metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.7D28, D28.10D4, C4.7(C2×D28), C4.94(D4×D7), D14⋊C8⋊14C2, (C7×Q8).2D4, C4⋊C4.26D14, Q8⋊C4⋊12D7, D14⋊2Q8⋊5C2, C28.123(C2×D4), (C2×C8).124D14, C7⋊3(D4.7D4), C14.D8⋊11C2, C14.25C22≀C2, C14.48(C4○D8), (C2×Q8).109D14, (C22×D7).17D4, C22.200(D4×D7), (C2×C56).135C22, (C2×C28).250C23, (C2×Dic7).154D4, (C2×D28).64C22, (Q8×C14).33C22, C2.28(C22⋊D28), C2.15(Q16⋊D7), C14.61(C8.C22), C2.17(SD16⋊3D7), (C2×Dic14).72C22, (C2×C7⋊Q16)⋊4C2, (C2×C56⋊C2)⋊18C2, (C2×C7⋊C8).41C22, (C2×C4×D7).23C22, (C7×Q8⋊C4)⋊12C2, (C2×C14).263(C2×D4), (C7×C4⋊C4).51C22, (C2×Q8⋊2D7).4C2, (C2×C4).357(C22×D7), SmallGroup(448,344)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.D28
G = < a,b,c,d | a4=c28=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 916 in 152 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, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×D7, D4.7D4, C56⋊C2, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C7⋊Q16, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, Q8⋊2D7, Q8×C14, C14.D8, D14⋊C8, C7×Q8⋊C4, D14⋊2Q8, C2×C56⋊C2, C2×C7⋊Q16, C2×Q8⋊2D7, Q8.D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C4○D8, C8.C22, D28, C22×D7, D4.7D4, C2×D28, D4×D7, C22⋊D28, SD16⋊3D7, Q16⋊D7, Q8.D28
►On 224 points
Generators in S224
(1 44 197 91)(2 92 198 45)(3 46 199 93)(4 94 200 47)(5 48 201 95)(6 96 202 49)(7 50 203 97)(8 98 204 51)(9 52 205 99)(10 100 206 53)(11 54 207 101)(12 102 208 55)(13 56 209 103)(14 104 210 29)(15 30 211 105)(16 106 212 31)(17 32 213 107)(18 108 214 33)(19 34 215 109)(20 110 216 35)(21 36 217 111)(22 112 218 37)(23 38 219 85)(24 86 220 39)(25 40 221 87)(26 88 222 41)(27 42 223 89)(28 90 224 43)(57 133 180 160)(58 161 181 134)(59 135 182 162)(60 163 183 136)(61 137 184 164)(62 165 185 138)(63 139 186 166)(64 167 187 140)(65 113 188 168)(66 141 189 114)(67 115 190 142)(68 143 191 116)(69 117 192 144)(70 145 193 118)(71 119 194 146)(72 147 195 120)(73 121 196 148)(74 149 169 122)(75 123 170 150)(76 151 171 124)(77 125 172 152)(78 153 173 126)(79 127 174 154)(80 155 175 128)(81 129 176 156)(82 157 177 130)(83 131 178 158)(84 159 179 132)
(1 163 197 136)(2 184 198 61)(3 165 199 138)(4 186 200 63)(5 167 201 140)(6 188 202 65)(7 141 203 114)(8 190 204 67)(9 143 205 116)(10 192 206 69)(11 145 207 118)(12 194 208 71)(13 147 209 120)(14 196 210 73)(15 149 211 122)(16 170 212 75)(17 151 213 124)(18 172 214 77)(19 153 215 126)(20 174 216 79)(21 155 217 128)(22 176 218 81)(23 157 219 130)(24 178 220 83)(25 159 221 132)(26 180 222 57)(27 161 223 134)(28 182 224 59)(29 148 104 121)(30 74 105 169)(31 150 106 123)(32 76 107 171)(33 152 108 125)(34 78 109 173)(35 154 110 127)(36 80 111 175)(37 156 112 129)(38 82 85 177)(39 158 86 131)(40 84 87 179)(41 160 88 133)(42 58 89 181)(43 162 90 135)(44 60 91 183)(45 164 92 137)(46 62 93 185)(47 166 94 139)(48 64 95 187)(49 168 96 113)(50 66 97 189)(51 142 98 115)(52 68 99 191)(53 144 100 117)(54 70 101 193)(55 146 102 119)(56 72 103 195)
(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)
