direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C28, C28⋊7(C22×C4), (C23×C28)⋊6C2, C23⋊5(C2×C28), (C2×C42)⋊7C14, (C23×C4)⋊5C14, C4⋊1(C22×C28), (C4×C28)⋊57C22, C42⋊16(C2×C14), C2.4(C23×C28), C24.33(C2×C14), C14.56(C23×C4), C22⋊1(C22×C28), C22.59(D4×C14), (C2×C14).335C24, (C2×C28).707C23, (C22×C28)⋊58C22, (C22×D4).13C14, C14.179(C22×D4), C22.8(C23×C14), (D4×C14).330C22, C23.28(C22×C14), (C23×C14).90C22, (C22×C14).252C23, (C2×C4×C28)⋊20C2, C2.3(D4×C2×C14), (C2×C4)⋊7(C2×C28), (C14×C4⋊C4)⋊52C2, (C2×C4⋊C4)⋊25C14, C4⋊C4⋊19(C2×C14), (C2×C28)⋊32(C2×C4), (D4×C2×C14).26C2, C2.2(C14×C4○D4), (C7×C4⋊C4)⋊76C22, (C2×C14)⋊5(C22×C4), (C2×C22⋊C4)⋊16C14, (C14×C22⋊C4)⋊36C2, C22⋊C4⋊17(C2×C14), (C22×C4)⋊16(C2×C14), (C22×C14)⋊13(C2×C4), (C2×D4).76(C2×C14), C14.221(C2×C4○D4), (C2×C14).681(C2×D4), C22.27(C7×C4○D4), (C7×C22⋊C4)⋊71C22, (C2×C4).54(C22×C14), (C2×C14).227(C4○D4), SmallGroup(448,1298)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 578 in 426 conjugacy classes, 274 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×6], C22, C22 [×14], C22 [×24], C7, C2×C4 [×18], C2×C4 [×22], D4 [×16], C23, C23 [×12], C23 [×8], C14 [×3], C14 [×4], C14 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×12], C24 [×2], C28 [×8], C28 [×6], C2×C14, C2×C14 [×14], C2×C14 [×24], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C2×C28 [×18], C2×C28 [×22], C7×D4 [×16], C22×C14, C22×C14 [×12], C22×C14 [×8], C2×C4×D4, C4×C28 [×4], C7×C22⋊C4 [×8], C7×C4⋊C4 [×4], C22×C28 [×3], C22×C28 [×10], C22×C28 [×8], D4×C14 [×12], C23×C14 [×2], C2×C4×C28, C14×C22⋊C4 [×2], C14×C4⋊C4, D4×C28 [×8], C23×C28 [×2], D4×C2×C14, D4×C2×C28
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C7, C2×C4 [×28], D4 [×4], C23 [×15], C14 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C28 [×8], C2×C14 [×35], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C28 [×28], C7×D4 [×4], C22×C14 [×15], C2×C4×D4, C22×C28 [×14], D4×C14 [×6], C7×C4○D4 [×2], C23×C14, D4×C28 [×4], C23×C28, D4×C2×C14, C14×C4○D4, D4×C2×C28
Generators and relations
G = < a,b,c,d | a2=b28=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 224 points
Generators in S224
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 57)(27 58)(28 59)(29 183)(30 184)(31 185)(32 186)(33 187)(34 188)(35 189)(36 190)(37 191)(38 192)(39 193)(40 194)(41 195)(42 196)(43 169)(44 170)(45 171)(46 172)(47 173)(48 174)(49 175)(50 176)(51 177)(52 178)(53 179)(54 180)(55 181)(56 182)(85 158)(86 159)(87 160)(88 161)(89 162)(90 163)(91 164)(92 165)(93 166)(94 167)(95 168)(96 141)(97 142)(98 143)(99 144)(100 145)(101 146)(102 147)(103 148)(104 149)(105 150)(106 151)(107 152)(108 153)(109 154)(110 155)(111 156)(112 157)(113 200)(114 201)(115 202)(116 203)(117 204)(118 205)(119 206)(120 207)(121 208)(122 209)(123 210)(124 211)(125 212)(126 213)(127 214)(128 215)(129 216)(130 217)(131 218)(132 219)(133 220)(134 221)(135 222)(136 223)(137 224)(138 197)(139 198)(140 199)
(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 97 215 35)(2 98 216 36)(3 99 217 37)(4 100 218 38)(5 101 219 39)(6 102 220 40)(7 103 221 41)(8 104 222 42)(9 105 223 43)(10 106 224 44)(11 107 197 45)(12 108 198 46)(13 109 199 47)(14 110 200 48)(15 111 201 49)(16 112 202 50)(17 85 203 51)(18 86 204 52)(19 87 205 53)(20 88 206 54)(21 89 207 55)(22 90 208 56)(23 91 209 29)(24 92 210 30)(25 93 211 31)(26 94 212 32)(27 95 213 33)(28 96 214 34)(57 167 125 186)(58 168 126 187)(59 141 127 188)(60 142 128 189)(61 143 129 190)(62 144 130 191)(63 145 131 192)(64 146 132 193)(65 147 133 194)(66 148 134 195)(67 149 135 196)(68 150 136 169)(69 151 137 170)(70 152 138 171)(71 153 139 172)(72 154 140 173)(73 155 113 174)(74 156 114 175)(75 157 115 176)(76 158 116 177)(77 159 117 178)(78 160 118 179)(79 161 119 180)(80 162 120 181)(81 163 121 182)(82 164 122 183)(83 165 123 184)(84 166 124 185)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 129)(17 130)(18 131)(19 132)(20 133)(21 134)(22 135)(23 136)(24 137)(25 138)(26 139)(27 140)(28 113)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 193)(54 194)(55 195)(56 196)(57 198)(58 199)(59 200)(60 201)(61 202)(62 203)(63 204)(64 205)(65 206)(66 207)(67 208)(68 209)(69 210)(70 211)(71 212)(72 213)(73 214)(74 215)(75 216)(76 217)(77 218)(78 219)(79 220)(80 221)(81 222)(82 223)(83 224)(84 197)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 168)(110 141)(111 142)(112 143)
G:=sub<Sym(224)| (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,57)(27,58)(28,59)(29,183)(30,184)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,191)(38,192)(39,193)(40,194)(41,195)(42,196)(43,169)(44,170)(45,171)(46,172)(47,173)(48,174)(49,175)(50,176)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,141)(97,142)(98,143)(99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,151)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,200)(114,201)(115,202)(116,203)(117,204)(118,205)(119,206)(120,207)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,197)(139,198)(140,199), (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), (1,97,215,35)(2,98,216,36)(3,99,217,37)(4,100,218,38)(5,101,219,39)(6,102,220,40)(7,103,221,41)(8,104,222,42)(9,105,223,43)(10,106,224,44)(11,107,197,45)(12,108,198,46)(13,109,199,47)(14,110,200,48)(15,111,201,49)(16,112,202,50)(17,85,203,51)(18,86,204,52)(19,87,205,53)(20,88,206,54)(21,89,207,55)(22,90,208,56)(23,91,209,29)(24,92,210,30)(25,93,211,31)(26,94,212,32)(27,95,213,33)(28,96,214,34)(57,167,125,186)(58,168,126,187)(59,141,127,188)(60,142,128,189)(61,143,129,190)(62,144,130,191)(63,145,131,192)(64,146,132,193)(65,147,133,194)(66,148,134,195)(67,149,135,196)(68,150,136,169)(69,151,137,170)(70,152,138,171)(71,153,139,172)(72,154,140,173)(73,155,113,174)(74,156,114,175)(75,157,115,176)(76,158,116,177)(77,159,117,178)(78,160,118,179)(79,161,119,180)(80,162,120,181)(81,163,121,182)(82,164,122,183)(83,165,123,184)(84,166,124,185), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,113)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,198)(58,199)(59,200)(60,201)(61,202)(62,203)(63,204)(64,205)(65,206)(66,207)(67,208)(68,209)(69,210)(70,211)(71,212)(72,213)(73,214)(74,215)(75,216)(76,217)(77,218)(78,219)(79,220)(80,221)(81,222)(82,223)(83,224)(84,197)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,141)(111,142)(112,143)>;
G:=Group( (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,57)(27,58)(28,59)(29,183)(30,184)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,191)(38,192)(39,193)(40,194)(41,195)(42,196)(43,169)(44,170)(45,171)(46,172)(47,173)(48,174)(49,175)(50,176)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,141)(97,142)(98,143)(99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,151)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,200)(114,201)(115,202)(116,203)(117,204)(118,205)(119,206)(120,207)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,197)(139,198)(140,199), (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), (1,97,215,35)(2,98,216,36)(3,99,217,37)(4,100,218,38)(5,101,219,39)(6,102,220,40)(7,103,221,41)(8,104,222,42)(9,105,223,43)(10,106,224,44)(11,107,197,45)(12,108,198,46)(13,109,199,47)(14,110,200,48)(15,111,201,49)(16,112,202,50)(17,85,203,51)(18,86,204,52)(19,87,205,53)(20,88,206,54)(21,89,207,55)(22,90,208,56)(23,91,209,29)(24,92,210,30)(25,93,211,31)(26,94,212,32)(27,95,213,33)(28,96,214,34)(57,167,125,186)(58,168,126,187)(59,141,127,188)(60,142,128,189)(61,143,129,190)(62,144,130,191)(63,145,131,192)(64,146,132,193)(65,147,133,194)(66,148,134,195)(67,149,135,196)(68,150,136,169)(69,151,137,170)(70,152,138,171)(71,153,139,172)(72,154,140,173)(73,155,113,174)(74,156,114,175)(75,157,115,176)(76,158,116,177)(77,159,117,178)(78,160,118,179)(79,161,119,180)(80,162,120,181)(81,163,121,182)(82,164,122,183)(83,165,123,184)(84,166,124,185), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,113)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,198)(58,199)(59,200)(60,201)(61,202)(62,203)(63,204)(64,205)(65,206)(66,207)(67,208)(68,209)(69,210)(70,211)(71,212)(72,213)(73,214)(74,215)(75,216)(76,217)(77,218)(78,219)(79,220)(80,221)(81,222)(82,223)(83,224)(84,197)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,141)(111,142)(112,143) );
G=PermutationGroup([(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,72),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,84),(26,57),(27,58),(28,59),(29,183),(30,184),(31,185),(32,186),(33,187),(34,188),(35,189),(36,190),(37,191),(38,192),(39,193),(40,194),(41,195),(42,196),(43,169),(44,170),(45,171),(46,172),(47,173),(48,174),(49,175),(50,176),(51,177),(52,178),(53,179),(54,180),(55,181),(56,182),(85,158),(86,159),(87,160),(88,161),(89,162),(90,163),(91,164),(92,165),(93,166),(94,167),(95,168),(96,141),(97,142),(98,143),(99,144),(100,145),(101,146),(102,147),(103,148),(104,149),(105,150),(106,151),(107,152),(108,153),(109,154),(110,155),(111,156),(112,157),(113,200),(114,201),(115,202),(116,203),(117,204),(118,205),(119,206),(120,207),(121,208),(122,209),(123,210),(124,211),(125,212),(126,213),(127,214),(128,215),(129,216),(130,217),(131,218),(132,219),(133,220),(134,221),(135,222),(136,223),(137,224),(138,197),(139,198),(140,199)], [(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)], [(1,97,215,35),(2,98,216,36),(3,99,217,37),(4,100,218,38),(5,101,219,39),(6,102,220,40),(7,103,221,41),(8,104,222,42),(9,105,223,43),(10,106,224,44),(11,107,197,45),(12,108,198,46),(13,109,199,47),(14,110,200,48),(15,111,201,49),(16,112,202,50),(17,85,203,51),(18,86,204,52),(19,87,205,53),(20,88,206,54),(21,89,207,55),(22,90,208,56),(23,91,209,29),(24,92,210,30),(25,93,211,31),(26,94,212,32),(27,95,213,33),(28,96,214,34),(57,167,125,186),(58,168,126,187),(59,141,127,188),(60,142,128,189),(61,143,129,190),(62,144,130,191),(63,145,131,192),(64,146,132,193),(65,147,133,194),(66,148,134,195),(67,149,135,196),(68,150,136,169),(69,151,137,170),(70,152,138,171),(71,153,139,172),(72,154,140,173),(73,155,113,174),(74,156,114,175),(75,157,115,176),(76,158,116,177),(77,159,117,178),(78,160,118,179),(79,161,119,180),(80,162,120,181),(81,163,121,182),(82,164,122,183),(83,165,123,184),(84,166,124,185)], [(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,127),(15,128),(16,129),(17,130),(18,131),(19,132),(20,133),(21,134),(22,135),(23,136),(24,137),(25,138),(26,139),(27,140),(28,113),(29,169),(30,170),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,177),(38,178),(39,179),(40,180),(41,181),(42,182),(43,183),(44,184),(45,185),(46,186),(47,187),(48,188),(49,189),(50,190),(51,191),(52,192),(53,193),(54,194),(55,195),(56,196),(57,198),(58,199),(59,200),(60,201),(61,202),(62,203),(63,204),(64,205),(65,206),(66,207),(67,208),(68,209),(69,210),(70,211),(71,212),(72,213),(73,214),(74,215),(75,216),(76,217),(77,218),(78,219),(79,220),(80,221),(81,222),(82,223),(83,224),(84,197),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163),(105,164),(106,165),(107,166),(108,167),(109,168),(110,141),(111,142),(112,143)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 19 | 0 |
| 0 | 0 | 0 | 19 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 1 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 28 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,28,0,0,0,0,19,0,0,0,0,19],[1,0,0,0,0,28,0,0,0,0,1,1,0,0,27,28],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,28] >;
280 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4X | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14CL | 28A | ··· | 28AV | 28AW | ··· | 28EN |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | C4○D4 | C7×D4 | C7×C4○D4 |
| kernel | D4×C2×C28 | C2×C4×C28 | C14×C22⋊C4 | C14×C4⋊C4 | D4×C28 | C23×C28 | D4×C2×C14 | D4×C14 | C2×C4×D4 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C4×D4 | C23×C4 | C22×D4 | C2×D4 | C2×C28 | C2×C14 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 16 | 6 | 6 | 12 | 6 | 48 | 12 | 6 | 96 | 4 | 4 | 24 | 24 |
In GAP, Magma, Sage, TeX
D_4\times C_2\times C_{28} % in TeX
G:=Group("D4xC2xC28"); // GroupNames label
G:=SmallGroup(448,1298);
// by ID
G=gap.SmallGroup(448,1298);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,1192]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=d^2=1,a*b=b*a,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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