metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊1D28, C28⋊13SD16, C42.57D14, (C4×Q8)⋊3D7, (C7×Q8)⋊8D4, C4⋊3(Q8⋊D7), (Q8×C28)⋊3C2, C28⋊C8⋊26C2, C7⋊3(C4⋊SD16), C4.16(C2×D28), (C2×C28).66D4, C28.20(C2×D4), C4⋊C4.253D14, C28⋊4D4.6C2, C14.D8⋊32C2, C28.60(C4○D4), C4.12(C4○D28), (C4×C28).98C22, (C2×Q8).160D14, C14.70(C2×SD16), C2.15(C28⋊7D4), C14.67(C4⋊D4), (C2×C28).347C23, (C2×D28).95C22, C2.10(D4⋊D14), C14.112(C8⋊C22), (Q8×C14).195C22, (C2×Q8⋊D7)⋊7C2, C2.7(C2×Q8⋊D7), (C2×C14).478(C2×D4), (C2×C7⋊C8).101C22, (C2×C4).249(C7⋊D4), (C7×C4⋊C4).284C22, (C2×C4).447(C22×D7), C22.155(C2×C7⋊D4), SmallGroup(448,561)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊D28
G = < a,b,c,d | a4=c28=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Subgroups: 836 in 128 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D7, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C28, C28, C28, D14, C2×C14, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C7⋊C8, D28, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C4⋊SD16, C2×C7⋊C8, Q8⋊D7, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×D28, C2×D28, Q8×C14, C28⋊C8, C14.D8, C28⋊4D4, C2×Q8⋊D7, Q8×C28, Q8⋊D28
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C4⋊D4, C2×SD16, C8⋊C22, D28, C7⋊D4, C22×D7, C4⋊SD16, Q8⋊D7, C2×D28, C4○D28, C2×C7⋊D4, C28⋊7D4, C2×Q8⋊D7, D4⋊D14, Q8⋊D28
►On 224 points
Generators in S224
(1 49 134 193)(2 50 135 194)(3 51 136 195)(4 52 137 196)(5 53 138 169)(6 54 139 170)(7 55 140 171)(8 56 113 172)(9 29 114 173)(10 30 115 174)(11 31 116 175)(12 32 117 176)(13 33 118 177)(14 34 119 178)(15 35 120 179)(16 36 121 180)(17 37 122 181)(18 38 123 182)(19 39 124 183)(20 40 125 184)(21 41 126 185)(22 42 127 186)(23 43 128 187)(24 44 129 188)(25 45 130 189)(26 46 131 190)(27 47 132 191)(28 48 133 192)(57 206 163 88)(58 207 164 89)(59 208 165 90)(60 209 166 91)(61 210 167 92)(62 211 168 93)(63 212 141 94)(64 213 142 95)(65 214 143 96)(66 215 144 97)(67 216 145 98)(68 217 146 99)(69 218 147 100)(70 219 148 101)(71 220 149 102)(72 221 150 103)(73 222 151 104)(74 223 152 105)(75 224 153 106)(76 197 154 107)(77 198 155 108)(78 199 156 109)(79 200 157 110)(80 201 158 111)(81 202 159 112)(82 203 160 85)(83 204 161 86)(84 205 162 87)
(1 87 134 205)(2 88 135 206)(3 89 136 207)(4 90 137 208)(5 91 138 209)(6 92 139 210)(7 93 140 211)(8 94 113 212)(9 95 114 213)(10 96 115 214)(11 97 116 215)(12 98 117 216)(13 99 118 217)(14 100 119 218)(15 101 120 219)(16 102 121 220)(17 103 122 221)(18 104 123 222)(19 105 124 223)(20 106 125 224)(21 107 126 197)(22 108 127 198)(23 109 128 199)(24 110 129 200)(25 111 130 201)(26 112 131 202)(27 85 132 203)(28 86 133 204)(29 142 173 64)(30 143 174 65)(31 144 175 66)(32 145 176 67)(33 146 177 68)(34 147 178 69)(35 148 179 70)(36 149 180 71)(37 150 181 72)(38 151 182 73)(39 152 183 74)(40 153 184 75)(41 154 185 76)(42 155 186 77)(43 156 187 78)(44 157 188 79)(45 158 189 80)(46 159 190 81)(47 160 191 82)(48 161 192 83)(49 162 193 84)(50 163 194 57)(51 164 195 58)(52 165 196 59)(53 166 169 60)(54 167 170 61)(55 168 171 62)(56 141 172 63)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 184)(30 183)(31 182)(32 181)(33 180)(34 179)(35 178)(36 177)(37 176)(38 175)(39 174)(40 173)(41 172)(42 171)(43 170)(44 169)(45 196)(46 195)(47 194)(48 193)(49 192)(50 191)(51 190)(52 189)(53 188)(54 187)(55 186)(56 185)(57 85)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 197)(142 224)(143 223)(144 222)(145 221)(146 220)(147 219)(148 218)(149 217)(150 216)(151 215)(152 214)(153 213)(154 212)(155 211)(156 210)(157 209)(158 208)(159 207)(160 206)(161 205)(162 204)(163 203)(164 202)(165 201)(166 200)(167 199)(168 198)
G:=sub<Sym(224)| (1,49,134,193)(2,50,135,194)(3,51,136,195)(4,52,137,196)(5,53,138,169)(6,54,139,170)(7,55,140,171)(8,56,113,172)(9,29,114,173)(10,30,115,174)(11,31,116,175)(12,32,117,176)(13,33,118,177)(14,34,119,178)(15,35,120,179)(16,36,121,180)(17,37,122,181)(18,38,123,182)(19,39,124,183)(20,40,125,184)(21,41,126,185)(22,42,127,186)(23,43,128,187)(24,44,129,188)(25,45,130,189)(26,46,131,190)(27,47,132,191)(28,48,133,192)(57,206,163,88)(58,207,164,89)(59,208,165,90)(60,209,166,91)(61,210,167,92)(62,211,168,93)(63,212,141,94)(64,213,142,95)(65,214,143,96)(66,215,144,97)(67,216,145,98)(68,217,146,99)(69,218,147,100)(70,219,148,101)(71,220,149,102)(72,221,150,103)(73,222,151,104)(74,223,152,105)(75,224,153,106)(76,197,154,107)(77,198,155,108)(78,199,156,109)(79,200,157,110)(80,201,158,111)(81,202,159,112)(82,203,160,85)(83,204,161,86)(84,205,162,87), (1,87,134,205)(2,88,135,206)(3,89,136,207)(4,90,137,208)(5,91,138,209)(6,92,139,210)(7,93,140,211)(8,94,113,212)(9,95,114,213)(10,96,115,214)(11,97,116,215)(12,98,117,216)(13,99,118,217)(14,100,119,218)(15,101,120,219)(16,102,121,220)(17,103,122,221)(18,104,123,222)(19,105,124,223)(20,106,125,224)(21,107,126,197)(22,108,127,198)(23,109,128,199)(24,110,129,200)(25,111,130,201)(26,112,131,202)(27,85,132,203)(28,86,133,204)(29,142,173,64)(30,143,174,65)(31,144,175,66)(32,145,176,67)(33,146,177,68)(34,147,178,69)(35,148,179,70)(36,149,180,71)(37,150,181,72)(38,151,182,73)(39,152,183,74)(40,153,184,75)(41,154,185,76)(42,155,186,77)(43,156,187,78)(44,157,188,79)(45,158,189,80)(46,159,190,81)(47,160,191,82)(48,161,192,83)(49,162,193,84)(50,163,194,57)(51,164,195,58)(52,165,196,59)(53,166,169,60)(54,167,170,61)(55,168,171,62)(56,141,172,63), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,184)(30,183)(31,182)(32,181)(33,180)(34,179)(35,178)(36,177)(37,176)(38,175)(39,174)(40,173)(41,172)(42,171)(43,170)(44,169)(45,196)(46,195)(47,194)(48,193)(49,192)(50,191)(51,190)(52,189)(53,188)(54,187)(55,186)(56,185)(57,85)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,197)(142,224)(143,223)(144,222)(145,221)(146,220)(147,219)(148,218)(149,217)(150,216)(151,215)(152,214)(153,213)(154,212)(155,211)(156,210)(157,209)(158,208)(159,207)(160,206)(161,205)(162,204)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198)>;
G:=Group( (1,49,134,193)(2,50,135,194)(3,51,136,195)(4,52,137,196)(5,53,138,169)(6,54,139,170)(7,55,140,171)(8,56,113,172)(9,29,114,173)(10,30,115,174)(11,31,116,175)(12,32,117,176)(13,33,118,177)(14,34,119,178)(15,35,120,179)(16,36,121,180)(17,37,122,181)(18,38,123,182)(19,39,124,183)(20,40,125,184)(21,41,126,185)(22,42,127,186)(23,43,128,187)(24,44,129,188)(25,45,130,189)(26,46,131,190)(27,47,132,191)(28,48,133,192)(57,206,163,88)(58,207,164,89)(59,208,165,90)(60,209,166,91)(61,210,167,92)(62,211,168,93)(63,212,141,94)(64,213,142,95)(65,214,143,96)(66,215,144,97)(67,216,145,98)(68,217,146,99)(69,218,147,100)(70,219,148,101)(71,220,149,102)(72,221,150,103)(73,222,151,104)(74,223,152,105)(75,224,153,106)(76,197,154,107)(77,198,155,108)(78,199,156,109)(79,200,157,110)(80,201,158,111)(81,202,159,112)(82,203,160,85)(83,204,161,86)(84,205,162,87), (1,87,134,205)(2,88,135,206)(3,89,136,207)(4,90,137,208)(5,91,138,209)(6,92,139,210)(7,93,140,211)(8,94,113,212)(9,95,114,213)(10,96,115,214)(11,97,116,215)(12,98,117,216)(13,99,118,217)(14,100,119,218)(15,101,120,219)(16,102,121,220)(17,103,122,221)(18,104,123,222)(19,105,124,223)(20,106,125,224)(21,107,126,197)(22,108,127,198)(23,109,128,199)(24,110,129,200)(25,111,130,201)(26,112,131,202)(27,85,132,203)(28,86,133,204)(29,142,173,64)(30,143,174,65)(31,144,175,66)(32,145,176,67)(33,146,177,68)(34,147,178,69)(35,148,179,70)(36,149,180,71)(37,150,181,72)(38,151,182,73)(39,152,183,74)(40,153,184,75)(41,154,185,76)(42,155,186,77)(43,156,187,78)(44,157,188,79)(45,158,189,80)(46,159,190,81)(47,160,191,82)(48,161,192,83)(49,162,193,84)(50,163,194,57)(51,164,195,58)(52,165,196,59)(53,166,169,60)(54,167,170,61)(55,168,171,62)(56,141,172,63), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,184)(30,183)(31,182)(32,181)(33,180)(34,179)(35,178)(36,177)(37,176)(38,175)(39,174)(40,173)(41,172)(42,171)(43,170)(44,169)(45,196)(46,195)(47,194)(48,193)(49,192)(50,191)(51,190)(52,189)(53,188)(54,187)(55,186)(56,185)(57,85)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,197)(142,224)(143,223)(144,222)(145,221)(146,220)(147,219)(148,218)(149,217)(150,216)(151,215)(152,214)(153,213)(154,212)(155,211)(156,210)(157,209)(158,208)(159,207)(160,206)(161,205)(162,204)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198) );
G=PermutationGroup([[(1,49,134,193),(2,50,135,194),(3,51,136,195),(4,52,137,196),(5,53,138,169),(6,54,139,170),(7,55,140,171),(8,56,113,172),(9,29,114,173),(10,30,115,174),(11,31,116,175),(12,32,117,176),(13,33,118,177),(14,34,119,178),(15,35,120,179),(16,36,121,180),(17,37,122,181),(18,38,123,182),(19,39,124,183),(20,40,125,184),(21,41,126,185),(22,42,127,186),(23,43,128,187),(24,44,129,188),(25,45,130,189),(26,46,131,190),(27,47,132,191),(28,48,133,192),(57,206,163,88),(58,207,164,89),(59,208,165,90),(60,209,166,91),(61,210,167,92),(62,211,168,93),(63,212,141,94),(64,213,142,95),(65,214,143,96),(66,215,144,97),(67,216,145,98),(68,217,146,99),(69,218,147,100),(70,219,148,101),(71,220,149,102),(72,221,150,103),(73,222,151,104),(74,223,152,105),(75,224,153,106),(76,197,154,107),(77,198,155,108),(78,199,156,109),(79,200,157,110),(80,201,158,111),(81,202,159,112),(82,203,160,85),(83,204,161,86),(84,205,162,87)], [(1,87,134,205),(2,88,135,206),(3,89,136,207),(4,90,137,208),(5,91,138,209),(6,92,139,210),(7,93,140,211),(8,94,113,212),(9,95,114,213),(10,96,115,214),(11,97,116,215),(12,98,117,216),(13,99,118,217),(14,100,119,218),(15,101,120,219),(16,102,121,220),(17,103,122,221),(18,104,123,222),(19,105,124,223),(20,106,125,224),(21,107,126,197),(22,108,127,198),(23,109,128,199),(24,110,129,200),(25,111,130,201),(26,112,131,202),(27,85,132,203),(28,86,133,204),(29,142,173,64),(30,143,174,65),(31,144,175,66),(32,145,176,67),(33,146,177,68),(34,147,178,69),(35,148,179,70),(36,149,180,71),(37,150,181,72),(38,151,182,73),(39,152,183,74),(40,153,184,75),(41,154,185,76),(42,155,186,77),(43,156,187,78),(44,157,188,79),(45,158,189,80),(46,159,190,81),(47,160,191,82),(48,161,192,83),(49,162,193,84),(50,163,194,57),(51,164,195,58),(52,165,196,59),(53,166,169,60),(54,167,170,61),(55,168,171,62),(56,141,172,63)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,184),(30,183),(31,182),(32,181),(33,180),(34,179),(35,178),(36,177),(37,176),(38,175),(39,174),(40,173),(41,172),(42,171),(43,170),(44,169),(45,196),(46,195),(47,194),(48,193),(49,192),(50,191),(51,190),(52,189),(53,188),(54,187),(55,186),(56,185),(57,85),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86),(113,126),(114,125),(115,124),(116,123),(117,122),(118,121),(119,120),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134),(141,197),(142,224),(143,223),(144,222),(145,221),(146,220),(147,219),(148,218),(149,217),(150,216),(151,215),(152,214),(153,213),(154,212),(155,211),(156,210),(157,209),(158,208),(159,207),(160,206),(161,205),(162,204),(163,203),(164,202),(165,201),(166,200),(167,199),(168,198)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 56 | 56 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | SD16 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | D28 | C4○D28 | C8⋊C22 | Q8⋊D7 | D4⋊D14 |
| kernel | Q8⋊D28 | C28⋊C8 | C14.D8 | C28⋊4D4 | C2×Q8⋊D7 | Q8×C28 | C2×C28 | C7×Q8 | C4×Q8 | C28 | C28 | C42 | C4⋊C4 | C2×Q8 | C2×C4 | Q8 | C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 12 | 12 | 12 | 1 | 6 | 6 |
Matrix representation of Q8⋊D28 ►in GL6(𝔽113)
| 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 | 111 |
| 0 | 0 | 0 | 0 | 1 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 87 |
| 0 | 0 | 0 | 0 | 13 | 87 |
| 0 | 89 | 0 | 0 | 0 | 0 |
| 33 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 104 | 41 | 0 | 0 |
| 0 | 0 | 111 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 112 | 10 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 104 | 41 | 0 | 0 |
| 0 | 0 | 89 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,111,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,26,13,0,0,0,0,87,87],[0,33,0,0,0,0,89,9,0,0,0,0,0,0,104,111,0,0,0,0,41,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,10,1,0,0,0,0,0,0,104,89,0,0,0,0,41,9,0,0,0,0,0,0,1,1,0,0,0,0,0,112] >;
Q8⋊D28 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{28} % in TeX
G:=Group("Q8:D28"); // GroupNames label
G:=SmallGroup(448,561);
// by ID
G=gap.SmallGroup(448,561);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^28=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations