metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊6D28, C42.129D14, C14.1102+ (1+4), (C7×Q8)⋊11D4, (C4×Q8)⋊10D7, (C4×D28)⋊39C2, (Q8×C28)⋊12C2, C7⋊2(Q8⋊6D4), C28.58(C2×D4), C4.26(C2×D28), C28⋊17(C4○D4), C4⋊D28⋊13C2, C28⋊1D4⋊18C2, C4⋊C4.296D14, C4⋊3(Q8⋊2D7), (C2×Q8).205D14, C14.20(C22×D4), C2.22(C22×D28), (C2×C14).121C24, (C4×C28).173C22, (C2×C28).170C23, (C2×D28).28C22, C2.22(D4⋊8D14), D14⋊C4.101C22, C4⋊Dic7.399C22, (Q8×C14).221C22, (C22×D7).46C23, C22.142(C23×D7), (C2×Dic7).215C23, (C2×Q8⋊2D7)⋊4C2, (C2×C4×D7).73C22, C14.112(C2×C4○D4), C2.11(C2×Q8⋊2D7), (C7×C4⋊C4).349C22, (C2×C4).734(C22×D7), SmallGroup(448,1030)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1796 in 312 conjugacy classes, 115 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×5], C22, C22 [×18], C7, C2×C4, C2×C4 [×6], C2×C4 [×14], D4 [×24], Q8 [×4], C23 [×6], D7 [×6], C14 [×3], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4, C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], Dic7 [×2], C28 [×8], C28 [×3], D14 [×18], C2×C14, C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C4⋊1D4 [×3], C2×C4○D4 [×2], C4×D7 [×12], D28 [×24], C2×Dic7 [×2], C2×C28, C2×C28 [×6], C7×Q8 [×4], C22×D7 [×6], Q8⋊6D4, C4⋊Dic7, D14⋊C4 [×6], C4×C28 [×3], C7×C4⋊C4 [×3], C2×C4×D7 [×6], C2×D28 [×15], Q8⋊2D7 [×8], Q8×C14, C4×D28 [×3], C4⋊D28 [×3], C28⋊1D4 [×6], Q8×C28, C2×Q8⋊2D7 [×2], Q8⋊6D28
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2+ (1+4), D28 [×4], C22×D7 [×7], Q8⋊6D4, C2×D28 [×6], Q8⋊2D7 [×2], C23×D7, C22×D28, C2×Q8⋊2D7, D4⋊8D14, Q8⋊6D28
Generators and relations
G = < a,b,c,d | a4=c28=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
►On 224 points
Generators in S224
(1 111 55 65)(2 112 56 66)(3 85 29 67)(4 86 30 68)(5 87 31 69)(6 88 32 70)(7 89 33 71)(8 90 34 72)(9 91 35 73)(10 92 36 74)(11 93 37 75)(12 94 38 76)(13 95 39 77)(14 96 40 78)(15 97 41 79)(16 98 42 80)(17 99 43 81)(18 100 44 82)(19 101 45 83)(20 102 46 84)(21 103 47 57)(22 104 48 58)(23 105 49 59)(24 106 50 60)(25 107 51 61)(26 108 52 62)(27 109 53 63)(28 110 54 64)(113 151 183 204)(114 152 184 205)(115 153 185 206)(116 154 186 207)(117 155 187 208)(118 156 188 209)(119 157 189 210)(120 158 190 211)(121 159 191 212)(122 160 192 213)(123 161 193 214)(124 162 194 215)(125 163 195 216)(126 164 196 217)(127 165 169 218)(128 166 170 219)(129 167 171 220)(130 168 172 221)(131 141 173 222)(132 142 174 223)(133 143 175 224)(134 144 176 197)(135 145 177 198)(136 146 178 199)(137 147 179 200)(138 148 180 201)(139 149 181 202)(140 150 182 203)
(1 125 55 195)(2 126 56 196)(3 127 29 169)(4 128 30 170)(5 129 31 171)(6 130 32 172)(7 131 33 173)(8 132 34 174)(9 133 35 175)(10 134 36 176)(11 135 37 177)(12 136 38 178)(13 137 39 179)(14 138 40 180)(15 139 41 181)(16 140 42 182)(17 113 43 183)(18 114 44 184)(19 115 45 185)(20 116 46 186)(21 117 47 187)(22 118 48 188)(23 119 49 189)(24 120 50 190)(25 121 51 191)(26 122 52 192)(27 123 53 193)(28 124 54 194)(57 155 103 208)(58 156 104 209)(59 157 105 210)(60 158 106 211)(61 159 107 212)(62 160 108 213)(63 161 109 214)(64 162 110 215)(65 163 111 216)(66 164 112 217)(67 165 85 218)(68 166 86 219)(69 167 87 220)(70 168 88 221)(71 141 89 222)(72 142 90 223)(73 143 91 224)(74 144 92 197)(75 145 93 198)(76 146 94 199)(77 147 95 200)(78 148 96 201)(79 149 97 202)(80 150 98 203)(81 151 99 204)(82 152 100 205)(83 153 101 206)(84 154 102 207)
(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 57)(2 84)(3 83)(4 82)(5 81)(6 80)(7 79)(8 78)(9 77)(10 76)(11 75)(12 74)(13 73)(14 72)(15 71)(16 70)(17 69)(18 68)(19 67)(20 66)(21 65)(22 64)(23 63)(24 62)(25 61)(26 60)(27 59)(28 58)(29 101)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 112)(47 111)(48 110)(49 109)(50 108)(51 107)(52 106)(53 105)(54 104)(55 103)(56 102)(113 167)(114 166)(115 165)(116 164)(117 163)(118 162)(119 161)(120 160)(121 159)(122 158)(123 157)(124 156)(125 155)(126 154)(127 153)(128 152)(129 151)(130 150)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)(140 168)(169 206)(170 205)(171 204)(172 203)(173 202)(174 201)(175 200)(176 199)(177 198)(178 197)(179 224)(180 223)(181 222)(182 221)(183 220)(184 219)(185 218)(186 217)(187 216)(188 215)(189 214)(190 213)(191 212)(192 211)(193 210)(194 209)(195 208)(196 207)
G:=sub<Sym(224)| (1,111,55,65)(2,112,56,66)(3,85,29,67)(4,86,30,68)(5,87,31,69)(6,88,32,70)(7,89,33,71)(8,90,34,72)(9,91,35,73)(10,92,36,74)(11,93,37,75)(12,94,38,76)(13,95,39,77)(14,96,40,78)(15,97,41,79)(16,98,42,80)(17,99,43,81)(18,100,44,82)(19,101,45,83)(20,102,46,84)(21,103,47,57)(22,104,48,58)(23,105,49,59)(24,106,50,60)(25,107,51,61)(26,108,52,62)(27,109,53,63)(28,110,54,64)(113,151,183,204)(114,152,184,205)(115,153,185,206)(116,154,186,207)(117,155,187,208)(118,156,188,209)(119,157,189,210)(120,158,190,211)(121,159,191,212)(122,160,192,213)(123,161,193,214)(124,162,194,215)(125,163,195,216)(126,164,196,217)(127,165,169,218)(128,166,170,219)(129,167,171,220)(130,168,172,221)(131,141,173,222)(132,142,174,223)(133,143,175,224)(134,144,176,197)(135,145,177,198)(136,146,178,199)(137,147,179,200)(138,148,180,201)(139,149,181,202)(140,150,182,203), (1,125,55,195)(2,126,56,196)(3,127,29,169)(4,128,30,170)(5,129,31,171)(6,130,32,172)(7,131,33,173)(8,132,34,174)(9,133,35,175)(10,134,36,176)(11,135,37,177)(12,136,38,178)(13,137,39,179)(14,138,40,180)(15,139,41,181)(16,140,42,182)(17,113,43,183)(18,114,44,184)(19,115,45,185)(20,116,46,186)(21,117,47,187)(22,118,48,188)(23,119,49,189)(24,120,50,190)(25,121,51,191)(26,122,52,192)(27,123,53,193)(28,124,54,194)(57,155,103,208)(58,156,104,209)(59,157,105,210)(60,158,106,211)(61,159,107,212)(62,160,108,213)(63,161,109,214)(64,162,110,215)(65,163,111,216)(66,164,112,217)(67,165,85,218)(68,166,86,219)(69,167,87,220)(70,168,88,221)(71,141,89,222)(72,142,90,223)(73,143,91,224)(74,144,92,197)(75,145,93,198)(76,146,94,199)(77,147,95,200)(78,148,96,201)(79,149,97,202)(80,150,98,203)(81,151,99,204)(82,152,100,205)(83,153,101,206)(84,154,102,207), (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), (1,57)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,73)(14,72)(15,71)(16,70)(17,69)(18,68)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,101)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102)(113,167)(114,166)(115,165)(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(140,168)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,224)(180,223)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,210)(194,209)(195,208)(196,207)>;
G:=Group( (1,111,55,65)(2,112,56,66)(3,85,29,67)(4,86,30,68)(5,87,31,69)(6,88,32,70)(7,89,33,71)(8,90,34,72)(9,91,35,73)(10,92,36,74)(11,93,37,75)(12,94,38,76)(13,95,39,77)(14,96,40,78)(15,97,41,79)(16,98,42,80)(17,99,43,81)(18,100,44,82)(19,101,45,83)(20,102,46,84)(21,103,47,57)(22,104,48,58)(23,105,49,59)(24,106,50,60)(25,107,51,61)(26,108,52,62)(27,109,53,63)(28,110,54,64)(113,151,183,204)(114,152,184,205)(115,153,185,206)(116,154,186,207)(117,155,187,208)(118,156,188,209)(119,157,189,210)(120,158,190,211)(121,159,191,212)(122,160,192,213)(123,161,193,214)(124,162,194,215)(125,163,195,216)(126,164,196,217)(127,165,169,218)(128,166,170,219)(129,167,171,220)(130,168,172,221)(131,141,173,222)(132,142,174,223)(133,143,175,224)(134,144,176,197)(135,145,177,198)(136,146,178,199)(137,147,179,200)(138,148,180,201)(139,149,181,202)(140,150,182,203), (1,125,55,195)(2,126,56,196)(3,127,29,169)(4,128,30,170)(5,129,31,171)(6,130,32,172)(7,131,33,173)(8,132,34,174)(9,133,35,175)(10,134,36,176)(11,135,37,177)(12,136,38,178)(13,137,39,179)(14,138,40,180)(15,139,41,181)(16,140,42,182)(17,113,43,183)(18,114,44,184)(19,115,45,185)(20,116,46,186)(21,117,47,187)(22,118,48,188)(23,119,49,189)(24,120,50,190)(25,121,51,191)(26,122,52,192)(27,123,53,193)(28,124,54,194)(57,155,103,208)(58,156,104,209)(59,157,105,210)(60,158,106,211)(61,159,107,212)(62,160,108,213)(63,161,109,214)(64,162,110,215)(65,163,111,216)(66,164,112,217)(67,165,85,218)(68,166,86,219)(69,167,87,220)(70,168,88,221)(71,141,89,222)(72,142,90,223)(73,143,91,224)(74,144,92,197)(75,145,93,198)(76,146,94,199)(77,147,95,200)(78,148,96,201)(79,149,97,202)(80,150,98,203)(81,151,99,204)(82,152,100,205)(83,153,101,206)(84,154,102,207), (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), (1,57)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,73)(14,72)(15,71)(16,70)(17,69)(18,68)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,101)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102)(113,167)(114,166)(115,165)(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(140,168)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,224)(180,223)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,210)(194,209)(195,208)(196,207) );
G=PermutationGroup([(1,111,55,65),(2,112,56,66),(3,85,29,67),(4,86,30,68),(5,87,31,69),(6,88,32,70),(7,89,33,71),(8,90,34,72),(9,91,35,73),(10,92,36,74),(11,93,37,75),(12,94,38,76),(13,95,39,77),(14,96,40,78),(15,97,41,79),(16,98,42,80),(17,99,43,81),(18,100,44,82),(19,101,45,83),(20,102,46,84),(21,103,47,57),(22,104,48,58),(23,105,49,59),(24,106,50,60),(25,107,51,61),(26,108,52,62),(27,109,53,63),(28,110,54,64),(113,151,183,204),(114,152,184,205),(115,153,185,206),(116,154,186,207),(117,155,187,208),(118,156,188,209),(119,157,189,210),(120,158,190,211),(121,159,191,212),(122,160,192,213),(123,161,193,214),(124,162,194,215),(125,163,195,216),(126,164,196,217),(127,165,169,218),(128,166,170,219),(129,167,171,220),(130,168,172,221),(131,141,173,222),(132,142,174,223),(133,143,175,224),(134,144,176,197),(135,145,177,198),(136,146,178,199),(137,147,179,200),(138,148,180,201),(139,149,181,202),(140,150,182,203)], [(1,125,55,195),(2,126,56,196),(3,127,29,169),(4,128,30,170),(5,129,31,171),(6,130,32,172),(7,131,33,173),(8,132,34,174),(9,133,35,175),(10,134,36,176),(11,135,37,177),(12,136,38,178),(13,137,39,179),(14,138,40,180),(15,139,41,181),(16,140,42,182),(17,113,43,183),(18,114,44,184),(19,115,45,185),(20,116,46,186),(21,117,47,187),(22,118,48,188),(23,119,49,189),(24,120,50,190),(25,121,51,191),(26,122,52,192),(27,123,53,193),(28,124,54,194),(57,155,103,208),(58,156,104,209),(59,157,105,210),(60,158,106,211),(61,159,107,212),(62,160,108,213),(63,161,109,214),(64,162,110,215),(65,163,111,216),(66,164,112,217),(67,165,85,218),(68,166,86,219),(69,167,87,220),(70,168,88,221),(71,141,89,222),(72,142,90,223),(73,143,91,224),(74,144,92,197),(75,145,93,198),(76,146,94,199),(77,147,95,200),(78,148,96,201),(79,149,97,202),(80,150,98,203),(81,151,99,204),(82,152,100,205),(83,153,101,206),(84,154,102,207)], [(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)], [(1,57),(2,84),(3,83),(4,82),(5,81),(6,80),(7,79),(8,78),(9,77),(10,76),(11,75),(12,74),(13,73),(14,72),(15,71),(16,70),(17,69),(18,68),(19,67),(20,66),(21,65),(22,64),(23,63),(24,62),(25,61),(26,60),(27,59),(28,58),(29,101),(30,100),(31,99),(32,98),(33,97),(34,96),(35,95),(36,94),(37,93),(38,92),(39,91),(40,90),(41,89),(42,88),(43,87),(44,86),(45,85),(46,112),(47,111),(48,110),(49,109),(50,108),(51,107),(52,106),(53,105),(54,104),(55,103),(56,102),(113,167),(114,166),(115,165),(116,164),(117,163),(118,162),(119,161),(120,160),(121,159),(122,158),(123,157),(124,156),(125,155),(126,154),(127,153),(128,152),(129,151),(130,150),(131,149),(132,148),(133,147),(134,146),(135,145),(136,144),(137,143),(138,142),(139,141),(140,168),(169,206),(170,205),(171,204),(172,203),(173,202),(174,201),(175,200),(176,199),(177,198),(178,197),(179,224),(180,223),(181,222),(182,221),(183,220),(184,219),(185,218),(186,217),(187,216),(188,215),(189,214),(190,213),(191,212),(192,211),(193,210),(194,209),(195,208),(196,207)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 5 |
| 0 | 0 | 14 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 19 | 2 |
| 0 | 0 | 22 | 10 |
| 3 | 25 | 0 | 0 |
| 8 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 23 | 19 | 0 | 0 |
| 18 | 6 | 0 | 0 |
| 0 | 0 | 4 | 5 |
| 0 | 0 | 26 | 25 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,4,14,0,0,5,25],[1,0,0,0,0,1,0,0,0,0,19,22,0,0,2,10],[3,8,0,0,25,9,0,0,0,0,1,0,0,0,0,1],[23,18,0,0,19,6,0,0,0,0,4,26,0,0,5,25] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 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 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D28 | 2+ (1+4) | Q8⋊2D7 | D4⋊8D14 |
| kernel | Q8⋊6D28 | C4×D28 | C4⋊D28 | C28⋊1D4 | Q8×C28 | C2×Q8⋊2D7 | C7×Q8 | C4×Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | Q8 | C14 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_6D_{28} % in TeX
G:=Group("Q8:6D28"); // GroupNames label
G:=SmallGroup(448,1030);
// by ID
G=gap.SmallGroup(448,1030);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,387,184,675,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^28=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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