(1 121 197 148)(2 147 198 120)(3 119 199 146)(4 145 200 118)(5 117 201 144)(6 143 202 116)(7 115 203 142)(8 141 204 114)(9 113 205 168)(10 167 206 140)(11 139 207 166)(12 165 208 138)(13 137 209 164)(14 163 210 136)(15 135 211 162)(16 161 212 134)(17 133 213 160)(18 159 214 132)(19 131 215 158)(20 157 216 130)(21 129 217 156)(22 155 218 128)(23 127 219 154)(24 153 220 126)(25 125 221 152)(26 151 222 124)(27 123 223 150)(28 149 224 122)(29 183 104 60)(30 59 105 182)(31 181 106 58)(32 57 107 180)(33 179 108 84)(34 83 109 178)(35 177 110 82)(36 81 111 176)(37 175 112 80)(38 79 85 174)(39 173 86 78)(40 77 87 172)(41 171 88 76)(42 75 89 170)(43 169 90 74)(44 73 91 196)(45 195 92 72)(46 71 93 194)(47 193 94 70)(48 69 95 192)(49 191 96 68)(50 67 97 190)(51 189 98 66)(52 65 99 188)(53 187 100 64)(54 63 101 186)(55 185 102 62)(56 61 103 184)
G:=sub<Sym(224)| (1,44,197,91)(2,92,198,45)(3,46,199,93)(4,94,200,47)(5,48,201,95)(6,96,202,49)(7,50,203,97)(8,98,204,51)(9,52,205,99)(10,100,206,53)(11,54,207,101)(12,102,208,55)(13,56,209,103)(14,104,210,29)(15,30,211,105)(16,106,212,31)(17,32,213,107)(18,108,214,33)(19,34,215,109)(20,110,216,35)(21,36,217,111)(22,112,218,37)(23,38,219,85)(24,86,220,39)(25,40,221,87)(26,88,222,41)(27,42,223,89)(28,90,224,43)(57,133,180,160)(58,161,181,134)(59,135,182,162)(60,163,183,136)(61,137,184,164)(62,165,185,138)(63,139,186,166)(64,167,187,140)(65,113,188,168)(66,141,189,114)(67,115,190,142)(68,143,191,116)(69,117,192,144)(70,145,193,118)(71,119,194,146)(72,147,195,120)(73,121,196,148)(74,149,169,122)(75,123,170,150)(76,151,171,124)(77,125,172,152)(78,153,173,126)(79,127,174,154)(80,155,175,128)(81,129,176,156)(82,157,177,130)(83,131,178,158)(84,159,179,132), (1,163,197,136)(2,184,198,61)(3,165,199,138)(4,186,200,63)(5,167,201,140)(6,188,202,65)(7,141,203,114)(8,190,204,67)(9,143,205,116)(10,192,206,69)(11,145,207,118)(12,194,208,71)(13,147,209,120)(14,196,210,73)(15,149,211,122)(16,170,212,75)(17,151,213,124)(18,172,214,77)(19,153,215,126)(20,174,216,79)(21,155,217,128)(22,176,218,81)(23,157,219,130)(24,178,220,83)(25,159,221,132)(26,180,222,57)(27,161,223,134)(28,182,224,59)(29,148,104,121)(30,74,105,169)(31,150,106,123)(32,76,107,171)(33,152,108,125)(34,78,109,173)(35,154,110,127)(36,80,111,175)(37,156,112,129)(38,82,85,177)(39,158,86,131)(40,84,87,179)(41,160,88,133)(42,58,89,181)(43,162,90,135)(44,60,91,183)(45,164,92,137)(46,62,93,185)(47,166,94,139)(48,64,95,187)(49,168,96,113)(50,66,97,189)(51,142,98,115)(52,68,99,191)(53,144,100,117)(54,70,101,193)(55,146,102,119)(56,72,103,195), (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), (1,121,197,148)(2,147,198,120)(3,119,199,146)(4,145,200,118)(5,117,201,144)(6,143,202,116)(7,115,203,142)(8,141,204,114)(9,113,205,168)(10,167,206,140)(11,139,207,166)(12,165,208,138)(13,137,209,164)(14,163,210,136)(15,135,211,162)(16,161,212,134)(17,133,213,160)(18,159,214,132)(19,131,215,158)(20,157,216,130)(21,129,217,156)(22,155,218,128)(23,127,219,154)(24,153,220,126)(25,125,221,152)(26,151,222,124)(27,123,223,150)(28,149,224,122)(29,183,104,60)(30,59,105,182)(31,181,106,58)(32,57,107,180)(33,179,108,84)(34,83,109,178)(35,177,110,82)(36,81,111,176)(37,175,112,80)(38,79,85,174)(39,173,86,78)(40,77,87,172)(41,171,88,76)(42,75,89,170)(43,169,90,74)(44,73,91,196)(45,195,92,72)(46,71,93,194)(47,193,94,70)(48,69,95,192)(49,191,96,68)(50,67,97,190)(51,189,98,66)(52,65,99,188)(53,187,100,64)(54,63,101,186)(55,185,102,62)(56,61,103,184)>;
