metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊10Q8, C42.130D14, C14.1112+ (1+4), (C4×Q8)⋊12D7, C4.50(Q8×D7), (Q8×C28)⋊14C2, C7⋊4(D4⋊3Q8), D14.8(C2×Q8), C4⋊C4.326D14, D14⋊3Q8⋊8C2, C28⋊2Q8⋊28C2, (C4×D28).21C2, C28.108(C2×Q8), D14⋊Q8⋊11C2, C4.67(C4○D28), (C2×Q8).178D14, C28.6Q8⋊18C2, C28.118(C4○D4), C14.31(C22×Q8), (C2×C14).123C24, (C4×C28).175C22, (C2×C28).590C23, C2.23(D4⋊8D14), D14⋊C4.103C22, (C2×D28).289C22, Dic7⋊C4.69C22, C4⋊Dic7.202C22, (Q8×C14).223C22, (C2×Dic7).55C23, C22.144(C23×D7), (C2×Dic14).31C22, (C22×D7).180C23, (D7×C4⋊C4)⋊18C2, C2.14(C2×Q8×D7), C14.55(C2×C4○D4), C2.62(C2×C4○D28), (C2×C4×D7).74C22, (C7×C4⋊C4).351C22, (C2×C4).169(C22×D7), SmallGroup(448,1032)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1028 in 228 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], C22, C22 [×8], C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], D7 [×4], C14 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×2], Dic7 [×6], C28 [×4], C28 [×5], D14 [×4], D14 [×4], C2×C14, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic14 [×2], C4×D7 [×8], D28 [×4], C2×Dic7 [×6], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×2], C22×D7 [×2], D4⋊3Q8, Dic7⋊C4 [×8], C4⋊Dic7, C4⋊Dic7 [×4], D14⋊C4 [×6], C4×C28, C4×C28 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14 [×2], C2×C4×D7 [×6], C2×D28, Q8×C14, C28⋊2Q8, C28.6Q8 [×2], C4×D28, C4×D28 [×2], D7×C4⋊C4 [×2], D14⋊Q8 [×4], D14⋊3Q8 [×2], Q8×C28, D28⋊10Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C4○D4 [×2], C24, D14 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C22×D7 [×7], D4⋊3Q8, C4○D28 [×2], Q8×D7 [×2], C23×D7, C2×C4○D28, C2×Q8×D7, D4⋊8D14, D28⋊10Q8
Generators and relations
G = < a,b,c,d | a28=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a14b, bd=db, dcd-1=c-1 >
►On 224 points
Generators in S224
(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 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 112)(10 111)(11 110)(12 109)(13 108)(14 107)(15 106)(16 105)(17 104)(18 103)(19 102)(20 101)(21 100)(22 99)(23 98)(24 97)(25 96)(26 95)(27 94)(28 93)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 60)(44 59)(45 58)(46 57)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(113 224)(114 223)(115 222)(116 221)(117 220)(118 219)(119 218)(120 217)(121 216)(122 215)(123 214)(124 213)(125 212)(126 211)(127 210)(128 209)(129 208)(130 207)(131 206)(132 205)(133 204)(134 203)(135 202)(136 201)(137 200)(138 199)(139 198)(140 197)(141 182)(142 181)(143 180)(144 179)(145 178)(146 177)(147 176)(148 175)(149 174)(150 173)(151 172)(152 171)(153 170)(154 169)(155 196)(156 195)(157 194)(158 193)(159 192)(160 191)(161 190)(162 189)(163 188)(164 187)(165 186)(166 185)(167 184)(168 183)
(1 39 86 72)(2 40 87 73)(3 41 88 74)(4 42 89 75)(5 43 90 76)(6 44 91 77)(7 45 92 78)(8 46 93 79)(9 47 94 80)(10 48 95 81)(11 49 96 82)(12 50 97 83)(13 51 98 84)(14 52 99 57)(15 53 100 58)(16 54 101 59)(17 55 102 60)(18 56 103 61)(19 29 104 62)(20 30 105 63)(21 31 106 64)(22 32 107 65)(23 33 108 66)(24 34 109 67)(25 35 110 68)(26 36 111 69)(27 37 112 70)(28 38 85 71)(113 190 210 161)(114 191 211 162)(115 192 212 163)(116 193 213 164)(117 194 214 165)(118 195 215 166)(119 196 216 167)(120 169 217 168)(121 170 218 141)(122 171 219 142)(123 172 220 143)(124 173 221 144)(125 174 222 145)(126 175 223 146)(127 176 224 147)(128 177 197 148)(129 178 198 149)(130 179 199 150)(131 180 200 151)(132 181 201 152)(133 182 202 153)(134 183 203 154)(135 184 204 155)(136 185 205 156)(137 186 206 157)(138 187 207 158)(139 188 208 159)(140 189 209 160)
(1 158 86 187)(2 159 87 188)(3 160 88 189)(4 161 89 190)(5 162 90 191)(6 163 91 192)(7 164 92 193)(8 165 93 194)(9 166 94 195)(10 167 95 196)(11 168 96 169)(12 141 97 170)(13 142 98 171)(14 143 99 172)(15 144 100 173)(16 145 101 174)(17 146 102 175)(18 147 103 176)(19 148 104 177)(20 149 105 178)(21 150 106 179)(22 151 107 180)(23 152 108 181)(24 153 109 182)(25 154 110 183)(26 155 111 184)(27 156 112 185)(28 157 85 186)(29 197 62 128)(30 198 63 129)(31 199 64 130)(32 200 65 131)(33 201 66 132)(34 202 67 133)(35 203 68 134)(36 204 69 135)(37 205 70 136)(38 206 71 137)(39 207 72 138)(40 208 73 139)(41 209 74 140)(42 210 75 113)(43 211 76 114)(44 212 77 115)(45 213 78 116)(46 214 79 117)(47 215 80 118)(48 216 81 119)(49 217 82 120)(50 218 83 121)(51 219 84 122)(52 220 57 123)(53 221 58 124)(54 222 59 125)(55 223 60 126)(56 224 61 127)
G:=sub<Sym(224)| (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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,112)(10,111)(11,110)(12,109)(13,108)(14,107)(15,106)(16,105)(17,104)(18,103)(19,102)(20,101)(21,100)(22,99)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(113,224)(114,223)(115,222)(116,221)(117,220)(118,219)(119,218)(120,217)(121,216)(122,215)(123,214)(124,213)(125,212)(126,211)(127,210)(128,209)(129,208)(130,207)(131,206)(132,205)(133,204)(134,203)(135,202)(136,201)(137,200)(138,199)(139,198)(140,197)(141,182)(142,181)(143,180)(144,179)(145,178)(146,177)(147,176)(148,175)(149,174)(150,173)(151,172)(152,171)(153,170)(154,169)(155,196)(156,195)(157,194)(158,193)(159,192)(160,191)(161,190)(162,189)(163,188)(164,187)(165,186)(166,185)(167,184)(168,183), (1,39,86,72)(2,40,87,73)(3,41,88,74)(4,42,89,75)(5,43,90,76)(6,44,91,77)(7,45,92,78)(8,46,93,79)(9,47,94,80)(10,48,95,81)(11,49,96,82)(12,50,97,83)(13,51,98,84)(14,52,99,57)(15,53,100,58)(16,54,101,59)(17,55,102,60)(18,56,103,61)(19,29,104,62)(20,30,105,63)(21,31,106,64)(22,32,107,65)(23,33,108,66)(24,34,109,67)(25,35,110,68)(26,36,111,69)(27,37,112,70)(28,38,85,71)(113,190,210,161)(114,191,211,162)(115,192,212,163)(116,193,213,164)(117,194,214,165)(118,195,215,166)(119,196,216,167)(120,169,217,168)(121,170,218,141)(122,171,219,142)(123,172,220,143)(124,173,221,144)(125,174,222,145)(126,175,223,146)(127,176,224,147)(128,177,197,148)(129,178,198,149)(130,179,199,150)(131,180,200,151)(132,181,201,152)(133,182,202,153)(134,183,203,154)(135,184,204,155)(136,185,205,156)(137,186,206,157)(138,187,207,158)(139,188,208,159)(140,189,209,160), (1,158,86,187)(2,159,87,188)(3,160,88,189)(4,161,89,190)(5,162,90,191)(6,163,91,192)(7,164,92,193)(8,165,93,194)(9,166,94,195)(10,167,95,196)(11,168,96,169)(12,141,97,170)(13,142,98,171)(14,143,99,172)(15,144,100,173)(16,145,101,174)(17,146,102,175)(18,147,103,176)(19,148,104,177)(20,149,105,178)(21,150,106,179)(22,151,107,180)(23,152,108,181)(24,153,109,182)(25,154,110,183)(26,155,111,184)(27,156,112,185)(28,157,85,186)(29,197,62,128)(30,198,63,129)(31,199,64,130)(32,200,65,131)(33,201,66,132)(34,202,67,133)(35,203,68,134)(36,204,69,135)(37,205,70,136)(38,206,71,137)(39,207,72,138)(40,208,73,139)(41,209,74,140)(42,210,75,113)(43,211,76,114)(44,212,77,115)(45,213,78,116)(46,214,79,117)(47,215,80,118)(48,216,81,119)(49,217,82,120)(50,218,83,121)(51,219,84,122)(52,220,57,123)(53,221,58,124)(54,222,59,125)(55,223,60,126)(56,224,61,127)>;
G:=Group( (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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,112)(10,111)(11,110)(12,109)(13,108)(14,107)(15,106)(16,105)(17,104)(18,103)(19,102)(20,101)(21,100)(22,99)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(113,224)(114,223)(115,222)(116,221)(117,220)(118,219)(119,218)(120,217)(121,216)(122,215)(123,214)(124,213)(125,212)(126,211)(127,210)(128,209)(129,208)(130,207)(131,206)(132,205)(133,204)(134,203)(135,202)(136,201)(137,200)(138,199)(139,198)(140,197)(141,182)(142,181)(143,180)(144,179)(145,178)(146,177)(147,176)(148,175)(149,174)(150,173)(151,172)(152,171)(153,170)(154,169)(155,196)(156,195)(157,194)(158,193)(159,192)(160,191)(161,190)(162,189)(163,188)(164,187)(165,186)(166,185)(167,184)(168,183), (1,39,86,72)(2,40,87,73)(3,41,88,74)(4,42,89,75)(5,43,90,76)(6,44,91,77)(7,45,92,78)(8,46,93,79)(9,47,94,80)(10,48,95,81)(11,49,96,82)(12,50,97,83)(13,51,98,84)(14,52,99,57)(15,53,100,58)(16,54,101,59)(17,55,102,60)(18,56,103,61)(19,29,104,62)(20,30,105,63)(21,31,106,64)(22,32,107,65)(23,33,108,66)(24,34,109,67)(25,35,110,68)(26,36,111,69)(27,37,112,70)(28,38,85,71)(113,190,210,161)(114,191,211,162)(115,192,212,163)(116,193,213,164)(117,194,214,165)(118,195,215,166)(119,196,216,167)(120,169,217,168)(121,170,218,141)(122,171,219,142)(123,172,220,143)(124,173,221,144)(125,174,222,145)(126,175,223,146)(127,176,224,147)(128,177,197,148)(129,178,198,149)(130,179,199,150)(131,180,200,151)(132,181,201,152)(133,182,202,153)(134,183,203,154)(135,184,204,155)(136,185,205,156)(137,186,206,157)(138,187,207,158)(139,188,208,159)(140,189,209,160), (1,158,86,187)(2,159,87,188)(3,160,88,189)(4,161,89,190)(5,162,90,191)(6,163,91,192)(7,164,92,193)(8,165,93,194)(9,166,94,195)(10,167,95,196)(11,168,96,169)(12,141,97,170)(13,142,98,171)(14,143,99,172)(15,144,100,173)(16,145,101,174)(17,146,102,175)(18,147,103,176)(19,148,104,177)(20,149,105,178)(21,150,106,179)(22,151,107,180)(23,152,108,181)(24,153,109,182)(25,154,110,183)(26,155,111,184)(27,156,112,185)(28,157,85,186)(29,197,62,128)(30,198,63,129)(31,199,64,130)(32,200,65,131)(33,201,66,132)(34,202,67,133)(35,203,68,134)(36,204,69,135)(37,205,70,136)(38,206,71,137)(39,207,72,138)(40,208,73,139)(41,209,74,140)(42,210,75,113)(43,211,76,114)(44,212,77,115)(45,213,78,116)(46,214,79,117)(47,215,80,118)(48,216,81,119)(49,217,82,120)(50,218,83,121)(51,219,84,122)(52,220,57,123)(53,221,58,124)(54,222,59,125)(55,223,60,126)(56,224,61,127) );
G=PermutationGroup([(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)], [(1,92),(2,91),(3,90),(4,89),(5,88),(6,87),(7,86),(8,85),(9,112),(10,111),(11,110),(12,109),(13,108),(14,107),(15,106),(16,105),(17,104),(18,103),(19,102),(20,101),(21,100),(22,99),(23,98),(24,97),(25,96),(26,95),(27,94),(28,93),(29,74),(30,73),(31,72),(32,71),(33,70),(34,69),(35,68),(36,67),(37,66),(38,65),(39,64),(40,63),(41,62),(42,61),(43,60),(44,59),(45,58),(46,57),(47,84),(48,83),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(113,224),(114,223),(115,222),(116,221),(117,220),(118,219),(119,218),(120,217),(121,216),(122,215),(123,214),(124,213),(125,212),(126,211),(127,210),(128,209),(129,208),(130,207),(131,206),(132,205),(133,204),(134,203),(135,202),(136,201),(137,200),(138,199),(139,198),(140,197),(141,182),(142,181),(143,180),(144,179),(145,178),(146,177),(147,176),(148,175),(149,174),(150,173),(151,172),(152,171),(153,170),(154,169),(155,196),(156,195),(157,194),(158,193),(159,192),(160,191),(161,190),(162,189),(163,188),(164,187),(165,186),(166,185),(167,184),(168,183)], [(1,39,86,72),(2,40,87,73),(3,41,88,74),(4,42,89,75),(5,43,90,76),(6,44,91,77),(7,45,92,78),(8,46,93,79),(9,47,94,80),(10,48,95,81),(11,49,96,82),(12,50,97,83),(13,51,98,84),(14,52,99,57),(15,53,100,58),(16,54,101,59),(17,55,102,60),(18,56,103,61),(19,29,104,62),(20,30,105,63),(21,31,106,64),(22,32,107,65),(23,33,108,66),(24,34,109,67),(25,35,110,68),(26,36,111,69),(27,37,112,70),(28,38,85,71),(113,190,210,161),(114,191,211,162),(115,192,212,163),(116,193,213,164),(117,194,214,165),(118,195,215,166),(119,196,216,167),(120,169,217,168),(121,170,218,141),(122,171,219,142),(123,172,220,143),(124,173,221,144),(125,174,222,145),(126,175,223,146),(127,176,224,147),(128,177,197,148),(129,178,198,149),(130,179,199,150),(131,180,200,151),(132,181,201,152),(133,182,202,153),(134,183,203,154),(135,184,204,155),(136,185,205,156),(137,186,206,157),(138,187,207,158),(139,188,208,159),(140,189,209,160)], [(1,158,86,187),(2,159,87,188),(3,160,88,189),(4,161,89,190),(5,162,90,191),(6,163,91,192),(7,164,92,193),(8,165,93,194),(9,166,94,195),(10,167,95,196),(11,168,96,169),(12,141,97,170),(13,142,98,171),(14,143,99,172),(15,144,100,173),(16,145,101,174),(17,146,102,175),(18,147,103,176),(19,148,104,177),(20,149,105,178),(21,150,106,179),(22,151,107,180),(23,152,108,181),(24,153,109,182),(25,154,110,183),(26,155,111,184),(27,156,112,185),(28,157,85,186),(29,197,62,128),(30,198,63,129),(31,199,64,130),(32,200,65,131),(33,201,66,132),(34,202,67,133),(35,203,68,134),(36,204,69,135),(37,205,70,136),(38,206,71,137),(39,207,72,138),(40,208,73,139),(41,209,74,140),(42,210,75,113),(43,211,76,114),(44,212,77,115),(45,213,78,116),(46,214,79,117),(47,215,80,118),(48,216,81,119),(49,217,82,120),(50,218,83,121),(51,219,84,122),(52,220,57,123),(53,221,58,124),(54,222,59,125),(55,223,60,126),(56,224,61,127)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 5 | 25 | 0 | 0 |
| 6 | 7 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 10 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 18 | 14 | 0 | 0 |
| 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 21 |
| 0 | 0 | 21 | 15 |
G:=sub<GL(4,GF(29))| [5,6,0,0,25,7,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,10,28,0,0,0,0,28,0,0,0,0,28],[18,8,0,0,14,11,0,0,0,0,0,1,0,0,28,0],[1,0,0,0,0,1,0,0,0,0,14,21,0,0,21,15] >;
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | 2+ (1+4) | Q8×D7 | D4⋊8D14 |
| kernel | D28⋊10Q8 | C28⋊2Q8 | C28.6Q8 | C4×D28 | D7×C4⋊C4 | D14⋊Q8 | D14⋊3Q8 | Q8×C28 | D28 | C4×Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 2 | 4 | 2 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
D_{28}\rtimes_{10}Q_8 % in TeX
G:=Group("D28:10Q8"); // GroupNames label
G:=SmallGroup(448,1032);
// by ID
G=gap.SmallGroup(448,1032);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,232,100,185,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^14*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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