Q8xD28 - GroupNames

G = Q₈×D₂₈ order 448 = 2⁶·7

Direct product of Q₈ and D₂₈

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈×D₂₈, C₄².₁₂₇D₁₄, C₁₄.₆₆2_-¹⁺⁴, C₇⋊₂(D₄×Q₈), C₄⋊₃(Q₈×D₇), (C₄×Q₈)⋊₈D₇, (C₇×Q₈)⋊₉D₄, C₂₈⋊₈(C₂×Q₈), D₁₄⋊₅(C₂×Q₈), (Q₈×C₂₈)⋊₁₀C₂, C₂₈.₅₆(C₂×D₄), C₄.₂₄(C₂×D₂₈), C₄⋊C₄.₂₉₄D₁₄, C₂₈⋊₂Q₈⋊₂₇C₂, (C₄×D₂₈).₂₀C₂, D₁₄⋊₂Q₈⋊₁₇C₂, (C₂×Q₈).₂₀₃D₁₄, C₂.₂₀(C₂²×D₂₈), C₁₄.₁₈(C₂²×D₄), C₁₄.₂₉(C₂²×Q₈), (C₂×C₁₄).₁₁₉C₂⁴, (C₂×C₂₈).₁₆₉C₂³, (C₄×C₂₈).₁₇₁C₂², D₁₄⋊C₄.₁₀₀C₂², (C₂×D₂₈).₂₈₈C₂², C₄⋊Dic₇.₃₀₅C₂², (Q₈×C₁₄).₂₁₉C₂², (C₂×Dic₇).₅₃C₂³, C₂².₁₄₀(C₂³×D₇), (C₂²×D₇).₁₇₈C₂³, C₂.₂₃(D₄.₁₀D₁₄), (C₂×Dic₁₄).₁₄₈C₂², (C₂×Q₈×D₇)⋊₃C₂, C₂.₁₂(C₂×Q₈×D₇), (C₂×C₄×D₇).₇₁C₂², (C₇×C₄⋊C₄).₃₄₇C₂², (C₂×C₄).₅₈₃(C₂²×D₇), SmallGroup(448,1028)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — Q₈×D₂₈

Chief series

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

Lower central

C₇ — C₂×C₁₄ — Q₈×D₂₈

Upper central

C₁ — C₂² — C₄×Q₈

Generators and relations for Q₈×D₂₈
G = < a,b,c,d | a⁴=c²⁸=d²=1, b²=a², bab^-1=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 1284 in 280 conjugacy classes, 123 normal (18 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, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₄⋊Q₈, C₂²×Q₈, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, D₄×Q₈, C₄⋊Dic₇, D₁₄⋊C₄, C₄×C₂₈, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×D₂₈, Q₈×D₇, Q₈×C₁₄, C₂₈⋊₂Q₈, C₄×D₂₈, D₁₄⋊₂Q₈, Q₈×C₂₈, C₂×Q₈×D₇, Q₈×D₂₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₇, C₂×D₄, C₂×Q₈, C₂⁴, D₁₄, C₂²×D₄, C₂²×Q₈, 2_-¹⁺⁴, D₂₈, C₂²×D₇, D₄×Q₈, C₂×D₂₈, Q₈×D₇, C₂³×D₇, C₂²×D₂₈, C₂×Q₈×D₇, D₄.₁₀D₁₄, Q₈×D₂₈

Smallest permutation representation of Q₈×D₂₈
►On 224 points

Generators in S₂₂₄

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

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

(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)

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

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

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

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

85 conjugacy classes

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

85 irreducible representations

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

Matrix representation of Q₈×D₂₈ ►in GL₆(𝔽₂₉)

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	0	1
0	0	0	0	28	0

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	21	14
0	0	0	0	14	8

28	22	0	0	0	0
7	19	0	0	0	0
0	0	28	27	0	0
0	0	1	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

28	0	0	0	0	0
7	1	0	0	0	0
0	0	28	0	0	0
0	0	1	1	0	0
0	0	0	0	28	0
0	0	0	0	0	28

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,14,0,0,0,0,14,8],[28,7,0,0,0,0,22,19,0,0,0,0,0,0,28,1,0,0,0,0,27,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,7,0,0,0,0,0,1,0,0,0,0,0,0,28,1,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;

Q₈×D₂₈ in GAP, Magma, Sage, TeX

Q_8\times D_{28}

% in TeX

G:=Group("Q8xD28");

// GroupNames label

G:=SmallGroup(448,1028);

// by ID

G=gap.SmallGroup(448,1028);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,675,80,18822]);

// 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,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = Q8×D28 order 448 = 26·7

Direct product of Q8 and D28

G = Q₈×D₂₈ order 448 = 2⁶·7

Direct product of Q₈ and D₂₈