G:=Group( (1,44,197,91)(2,92,198,45)(3,46,199,93)(4,94,200,47)(5,48,201,95)(6,96,202,49)(7,50,203,97)(8,98,204,51)(9,52,205,99)(10,100,206,53)(11,54,207,101)(12,102,208,55)(13,56,209,103)(14,104,210,29)(15,30,211,105)(16,106,212,31)(17,32,213,107)(18,108,214,33)(19,34,215,109)(20,110,216,35)(21,36,217,111)(22,112,218,37)(23,38,219,85)(24,86,220,39)(25,40,221,87)(26,88,222,41)(27,42,223,89)(28,90,224,43)(57,133,180,160)(58,161,181,134)(59,135,182,162)(60,163,183,136)(61,137,184,164)(62,165,185,138)(63,139,186,166)(64,167,187,140)(65,113,188,168)(66,141,189,114)(67,115,190,142)(68,143,191,116)(69,117,192,144)(70,145,193,118)(71,119,194,146)(72,147,195,120)(73,121,196,148)(74,149,169,122)(75,123,170,150)(76,151,171,124)(77,125,172,152)(78,153,173,126)(79,127,174,154)(80,155,175,128)(81,129,176,156)(82,157,177,130)(83,131,178,158)(84,159,179,132), (1,163,197,136)(2,184,198,61)(3,165,199,138)(4,186,200,63)(5,167,201,140)(6,188,202,65)(7,141,203,114)(8,190,204,67)(9,143,205,116)(10,192,206,69)(11,145,207,118)(12,194,208,71)(13,147,209,120)(14,196,210,73)(15,149,211,122)(16,170,212,75)(17,151,213,124)(18,172,214,77)(19,153,215,126)(20,174,216,79)(21,155,217,128)(22,176,218,81)(23,157,219,130)(24,178,220,83)(25,159,221,132)(26,180,222,57)(27,161,223,134)(28,182,224,59)(29,148,104,121)(30,74,105,169)(31,150,106,123)(32,76,107,171)(33,152,108,125)(34,78,109,173)(35,154,110,127)(36,80,111,175)(37,156,112,129)(38,82,85,177)(39,158,86,131)(40,84,87,179)(41,160,88,133)(42,58,89,181)(43,162,90,135)(44,60,91,183)(45,164,92,137)(46,62,93,185)(47,166,94,139)(48,64,95,187)(49,168,96,113)(50,66,97,189)(51,142,98,115)(52,68,99,191)(53,144,100,117)(54,70,101,193)(55,146,102,119)(56,72,103,195), (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), (1,121,197,148)(2,147,198,120)(3,119,199,146)(4,145,200,118)(5,117,201,144)(6,143,202,116)(7,115,203,142)(8,141,204,114)(9,113,205,168)(10,167,206,140)(11,139,207,166)(12,165,208,138)(13,137,209,164)(14,163,210,136)(15,135,211,162)(16,161,212,134)(17,133,213,160)(18,159,214,132)(19,131,215,158)(20,157,216,130)(21,129,217,156)(22,155,218,128)(23,127,219,154)(24,153,220,126)(25,125,221,152)(26,151,222,124)(27,123,223,150)(28,149,224,122)(29,183,104,60)(30,59,105,182)(31,181,106,58)(32,57,107,180)(33,179,108,84)(34,83,109,178)(35,177,110,82)(36,81,111,176)(37,175,112,80)(38,79,85,174)(39,173,86,78)(40,77,87,172)(41,171,88,76)(42,75,89,170)(43,169,90,74)(44,73,91,196)(45,195,92,72)(46,71,93,194)(47,193,94,70)(48,69,95,192)(49,191,96,68)(50,67,97,190)(51,189,98,66)(52,65,99,188)(53,187,100,64)(54,63,101,186)(55,185,102,62)(56,61,103,184) );
G=PermutationGroup([[(1,44,197,91),(2,92,198,45),(3,46,199,93),(4,94,200,47),(5,48,201,95),(6,96,202,49),(7,50,203,97),(8,98,204,51),(9,52,205,99),(10,100,206,53),(11,54,207,101),(12,102,208,55),(13,56,209,103),(14,104,210,29),(15,30,211,105),(16,106,212,31),(17,32,213,107),(18,108,214,33),(19,34,215,109),(20,110,216,35),(21,36,217,111),(22,112,218,37),(23,38,219,85),(24,86,220,39),(25,40,221,87),(26,88,222,41),(27,42,223,89),(28,90,224,43),(57,133,180,160),(58,161,181,134),(59,135,182,162),(60,163,183,136),(61,137,184,164),(62,165,185,138),(63,139,186,166),(64,167,187,140),(65,113,188,168),(66,141,189,114),(67,115,190,142),(68,143,191,116),(69,117,192,144),(70,145,193,118),(71,119,194,146),(72,147,195,120),(73,121,196,148),(74,149,169,122),(75,123,170,150),(76,151,171,124),(77,125,172,152),(78,153,173,126),(79,127,174,154),(80,155,175,128),(81,129,176,156),(82,157,177,130),(83,131,178,158),(84,159,179,132)], [(1,163,197,136),(2,184,198,61),(3,165,199,138),(4,186,200,63),(5,167,201,140),(6,188,202,65),(7,141,203,114),(8,190,204,67),(9,143,205,116),(10,192,206,69),(11,145,207,118),(12,194,208,71),(13,147,209,120),(14,196,210,73),(15,149,211,122),(16,170,212,75),(17,151,213,124),(18,172,214,77),(19,153,215,126),(20,174,216,79),(21,155,217,128),(22,176,218,81),(23,157,219,130),(24,178,220,83),(25,159,221,132),(26,180,222,57),(27,161,223,134),(28,182,224,59),(29,148,104,121),(30,74,105,169),(31,150,106,123),(32,76,107,171),(33,152,108,125),(34,78,109,173),(35,154,110,127),(36,80,111,175),(37,156,112,129),(38,82,85,177),(39,158,86,131),(40,84,87,179),(41,160,88,133),(42,58,89,181),(43,162,90,135),(44,60,91,183),(45,164,92,137),(46,62,93,185),(47,166,94,139),(48,64,95,187),(49,168,96,113),(50,66,97,189),(51,142,98,115),(52,68,99,191),(53,144,100,117),(54,70,101,193),(55,146,102,119),(56,72,103,195)], [(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)], [(1,121,197,148),(2,147,198,120),(3,119,199,146),(4,145,200,118),(5,117,201,144),(6,143,202,116),(7,115,203,142),(8,141,204,114),(9,113,205,168),(10,167,206,140),(11,139,207,166),(12,165,208,138),(13,137,209,164),(14,163,210,136),(15,135,211,162),(16,161,212,134),(17,133,213,160),(18,159,214,132),(19,131,215,158),(20,157,216,130),(21,129,217,156),(22,155,218,128),(23,127,219,154),(24,153,220,126),(25,125,221,152),(26,151,222,124),(27,123,223,150),(28,149,224,122),(29,183,104,60),(30,59,105,182),(31,181,106,58),(32,57,107,180),(33,179,108,84),(34,83,109,178),(35,177,110,82),(36,81,111,176),(37,175,112,80),(38,79,85,174),(39,173,86,78),(40,77,87,172),(41,171,88,76),(42,75,89,170),(43,169,90,74),(44,73,91,196),(45,195,92,72),(46,71,93,194),(47,193,94,70),(48,69,95,192),(49,191,96,68),(50,67,97,190),(51,189,98,66),(52,65,99,188),(53,187,100,64),(54,63,101,186),(55,185,102,62),(56,61,103,184)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 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 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | D28 | C8.C22 | D4×D7 | D4×D7 | SD16⋊3D7 | Q16⋊D7 |
| kernel | Q8.D28 | C14.D8 | D14⋊C8 | C7×Q8⋊C4 | D14⋊2Q8 | C2×C56⋊C2 | C2×C7⋊Q16 | C2×Q8⋊2D7 | D28 | C2×Dic7 | C7×Q8 | C22×D7 | Q8⋊C4 | C4⋊C4 | C2×C8 | C2×Q8 | C14 | Q8 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Q8.D28 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 22 |
| 0 | 0 | 41 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 9 |
| 0 | 0 | 0 | 15 |
| 23 | 36 | 0 | 0 |
| 77 | 96 | 0 | 0 |
| 0 | 0 | 62 | 4 |
| 0 | 0 | 28 | 51 |
| 90 | 77 | 0 | 0 |
| 90 | 23 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 32 | 0 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,1,41,0,0,22,112],[1,0,0,0,0,1,0,0,0,0,98,0,0,0,9,15],[23,77,0,0,36,96,0,0,0,0,62,28,0,0,4,51],[90,90,0,0,77,23,0,0,0,0,0,32,0,0,60,0] >;
Q8.D28 in GAP, Magma, Sage, TeX
Q_8.D_{28} % in TeX
G:=Group("Q8.D28"); // GroupNames label
G:=SmallGroup(448,344);
// by ID
G=gap.SmallGroup(448,344);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,758,135,184,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^28=1,b^2=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations