metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊8Q8, Dic14⋊12D4, C42.173D14, C14.362- 1+4, C7⋊4(D4×Q8), C4⋊1(Q8×D7), C4⋊Q8⋊11D7, C28⋊3(C2×Q8), C28⋊Q8⋊44C2, D14⋊7(C2×Q8), C4.74(D4×D7), C28.72(C2×D4), C4⋊C4.218D14, (C4×D28).26C2, D14⋊3Q8⋊36C2, D14⋊Q8⋊48C2, (C4×Dic14)⋊52C2, (C2×Q8).146D14, Dic7.30(C2×D4), D28⋊C4.13C2, C14.47(C22×Q8), (C2×C28).104C23, (C2×C14).271C24, (C4×C28).212C22, C14.101(C22×D4), D14⋊C4.152C22, (C2×D28).272C22, C4⋊Dic7.385C22, (Q8×C14).138C22, C22.292(C23×D7), Dic7⋊C4.166C22, (C4×Dic7).160C22, (C2×Dic7).142C23, (C22×D7).232C23, C2.37(Q8.10D14), (C2×Dic14).189C22, (C2×Q8×D7)⋊13C2, C2.74(C2×D4×D7), C2.30(C2×Q8×D7), (C7×C4⋊Q8)⋊13C2, (C2×C4×D7).145C22, (C7×C4⋊C4).214C22, (C2×C4).218(C22×D7), SmallGroup(448,1180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊8Q8
G = < a,b,c,d | a28=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a15, bc=cb, dbd-1=a14b, dcd-1=c-1 >
Subgroups: 1260 in 280 conjugacy classes, 115 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C22×Q8, Dic14, Dic14, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, D4×Q8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C4×C28, C7×C4⋊C4, C2×Dic14, C2×Dic14, C2×C4×D7, C2×D28, Q8×D7, Q8×C14, C4×Dic14, C4×D28, C28⋊Q8, D28⋊C4, D14⋊Q8, D14⋊3Q8, C7×C4⋊Q8, C2×Q8×D7, D28⋊8Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C24, D14, C22×D4, C22×Q8, 2- 1+4, C22×D7, D4×Q8, D4×D7, Q8×D7, C23×D7, C2×D4×D7, C2×Q8×D7, Q8.10D14, D28⋊8Q8
►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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(50 56)(51 55)(52 54)(57 65)(58 64)(59 63)(60 62)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(104 112)(105 111)(106 110)(107 109)(113 125)(114 124)(115 123)(116 122)(117 121)(118 120)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(141 143)(144 168)(145 167)(146 166)(147 165)(148 164)(149 163)(150 162)(151 161)(152 160)(153 159)(154 158)(155 157)(169 175)(170 174)(171 173)(176 196)(177 195)(178 194)(179 193)(180 192)(181 191)(182 190)(183 189)(184 188)(185 187)(197 199)(200 224)(201 223)(202 222)(203 221)(204 220)(205 219)(206 218)(207 217)(208 216)(209 215)(210 214)(211 213)
(1 169 130 153)(2 170 131 154)(3 171 132 155)(4 172 133 156)(5 173 134 157)(6 174 135 158)(7 175 136 159)(8 176 137 160)(9 177 138 161)(10 178 139 162)(11 179 140 163)(12 180 113 164)(13 181 114 165)(14 182 115 166)(15 183 116 167)(16 184 117 168)(17 185 118 141)(18 186 119 142)(19 187 120 143)(20 188 121 144)(21 189 122 145)(22 190 123 146)(23 191 124 147)(24 192 125 148)(25 193 126 149)(26 194 127 150)(27 195 128 151)(28 196 129 152)(29 79 98 202)(30 80 99 203)(31 81 100 204)(32 82 101 205)(33 83 102 206)(34 84 103 207)(35 57 104 208)(36 58 105 209)(37 59 106 210)(38 60 107 211)(39 61 108 212)(40 62 109 213)(41 63 110 214)(42 64 111 215)(43 65 112 216)(44 66 85 217)(45 67 86 218)(46 68 87 219)(47 69 88 220)(48 70 89 221)(49 71 90 222)(50 72 91 223)(51 73 92 224)(52 74 93 197)(53 75 94 198)(54 76 95 199)(55 77 96 200)(56 78 97 201)
(1 112 130 43)(2 99 131 30)(3 86 132 45)(4 101 133 32)(5 88 134 47)(6 103 135 34)(7 90 136 49)(8 105 137 36)(9 92 138 51)(10 107 139 38)(11 94 140 53)(12 109 113 40)(13 96 114 55)(14 111 115 42)(15 98 116 29)(16 85 117 44)(17 100 118 31)(18 87 119 46)(19 102 120 33)(20 89 121 48)(21 104 122 35)(22 91 123 50)(23 106 124 37)(24 93 125 52)(25 108 126 39)(26 95 127 54)(27 110 128 41)(28 97 129 56)(57 145 208 189)(58 160 209 176)(59 147 210 191)(60 162 211 178)(61 149 212 193)(62 164 213 180)(63 151 214 195)(64 166 215 182)(65 153 216 169)(66 168 217 184)(67 155 218 171)(68 142 219 186)(69 157 220 173)(70 144 221 188)(71 159 222 175)(72 146 223 190)(73 161 224 177)(74 148 197 192)(75 163 198 179)(76 150 199 194)(77 165 200 181)(78 152 201 196)(79 167 202 183)(80 154 203 170)(81 141 204 185)(82 156 205 172)(83 143 206 187)(84 158 207 174)
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(50,56)(51,55)(52,54)(57,65)(58,64)(59,63)(60,62)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(104,112)(105,111)(106,110)(107,109)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,143)(144,168)(145,167)(146,166)(147,165)(148,164)(149,163)(150,162)(151,161)(152,160)(153,159)(154,158)(155,157)(169,175)(170,174)(171,173)(176,196)(177,195)(178,194)(179,193)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187)(197,199)(200,224)(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)(208,216)(209,215)(210,214)(211,213), (1,169,130,153)(2,170,131,154)(3,171,132,155)(4,172,133,156)(5,173,134,157)(6,174,135,158)(7,175,136,159)(8,176,137,160)(9,177,138,161)(10,178,139,162)(11,179,140,163)(12,180,113,164)(13,181,114,165)(14,182,115,166)(15,183,116,167)(16,184,117,168)(17,185,118,141)(18,186,119,142)(19,187,120,143)(20,188,121,144)(21,189,122,145)(22,190,123,146)(23,191,124,147)(24,192,125,148)(25,193,126,149)(26,194,127,150)(27,195,128,151)(28,196,129,152)(29,79,98,202)(30,80,99,203)(31,81,100,204)(32,82,101,205)(33,83,102,206)(34,84,103,207)(35,57,104,208)(36,58,105,209)(37,59,106,210)(38,60,107,211)(39,61,108,212)(40,62,109,213)(41,63,110,214)(42,64,111,215)(43,65,112,216)(44,66,85,217)(45,67,86,218)(46,68,87,219)(47,69,88,220)(48,70,89,221)(49,71,90,222)(50,72,91,223)(51,73,92,224)(52,74,93,197)(53,75,94,198)(54,76,95,199)(55,77,96,200)(56,78,97,201), (1,112,130,43)(2,99,131,30)(3,86,132,45)(4,101,133,32)(5,88,134,47)(6,103,135,34)(7,90,136,49)(8,105,137,36)(9,92,138,51)(10,107,139,38)(11,94,140,53)(12,109,113,40)(13,96,114,55)(14,111,115,42)(15,98,116,29)(16,85,117,44)(17,100,118,31)(18,87,119,46)(19,102,120,33)(20,89,121,48)(21,104,122,35)(22,91,123,50)(23,106,124,37)(24,93,125,52)(25,108,126,39)(26,95,127,54)(27,110,128,41)(28,97,129,56)(57,145,208,189)(58,160,209,176)(59,147,210,191)(60,162,211,178)(61,149,212,193)(62,164,213,180)(63,151,214,195)(64,166,215,182)(65,153,216,169)(66,168,217,184)(67,155,218,171)(68,142,219,186)(69,157,220,173)(70,144,221,188)(71,159,222,175)(72,146,223,190)(73,161,224,177)(74,148,197,192)(75,163,198,179)(76,150,199,194)(77,165,200,181)(78,152,201,196)(79,167,202,183)(80,154,203,170)(81,141,204,185)(82,156,205,172)(83,143,206,187)(84,158,207,174)>;
