Q8:5D28 - GroupNames

G = Q₈⋊₅D₂₈ order 448 = 2⁶·7

1^st semidirect product of Q₈ and D₂₈ acting through Inn(Q₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈⋊₅D₂₈, C₄².₁₂₈D₁₄, C₁₄.₁₁2_-¹⁺⁴, (C₄×Q₈)⋊₉D₇, (C₇×Q₈)⋊₁₀D₄, (C₄×D₂₈)⋊₃₈C₂, (Q₈×C₂₈)⋊₁₁C₂, C₇⋊₂(Q₈⋊₅D₄), C₂₈.₅₇(C₂×D₄), C₄.₂₅(C₂×D₂₈), C₄⋊D₂₈⋊₁₇C₂, C₄⋊C₄.₂₉₅D₁₄, D₁₄⋊₁₃(C₄○D₄), D₁₄⋊₂Q₈⋊₁₈C₂, C₄.D₂₈⋊₂₀C₂, D₁₄⋊C₄.₆C₂², (C₂×Q₈).₂₀₄D₁₄, C₁₄.₁₉(C₂²×D₄), C₂.₂₁(C₂²×D₂₈), (C₄×C₂₈).₁₇₂C₂², (C₂×C₁₄).₁₂₀C₂⁴, (C₂×C₂₈).₄₉₈C₂³, (C₂×D₂₈).₂₁₆C₂², C₄⋊Dic₇.₃₀₆C₂², (Q₈×C₁₄).₂₂₀C₂², (C₂×Dic₇).₅₄C₂³, (C₂²×D₇).₄₅C₂³, C₂².₁₄₁(C₂³×D₇), C₂.₁₂(Q₈.₁₀D₁₄), (C₂×Dic₁₄).₁₄₉C₂², (C₂×Q₈×D₇)⋊₄C₂, C₂.₂₉(D₇×C₄○D₄), (C₂×Q₈⋊₂D₇)⋊₃C₂, (C₂×C₄×D₇).₇₂C₂², C₁₄.₁₄₅(C₂×C₄○D₄), (C₇×C₄⋊C₄).₃₄₈C₂², (C₂×C₄).₁₆₈(C₂²×D₇), SmallGroup(448,1029)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — Q₈⋊₅D₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — C₂×Q₈×D₇ — Q₈⋊₅D₂₈

Lower central

C₇ — C₂×C₁₄ — Q₈⋊₅D₂₈

Upper central

C₁ — C₂² — C₄×Q₈

Generators and relations for Q₈⋊₅D₂₈
G = < a,b,c,d | a⁴=c²⁸=d²=1, b²=a², bab^-1=a^-1, ac=ca, ad=da, cbc^-1=dbd=a²b, dcd=c^-1 >

Subgroups: 1476 in 290 conjugacy classes, 113 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, D₇, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₄.₄D₄, C₂²×Q₈, C₂×C₄○D₄, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂²×D₇, Q₈⋊₅D₄, C₄⋊Dic₇, D₁₄⋊C₄, D₁₄⋊C₄, C₄×C₂₈, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×D₂₈, Q₈×D₇, Q₈⋊₂D₇, Q₈×C₁₄, C₄×D₂₈, C₄.D₂₈, C₄⋊D₂₈, D₁₄⋊₂Q₈, Q₈×C₂₈, C₂×Q₈×D₇, C₂×Q₈⋊₂D₇, Q₈⋊₅D₂₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₄○D₄, C₂⁴, D₁₄, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, D₂₈, C₂²×D₇, Q₈⋊₅D₄, C₂×D₂₈, C₂³×D₇, C₂²×D₂₈, Q₈.₁₀D₁₄, D₇×C₄○D₄, Q₈⋊₅D₂₈

Smallest permutation representation of Q₈⋊₅D₂₈
►On 224 points

Generators in S₂₂₄

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

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

(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)

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

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

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

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

85 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	4A	···	4H	4I	4J	4K	4L	4M	4N	4O	4P	7A	7B	7C	14A	···	14I	28A	···	28L	28M	···	28AV
order	1	2	2	2	2	2	2	2	2	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	28	28	28	2	···	2	4	4	4	14	14	28	28	28	2	2	2	2	···	2	2	···	2	4	···	4

85 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+		+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₇	C₄○D₄	D₁₄	D₁₄	D₁₄	D₂₈	2_-¹⁺⁴	Q₈.₁₀D₁₄	D₇×C₄○D₄
kernel	Q₈⋊₅D₂₈	C₄×D₂₈	C₄.D₂₈	C₄⋊D₂₈	D₁₄⋊₂Q₈	Q₈×C₂₈	C₂×Q₈×D₇	C₂×Q₈⋊₂D₇	C₇×Q₈	C₄×Q₈	D₁₄	C₄²	C₄⋊C₄	C₂×Q₈	Q₈	C₁₄	C₂	C₂
# reps	1	3	3	3	3	1	1	1	4	3	4	9	9	3	24	1	6	6

Matrix representation of Q₈⋊₅D₂₈ ►in GL₆(𝔽₂₉)

28	0	0	0	0	0
0	28	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	1	27
0	0	0	0	1	28

28	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	27	22
0	0	0	0	9	2

10	11	0	0	0	0
12	22	0	0	0	0
0	0	0	28	0	0
0	0	1	0	0	0
0	0	0	0	17	24
0	0	0	0	17	12

19	1	0	0	0	0
17	10	0	0	0	0
0	0	0	28	0	0
0	0	28	0	0	0
0	0	0	0	12	5
0	0	0	0	12	17

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,1,0,0,0,0,27,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,9,0,0,0,0,22,2],[10,12,0,0,0,0,11,22,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,17,17,0,0,0,0,24,12],[19,17,0,0,0,0,1,10,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,12,12,0,0,0,0,5,17] >;

Q₈⋊₅D₂₈ in GAP, Magma, Sage, TeX

Q_8\rtimes_5D_{28}

% in TeX

G:=Group("Q8:5D28");

// GroupNames label

G:=SmallGroup(448,1029);

// by ID

G=gap.SmallGroup(448,1029);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,675,192,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^28=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;

// generators/relations

G = Q8⋊5D28 order 448 = 26·7

1st semidirect product of Q8 and D28 acting through Inn(Q8)

G = Q₈⋊₅D₂₈ order 448 = 2⁶·7

1^st semidirect product of Q₈ and D₂₈ acting through Inn(Q₈)