direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×D28, C42.127D14, C14.662- (1+4), C7⋊2(D4×Q8), C4⋊3(Q8×D7), (C7×Q8)⋊9D4, (C4×Q8)⋊8D7, C28⋊8(C2×Q8), D14⋊5(C2×Q8), (Q8×C28)⋊10C2, C4.24(C2×D28), C28.56(C2×D4), 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
Subgroups: 1284 in 280 conjugacy classes, 123 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×9], C22, C22 [×8], C7, C2×C4, C2×C4 [×6], C2×C4 [×18], D4 [×4], Q8 [×4], Q8 [×12], C23 [×2], D7 [×4], C14 [×3], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×14], Dic7 [×6], C28 [×8], C28 [×3], D14 [×4], D14 [×4], C2×C14, C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], Dic14 [×12], C4×D7 [×12], D28 [×4], C2×Dic7 [×6], C2×C28, C2×C28 [×6], C7×Q8 [×4], C22×D7 [×2], D4×Q8, C4⋊Dic7 [×9], D14⋊C4 [×6], C4×C28 [×3], C7×C4⋊C4 [×3], C2×Dic14 [×6], C2×C4×D7 [×6], C2×D28, Q8×D7 [×8], Q8×C14, C28⋊2Q8 [×3], C4×D28 [×3], D14⋊2Q8 [×6], Q8×C28, C2×Q8×D7 [×2], Q8×D28
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D7, C2×D4 [×6], C2×Q8 [×6], C24, D14 [×7], C22×D4, C22×Q8, 2- (1+4), D28 [×4], C22×D7 [×7], D4×Q8, C2×D28 [×6], Q8×D7 [×2], C23×D7, C22×D28, C2×Q8×D7, D4.10D14, Q8×D28
Generators and relations
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 >
►On 224 points
Generators in S224
(1 35 101 186)(2 36 102 187)(3 37 103 188)(4 38 104 189)(5 39 105 190)(6 40 106 191)(7 41 107 192)(8 42 108 193)(9 43 109 194)(10 44 110 195)(11 45 111 196)(12 46 112 169)(13 47 85 170)(14 48 86 171)(15 49 87 172)(16 50 88 173)(17 51 89 174)(18 52 90 175)(19 53 91 176)(20 54 92 177)(21 55 93 178)(22 56 94 179)(23 29 95 180)(24 30 96 181)(25 31 97 182)(26 32 98 183)(27 33 99 184)(28 34 100 185)(57 127 158 201)(58 128 159 202)(59 129 160 203)(60 130 161 204)(61 131 162 205)(62 132 163 206)(63 133 164 207)(64 134 165 208)(65 135 166 209)(66 136 167 210)(67 137 168 211)(68 138 141 212)(69 139 142 213)(70 140 143 214)(71 113 144 215)(72 114 145 216)(73 115 146 217)(74 116 147 218)(75 117 148 219)(76 118 149 220)(77 119 150 221)(78 120 151 222)(79 121 152 223)(80 122 153 224)(81 123 154 197)(82 124 155 198)(83 125 156 199)(84 126 157 200)
(1 128 101 202)(2 129 102 203)(3 130 103 204)(4 131 104 205)(5 132 105 206)(6 133 106 207)(7 134 107 208)(8 135 108 209)(9 136 109 210)(10 137 110 211)(11 138 111 212)(12 139 112 213)(13 140 85 214)(14 113 86 215)(15 114 87 216)(16 115 88 217)(17 116 89 218)(18 117 90 219)(19 118 91 220)(20 119 92 221)(21 120 93 222)(22 121 94 223)(23 122 95 224)(24 123 96 197)(25 124 97 198)(26 125 98 199)(27 126 99 200)(28 127 100 201)(29 80 180 153)(30 81 181 154)(31 82 182 155)(32 83 183 156)(33 84 184 157)(34 57 185 158)(35 58 186 159)(36 59 187 160)(37 60 188 161)(38 61 189 162)(39 62 190 163)(40 63 191 164)(41 64 192 165)(42 65 193 166)(43 66 194 167)(44 67 195 168)(45 68 196 141)(46 69 169 142)(47 70 170 143)(48 71 171 144)(49 72 172 145)(50 73 173 146)(51 74 174 147)(52 75 175 148)(53 76 176 149)(54 77 177 150)(55 78 178 151)(56 79 179 152)
(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)
(1 107)(2 106)(3 105)(4 104)(5 103)(6 102)(7 101)(8 100)(9 99)(10 98)(11 97)(12 96)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 112)(25 111)(26 110)(27 109)(28 108)(29 170)(30 169)(31 196)(32 195)(33 194)(34 193)(35 192)(36 191)(37 190)(38 189)(39 188)(40 187)(41 186)(42 185)(43 184)(44 183)(45 182)(46 181)(47 180)(48 179)(49 178)(50 177)(51 176)(52 175)(53 174)(54 173)(55 172)(56 171)(57 166)(58 165)(59 164)(60 163)(61 162)(62 161)(63 160)(64 159)(65 158)(66 157)(67 156)(68 155)(69 154)(70 153)(71 152)(72 151)(73 150)(74 149)(75 148)(76 147)(77 146)(78 145)(79 144)(80 143)(81 142)(82 141)(83 168)(84 167)(113 223)(114 222)(115 221)(116 220)(117 219)(118 218)(119 217)(120 216)(121 215)(122 214)(123 213)(124 212)(125 211)(126 210)(127 209)(128 208)(129 207)(130 206)(131 205)(132 204)(133 203)(134 202)(135 201)(136 200)(137 199)(138 198)(139 197)(140 224)
G:=sub<Sym(224)| (1,35,101,186)(2,36,102,187)(3,37,103,188)(4,38,104,189)(5,39,105,190)(6,40,106,191)(7,41,107,192)(8,42,108,193)(9,43,109,194)(10,44,110,195)(11,45,111,196)(12,46,112,169)(13,47,85,170)(14,48,86,171)(15,49,87,172)(16,50,88,173)(17,51,89,174)(18,52,90,175)(19,53,91,176)(20,54,92,177)(21,55,93,178)(22,56,94,179)(23,29,95,180)(24,30,96,181)(25,31,97,182)(26,32,98,183)(27,33,99,184)(28,34,100,185)(57,127,158,201)(58,128,159,202)(59,129,160,203)(60,130,161,204)(61,131,162,205)(62,132,163,206)(63,133,164,207)(64,134,165,208)(65,135,166,209)(66,136,167,210)(67,137,168,211)(68,138,141,212)(69,139,142,213)(70,140,143,214)(71,113,144,215)(72,114,145,216)(73,115,146,217)(74,116,147,218)(75,117,148,219)(76,118,149,220)(77,119,150,221)(78,120,151,222)(79,121,152,223)(80,122,153,224)(81,123,154,197)(82,124,155,198)(83,125,156,199)(84,126,157,200), (1,128,101,202)(2,129,102,203)(3,130,103,204)(4,131,104,205)(5,132,105,206)(6,133,106,207)(7,134,107,208)(8,135,108,209)(9,136,109,210)(10,137,110,211)(11,138,111,212)(12,139,112,213)(13,140,85,214)(14,113,86,215)(15,114,87,216)(16,115,88,217)(17,116,89,218)(18,117,90,219)(19,118,91,220)(20,119,92,221)(21,120,93,222)(22,121,94,223)(23,122,95,224)(24,123,96,197)(25,124,97,198)(26,125,98,199)(27,126,99,200)(28,127,100,201)(29,80,180,153)(30,81,181,154)(31,82,182,155)(32,83,183,156)(33,84,184,157)(34,57,185,158)(35,58,186,159)(36,59,187,160)(37,60,188,161)(38,61,189,162)(39,62,190,163)(40,63,191,164)(41,64,192,165)(42,65,193,166)(43,66,194,167)(44,67,195,168)(45,68,196,141)(46,69,169,142)(47,70,170,143)(48,71,171,144)(49,72,172,145)(50,73,173,146)(51,74,174,147)(52,75,175,148)(53,76,176,149)(54,77,177,150)(55,78,178,151)(56,79,179,152), (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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,96)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,112)(25,111)(26,110)(27,109)(28,108)(29,170)(30,169)(31,196)(32,195)(33,194)(34,193)(35,192)(36,191)(37,190)(38,189)(39,188)(40,187)(41,186)(42,185)(43,184)(44,183)(45,182)(46,181)(47,180)(48,179)(49,178)(50,177)(51,176)(52,175)(53,174)(54,173)(55,172)(56,171)(57,166)(58,165)(59,164)(60,163)(61,162)(62,161)(63,160)(64,159)(65,158)(66,157)(67,156)(68,155)(69,154)(70,153)(71,152)(72,151)(73,150)(74,149)(75,148)(76,147)(77,146)(78,145)(79,144)(80,143)(81,142)(82,141)(83,168)(84,167)(113,223)(114,222)(115,221)(116,220)(117,219)(118,218)(119,217)(120,216)(121,215)(122,214)(123,213)(124,212)(125,211)(126,210)(127,209)(128,208)(129,207)(130,206)(131,205)(132,204)(133,203)(134,202)(135,201)(136,200)(137,199)(138,198)(139,197)(140,224)>;
G:=Group( (1,35,101,186)(2,36,102,187)(3,37,103,188)(4,38,104,189)(5,39,105,190)(6,40,106,191)(7,41,107,192)(8,42,108,193)(9,43,109,194)(10,44,110,195)(11,45,111,196)(12,46,112,169)(13,47,85,170)(14,48,86,171)(15,49,87,172)(16,50,88,173)(17,51,89,174)(18,52,90,175)(19,53,91,176)(20,54,92,177)(21,55,93,178)(22,56,94,179)(23,29,95,180)(24,30,96,181)(25,31,97,182)(26,32,98,183)(27,33,99,184)(28,34,100,185)(57,127,158,201)(58,128,159,202)(59,129,160,203)(60,130,161,204)(61,131,162,205)(62,132,163,206)(63,133,164,207)(64,134,165,208)(65,135,166,209)(66,136,167,210)(67,137,168,211)(68,138,141,212)(69,139,142,213)(70,140,143,214)(71,113,144,215)(72,114,145,216)(73,115,146,217)(74,116,147,218)(75,117,148,219)(76,118,149,220)(77,119,150,221)(78,120,151,222)(79,121,152,223)(80,122,153,224)(81,123,154,197)(82,124,155,198)(83,125,156,199)(84,126,157,200), (1,128,101,202)(2,129,102,203)(3,130,103,204)(4,131,104,205)(5,132,105,206)(6,133,106,207)(7,134,107,208)(8,135,108,209)(9,136,109,210)(10,137,110,211)(11,138,111,212)(12,139,112,213)(13,140,85,214)(14,113,86,215)(15,114,87,216)(16,115,88,217)(17,116,89,218)(18,117,90,219)(19,118,91,220)(20,119,92,221)(21,120,93,222)(22,121,94,223)(23,122,95,224)(24,123,96,197)(25,124,97,198)(26,125,98,199)(27,126,99,200)(28,127,100,201)(29,80,180,153)(30,81,181,154)(31,82,182,155)(32,83,183,156)(33,84,184,157)(34,57,185,158)(35,58,186,159)(36,59,187,160)(37,60,188,161)(38,61,189,162)(39,62,190,163)(40,63,191,164)(41,64,192,165)(42,65,193,166)(43,66,194,167)(44,67,195,168)(45,68,196,141)(46,69,169,142)(47,70,170,143)(48,71,171,144)(49,72,172,145)(50,73,173,146)(51,74,174,147)(52,75,175,148)(53,76,176,149)(54,77,177,150)(55,78,178,151)(56,79,179,152), (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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,96)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,112)(25,111)(26,110)(27,109)(28,108)(29,170)(30,169)(31,196)(32,195)(33,194)(34,193)(35,192)(36,191)(37,190)(38,189)(39,188)(40,187)(41,186)(42,185)(43,184)(44,183)(45,182)(46,181)(47,180)(48,179)(49,178)(50,177)(51,176)(52,175)(53,174)(54,173)(55,172)(56,171)(57,166)(58,165)(59,164)(60,163)(61,162)(62,161)(63,160)(64,159)(65,158)(66,157)(67,156)(68,155)(69,154)(70,153)(71,152)(72,151)(73,150)(74,149)(75,148)(76,147)(77,146)(78,145)(79,144)(80,143)(81,142)(82,141)(83,168)(84,167)(113,223)(114,222)(115,221)(116,220)(117,219)(118,218)(119,217)(120,216)(121,215)(122,214)(123,213)(124,212)(125,211)(126,210)(127,209)(128,208)(129,207)(130,206)(131,205)(132,204)(133,203)(134,202)(135,201)(136,200)(137,199)(138,198)(139,197)(140,224) );
G=PermutationGroup([(1,35,101,186),(2,36,102,187),(3,37,103,188),(4,38,104,189),(5,39,105,190),(6,40,106,191),(7,41,107,192),(8,42,108,193),(9,43,109,194),(10,44,110,195),(11,45,111,196),(12,46,112,169),(13,47,85,170),(14,48,86,171),(15,49,87,172),(16,50,88,173),(17,51,89,174),(18,52,90,175),(19,53,91,176),(20,54,92,177),(21,55,93,178),(22,56,94,179),(23,29,95,180),(24,30,96,181),(25,31,97,182),(26,32,98,183),(27,33,99,184),(28,34,100,185),(57,127,158,201),(58,128,159,202),(59,129,160,203),(60,130,161,204),(61,131,162,205),(62,132,163,206),(63,133,164,207),(64,134,165,208),(65,135,166,209),(66,136,167,210),(67,137,168,211),(68,138,141,212),(69,139,142,213),(70,140,143,214),(71,113,144,215),(72,114,145,216),(73,115,146,217),(74,116,147,218),(75,117,148,219),(76,118,149,220),(77,119,150,221),(78,120,151,222),(79,121,152,223),(80,122,153,224),(81,123,154,197),(82,124,155,198),(83,125,156,199),(84,126,157,200)], [(1,128,101,202),(2,129,102,203),(3,130,103,204),(4,131,104,205),(5,132,105,206),(6,133,106,207),(7,134,107,208),(8,135,108,209),(9,136,109,210),(10,137,110,211),(11,138,111,212),(12,139,112,213),(13,140,85,214),(14,113,86,215),(15,114,87,216),(16,115,88,217),(17,116,89,218),(18,117,90,219),(19,118,91,220),(20,119,92,221),(21,120,93,222),(22,121,94,223),(23,122,95,224),(24,123,96,197),(25,124,97,198),(26,125,98,199),(27,126,99,200),(28,127,100,201),(29,80,180,153),(30,81,181,154),(31,82,182,155),(32,83,183,156),(33,84,184,157),(34,57,185,158),(35,58,186,159),(36,59,187,160),(37,60,188,161),(38,61,189,162),(39,62,190,163),(40,63,191,164),(41,64,192,165),(42,65,193,166),(43,66,194,167),(44,67,195,168),(45,68,196,141),(46,69,169,142),(47,70,170,143),(48,71,171,144),(49,72,172,145),(50,73,173,146),(51,74,174,147),(52,75,175,148),(53,76,176,149),(54,77,177,150),(55,78,178,151),(56,79,179,152)], [(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)], [(1,107),(2,106),(3,105),(4,104),(5,103),(6,102),(7,101),(8,100),(9,99),(10,98),(11,97),(12,96),(13,95),(14,94),(15,93),(16,92),(17,91),(18,90),(19,89),(20,88),(21,87),(22,86),(23,85),(24,112),(25,111),(26,110),(27,109),(28,108),(29,170),(30,169),(31,196),(32,195),(33,194),(34,193),(35,192),(36,191),(37,190),(38,189),(39,188),(40,187),(41,186),(42,185),(43,184),(44,183),(45,182),(46,181),(47,180),(48,179),(49,178),(50,177),(51,176),(52,175),(53,174),(54,173),(55,172),(56,171),(57,166),(58,165),(59,164),(60,163),(61,162),(62,161),(63,160),(64,159),(65,158),(66,157),(67,156),(68,155),(69,154),(70,153),(71,152),(72,151),(73,150),(74,149),(75,148),(76,147),(77,146),(78,145),(79,144),(80,143),(81,142),(82,141),(83,168),(84,167),(113,223),(114,222),(115,221),(116,220),(117,219),(118,218),(119,217),(120,216),(121,215),(122,214),(123,213),(124,212),(125,211),(126,210),(127,209),(128,208),(129,207),(130,206),(131,205),(132,204),(133,203),(134,202),(135,201),(136,200),(137,199),(138,198),(139,197),(140,224)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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ℤ
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