13,27,55,82)(14,28,56,83)(29,188,113,214)(30,189,114,215)(31,190,115,216)(32,191,116,217)(33,192,117,218)(34,193,118,219)(35,194,119,220)(36,195,120,221)(37,196,121,222)(38,183,122,223)(39,184,123,224)(40,185,124,211)(41,186,125,212)(42,187,126,213)(57,132,164,108)(58,133,165,109)(59,134,166,110)(60,135,167,111)(61,136,168,112)(62,137,155,99)(63,138,156,100)(64,139,157,101)(65,140,158,102)(66,127,159,103)(67,128,160,104)(68,129,161,105)(69,130,162,106)(70,131,163,107)(85,197,151,176)(86,198,152,177)(87,199,153,178)(88,200,154,179)(89,201,141,180)(90,202,142,181)(91,203,143,182)(92,204,144,169)(93,205,145,170)(94,206,146,171)(95,207,147,172)(96,208,148,173)(97,209,149,174)(98,210,150,175), (1,93,43,145)(2,92,44,144)(3,91,45,143)(4,90,46,142)(5,89,47,141)(6,88,48,154)(7,87,49,153)(8,86,50,152)(9,85,51,151)(10,98,52,150)(11,97,53,149)(12,96,54,148)(13,95,55,147)(14,94,56,146)(15,177,84,198)(16,176,71,197)(17,175,72,210)(18,174,73,209)(19,173,74,208)(20,172,75,207)(21,171,76,206)(22,170,77,205)(23,169,78,204)(24,182,79,203)(25,181,80,202)(26,180,81,201)(27,179,82,200)(28,178,83,199)(29,70,113,163)(30,69,114,162)(31,68,115,161)(32,67,116,160)(33,66,117,159)(34,65,118,158)(35,64,119,157)(36,63,120,156)(37,62,121,155)(38,61,122,168)(39,60,123,167)(40,59,124,166)(41,58,125,165)(42,57,126,164)(99,215,137,189)(100,214,138,188)(101,213,139,187)(102,212,140,186)(103,211,127,185)(104,224,128,184)(105,223,129,183)(106,222,130,196)(107,221,131,195)(108,220,132,194)(109,219,133,193)(110,218,134,192)(111,217,135,191)(112,216,136,190) );
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)], [(1,184,8,191),(2,183,9,190),(3,196,10,189),(4,195,11,188),(5,194,12,187),(6,193,13,186),(7,192,14,185),(15,116,22,123),(16,115,23,122),(17,114,24,121),(18,113,25,120),(19,126,26,119),(20,125,27,118),(21,124,28,117),(29,80,36,73),(30,79,37,72),(31,78,38,71),(32,77,39,84),(33,76,40,83),(34,75,41,82),(35,74,42,81),(43,224,50,217),(44,223,51,216),(45,222,52,215),(46,221,53,214),(47,220,54,213),(48,219,55,212),(49,218,56,211),(57,201,64,208),(58,200,65,207),(59,199,66,206),(60,198,67,205),(61,197,68,204),(62,210,69,203),(63,209,70,202),(85,112,92,105),(86,111,93,104),(87,110,94,103),(88,109,95,102),(89,108,96,101),(90,107,97,100),(91,106,98,99),(127,153,134,146),(128,152,135,145),(129,151,136,144),(130,150,137,143),(131,149,138,142),(132,148,139,141),(133,147,140,154),(155,175,162,182),(156,174,163,181),(157,173,164,180),(158,172,165,179),(159,171,166,178),(160,170,167,177),(161,169,168,176)], [(1,15,43,84),(2,16,44,71),(3,17,45,72),(4,18,46,73),(5,19,47,74),(6,20,48,75),(7,21,49,76),(8,22,50,77),(9,23,51,78),(10,24,52,79),(11,25,53,80),(12,26,54,81),(13,27,55,82),(14,28,56,83),(29,188,113,214),(30,189,114,215),(31,190,115,216),(32,191,116,217),(33,192,117,218),(34,193,118,219),(35,194,119,220),(36,195,120,221),(37,196,121,222),(38,183,122,223),(39,184,123,224),(40,185,124,211),(41,186,125,212),(42,187,126,213),(57,132,164,108),(58,133,165,109),(59,134,166,110),(60,135,167,111),(61,136,168,112),(62,137,155,99),(63,138,156,100),(64,139,157,101),(65,140,158,102),(66,127,159,103),(67,128,160,104),(68,129,161,105),(69,130,162,106),(70,131,163,107),(85,197,151,176),(86,198,152,177),(87,199,153,178),(88,200,154,179),(89,201,141,180),(90,202,142,181),(91,203,143,182),(92,204,144,169),(93,205,145,170),(94,206,146,171),(95,207,147,172),(96,208,148,173),(97,209,149,174),(98,210,150,175)], [(1,93,43,145),(2,92,44,144),(3,91,45,143),(4,90,46,142),(5,89,47,141),(6,88,48,154),(7,87,49,153),(8,86,50,152),(9,85,51,151),(10,98,52,150),(11,97,53,149),(12,96,54,148),(13,95,55,147),(14,94,56,146),(15,177,84,198),(16,176,71,197),(17,175,72,210),(18,174,73,209),(19,173,74,208),(20,172,75,207),(21,171,76,206),(22,170,77,205),(23,169,78,204),(24,182,79,203),(25,181,80,202),(26,180,81,201),(27,179,82,200),(28,178,83,199),(29,70,113,163),(30,69,114,162),(31,68,115,161),(32,67,116,160),(33,66,117,159),(34,65,118,158),(35,64,119,157),(36,63,120,156),(37,62,121,155),(38,61,122,168),(39,60,123,167),(40,59,124,166),(41,58,125,165),(42,57,126,164),(99,215,137,189),(100,214,138,188),(101,213,139,187),(102,212,140,186),(103,211,127,185),(104,224,128,184),(105,223,129,183),(106,222,130,196),(107,221,131,195),(108,220,132,194),(109,219,133,193),(110,218,134,192),(111,217,135,191),(112,216,136,190)]])
Dic7.Q8 is a maximal subgroup of
C14.102+ 1+4 C14.52- 1+4 C14.62- 1+4 C42.89D14 C42.93D14 C42.96D14 C42.102D14 C42.104D14 C42.105D14 C42.118D14 Dic14⋊10Q8 C42.232D14 C42.132D14 C42.134D14 C14.342+ 1+4 C14.352+ 1+4 C14.442+ 1+4 C14.492+ 1+4 (Q8×Dic7)⋊C2 C14.752- 1+4 C14.152- 1+4 C14.1182+ 1+4 C14.522+ 1+4 C14.202- 1+4 C14.212- 1+4 C14.582+ 1+4 C14.262- 1+4 C4⋊C4.197D14 C14.802- 1+4 C14.602+ 1+4 C14.832- 1+4 C14.642+ 1+4 C14.662+ 1+4 C14.852- 1+4 Dic14⋊7Q8 C42.147D14 D7×C42.C2 C42.148D14 C42.150D14 C42.151D14 C42.154D14 C42.159D14 C42.189D14 C42.162D14 C42.163D14 C42.165D14 Dic14⋊8Q8 C42.174D14 C42.176D14 C42.180D14
Dic7.Q8 is a maximal quotient of
C14.(C4×Q8) C7⋊(C42⋊8C4) Dic7⋊C4⋊C4 C4⋊Dic7⋊7C4 C2.(C28⋊Q8) (C2×Dic7).Q8 (C2×C28).28D4 Dic7⋊(C4⋊C4) C22.23(Q8×D7) (C2×C28).287D4 C4⋊C4⋊5Dic7 (C2×C28).288D4 (C2×C4).44D28 (C2×C28).54D4
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | C4○D28 | D4⋊2D7 | Q8×D7 |
| kernel | Dic7.Q8 | C4×Dic7 | Dic7⋊C4 | C4⋊Dic7 | C7×C4⋊C4 | Dic7 | C4⋊C4 | C14 | C2×C4 | C2 | C2 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 3 | 4 | 9 | 12 | 3 | 3 |
Matrix representation of Dic7.Q8 ►in GL4(𝔽29) generated by
| 1 | 28 | 0 | 0 |
| 5 | 25 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 18 | 24 | 0 | 0 |
| 7 | 11 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 16 | 26 |
| 11 | 13 | 0 | 0 |
| 22 | 18 | 0 | 0 |
| 0 | 0 | 22 | 22 |
| 0 | 0 | 3 | 7 |
| 16 | 2 | 0 | 0 |
| 3 | 13 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
G:=sub<GL(4,GF(29))| [1,5,0,0,28,25,0,0,0,0,28,0,0,0,0,28],[18,7,0,0,24,11,0,0,0,0,3,16,0,0,3,26],[11,22,0,0,13,18,0,0,0,0,22,3,0,0,22,7],[16,3,0,0,2,13,0,0,0,0,17,0,0,0,0,17] >;
Dic7.Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_7.Q_8 % in TeX
G:=Group("Dic7.Q8"); // GroupNames label
G:=SmallGroup(224,84);
// by ID
G=gap.SmallGroup(224,84);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,55,218,188,86,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^4=1,b^2=a^7,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=a^7*c^-1>;
// generators/relations