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(50,56)(51,55)(52,54)(57,65)(58,64)(59,63)(60,62)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(104,112)(105,111)(106,110)(107,109)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,143)(144,168)(145,167)(146,166)(147,165)(148,164)(149,163)(150,162)(151,161)(152,160)(153,159)(154,158)(155,157)(169,175)(170,174)(171,173)(176,196)(177,195)(178,194)(179,193)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187)(197,199)(200,224)(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)(208,216)(209,215)(210,214)(211,213), (1,169,130,153)(2,170,131,154)(3,171,132,155)(4,172,133,156)(5,173,134,157)(6,174,135,158)(7,175,136,159)(8,176,137,160)(9,177,138,161)(10,178,139,162)(11,179,140,163)(12,180,113,164)(13,181,114,165)(14,182,115,166)(15,183,116,167)(16,184,117,168)(17,185,118,141)(18,186,119,142)(19,187,120,143)(20,188,121,144)(21,189,122,145)(22,190,123,146)(23,191,124,147)(24,192,125,148)(25,193,126,149)(26,194,127,150)(27,195,128,151)(28,196,129,152)(29,79,98,202)(30,80,99,203)(31,81,100,204)(32,82,101,205)(33,83,102,206)(34,84,103,207)(35,57,104,208)(36,58,105,209)(37,59,106,210)(38,60,107,211)(39,61,108,212)(40,62,109,213)(41,63,110,214)(42,64,111,215)(43,65,112,216)(44,66,85,217)(45,67,86,218)(46,68,87,219)(47,69,88,220)(48,70,89,221)(49,71,90,222)(50,72,91,223)(51,73,92,224)(52,74,93,197)(53,75,94,198)(54,76,95,199)(55,77,96,200)(56,78,97,201), (1,112,130,43)(2,99,131,30)(3,86,132,45)(4,101,133,32)(5,88,134,47)(6,103,135,34)(7,90,136,49)(8,105,137,36)(9,92,138,51)(10,107,139,38)(11,94,140,53)(12,109,113,40)(13,96,114,55)(14,111,115,42)(15,98,116,29)(16,85,117,44)(17,100,118,31)(18,87,119,46)(19,102,120,33)(20,89,121,48)(21,104,122,35)(22,91,123,50)(23,106,124,37)(24,93,125,52)(25,108,126,39)(26,95,127,54)(27,110,128,41)(28,97,129,56)(57,145,208,189)(58,160,209,176)(59,147,210,191)(60,162,211,178)(61,149,212,193)(62,164,213,180)(63,151,214,195)(64,166,215,182)(65,153,216,169)(66,168,217,184)(67,155,218,171)(68,142,219,186)(69,157,220,173)(70,144,221,188)(71,159,222,175)(72,146,223,190)(73,161,224,177)(74,148,197,192)(75,163,198,179)(76,150,199,194)(77,165,200,181)(78,152,201,196)(79,167,202,183)(80,154,203,170)(81,141,204,185)(82,156,205,172)(83,143,206,187)(84,158,207,174) );
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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(50,56),(51,55),(52,54),(57,65),(58,64),(59,63),(60,62),(66,84),(67,83),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(85,103),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(104,112),(105,111),(106,110),(107,109),(113,125),(114,124),(115,123),(116,122),(117,121),(118,120),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(141,143),(144,168),(145,167),(146,166),(147,165),(148,164),(149,163),(150,162),(151,161),(152,160),(153,159),(154,158),(155,157),(169,175),(170,174),(171,173),(176,196),(177,195),(178,194),(179,193),(180,192),(181,191),(182,190),(183,189),(184,188),(185,187),(197,199),(200,224),(201,223),(202,222),(203,221),(204,220),(205,219),(206,218),(207,217),(208,216),(209,215),(210,214),(211,213)], [(1,169,130,153),(2,170,131,154),(3,171,132,155),(4,172,133,156),(5,173,134,157),(6,174,135,158),(7,175,136,159),(8,176,137,160),(9,177,138,161),(10,178,139,162),(11,179,140,163),(12,180,113,164),(13,181,114,165),(14,182,115,166),(15,183,116,167),(16,184,117,168),(17,185,118,141),(18,186,119,142),(19,187,120,143),(20,188,121,144),(21,189,122,145),(22,190,123,146),(23,191,124,147),(24,192,125,148),(25,193,126,149),(26,194,127,150),(27,195,128,151),(28,196,129,152),(29,79,98,202),(30,80,99,203),(31,81,100,204),(32,82,101,205),(33,83,102,206),(34,84,103,207),(35,57,104,208),(36,58,105,209),(37,59,106,210),(38,60,107,211),(39,61,108,212),(40,62,109,213),(41,63,110,214),(42,64,111,215),(43,65,112,216),(44,66,85,217),(45,67,86,218),(46,68,87,219),(47,69,88,220),(48,70,89,221),(49,71,90,222),(50,72,91,223),(51,73,92,224),(52,74,93,197),(53,75,94,198),(54,76,95,199),(55,77,96,200),(56,78,97,201)], [(1,112,130,43),(2,99,131,30),(3,86,132,45),(4,101,133,32),(5,88,134,47),(6,103,135,34),(7,90,136,49),(8,105,137,36),(9,92,138,51),(10,107,139,38),(11,94,140,53),(12,109,113,40),(13,96,114,55),(14,111,115,42),(15,98,116,29),(16,85,117,44),(17,100,118,31),(18,87,119,46),(19,102,120,33),(20,89,121,48),(21,104,122,35),(22,91,123,50),(23,106,124,37),(24,93,125,52),(25,108,126,39),(26,95,127,54),(27,110,128,41),(28,97,129,56),(57,145,208,189),(58,160,209,176),(59,147,210,191),(60,162,211,178),(61,149,212,193),(62,164,213,180),(63,151,214,195),(64,166,215,182),(65,153,216,169),(66,168,217,184),(67,155,218,171),(68,142,219,186),(69,157,220,173),(70,144,221,188),(71,159,222,175),(72,146,223,190),(73,161,224,177),(74,148,197,192),(75,163,198,179),(76,150,199,194),(77,165,200,181),(78,152,201,196),(79,167,202,183),(80,154,203,170),(81,141,204,185),(82,156,205,172),(83,143,206,187),(84,158,207,174)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | D14 | D14 | D14 | 2- 1+4 | D4×D7 | Q8×D7 | Q8.10D14 |
| kernel | D28⋊8Q8 | C4×Dic14 | C4×D28 | C28⋊Q8 | D28⋊C4 | D14⋊Q8 | D14⋊3Q8 | C7×C4⋊Q8 | C2×Q8×D7 | Dic14 | D28 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C14 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 4 | 3 | 3 | 12 | 6 | 1 | 6 | 6 | 6 |
Matrix representation of D28⋊8Q8 ►in GL6(𝔽29)
| 15 | 18 | 0 | 0 | 0 | 0 |
| 22 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 15 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 5 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 26 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [15,22,0,0,0,0,18,11,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,15,0,0,0,0,4,17,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,23,10,0,0,0,0,5,6],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,26,17] >;
D28⋊8Q8 in GAP, Magma, Sage, TeX
D_{28}\rtimes_8Q_8 % in TeX
G:=Group("D28:8Q8"); // GroupNames label
G:=SmallGroup(448,1180);
// by ID
G=gap.SmallGroup(448,1180);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,100,1571,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^15,b*c=c*b,d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations