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), C42⋊16(C2×C14), (C4×C28)⋊57C22, C2.4(C23×C28), C24.33(C2×C14), C14.56(C23×C4), C22⋊1(C22×C28), C22.59(D4×C14), (C2×C28).707C23, (C2×C14).335C24, (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), C22⋊C4⋊17(C2×C14), (C14×C22⋊C4)⋊36C2, (C2×C22⋊C4)⋊16C14, (C22×C14)⋊13(C2×C4), (C22×C4)⋊16(C2×C14), (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
Generators and relations for D4×C2×C28
G = < a,b,c,d | a2=b28=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 578 in 426 conjugacy classes, 274 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C4×D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, C22×C28, D4×C14, C23×C14, C2×C4×C28, C14×C22⋊C4, C14×C4⋊C4, D4×C28, C23×C28, D4×C2×C14, D4×C2×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22×C4, C2×D4, C4○D4, C24, C28, C2×C14, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C28, C7×D4, C22×C14, C2×C4×D4, C22×C28, D4×C14, C7×C4○D4, C23×C14, D4×C28, C23×C28, D4×C2×C14, C14×C4○D4, D4×C2×C28
►On 224 points
Generators in S224
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 202)(30 203)(31 204)(32 205)(33 206)(34 207)(35 208)(36 209)(37 210)(38 211)(39 212)(40 213)(41 214)(42 215)(43 216)(44 217)(45 218)(46 219)(47 220)(48 221)(49 222)(50 223)(51 224)(52 197)(53 198)(54 199)(55 200)(56 201)(57 155)(58 156)(59 157)(60 158)(61 159)(62 160)(63 161)(64 162)(65 163)(66 164)(67 165)(68 166)(69 167)(70 168)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(81 151)(82 152)(83 153)(84 154)(85 177)(86 178)(87 179)(88 180)(89 181)(90 182)(91 183)(92 184)(93 185)(94 186)(95 187)(96 188)(97 189)(98 190)(99 191)(100 192)(101 193)(102 194)(103 195)(104 196)(105 169)(106 170)(107 171)(108 172)(109 173)(110 174)(111 175)(112 176)
(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)
(1 218 66 175)(2 219 67 176)(3 220 68 177)(4 221 69 178)(5 222 70 179)(6 223 71 180)(7 224 72 181)(8 197 73 182)(9 198 74 183)(10 199 75 184)(11 200 76 185)(12 201 77 186)(13 202 78 187)(14 203 79 188)(15 204 80 189)(16 205 81 190)(17 206 82 191)(18 207 83 192)(19 208 84 193)(20 209 57 194)(21 210 58 195)(22 211 59 196)(23 212 60 169)(24 213 61 170)(25 214 62 171)(26 215 63 172)(27 216 64 173)(28 217 65 174)(29 148 95 115)(30 149 96 116)(31 150 97 117)(32 151 98 118)(33 152 99 119)(34 153 100 120)(35 154 101 121)(36 155 102 122)(37 156 103 123)(38 157 104 124)(39 158 105 125)(40 159 106 126)(41 160 107 127)(42 161 108 128)(43 162 109 129)(44 163 110 130)(45 164 111 131)(46 165 112 132)(47 166 85 133)(48 167 86 134)(49 168 87 135)(50 141 88 136)(51 142 89 137)(52 143 90 138)(53 144 91 139)(54 145 92 140)(55 146 93 113)(56 147 94 114)
(1 150)(2 151)(3 152)(4 153)(5 154)(6 155)(7 156)(8 157)(9 158)(10 159)(11 160)(12 161)(13 162)(14 163)(15 164)(16 165)(17 166)(18 167)(19 168)(20 141)(21 142)(22 143)(23 144)(24 145)(25 146)(26 147)(27 148)(28 149)(29 216)(30 217)(31 218)(32 219)(33 220)(34 221)(35 222)(36 223)(37 224)(38 197)(39 198)(40 199)(41 200)(42 201)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 208)(50 209)(51 210)(52 211)(53 212)(54 213)(55 214)(56 215)(57 136)(58 137)(59 138)(60 139)(61 140)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 121)(71 122)(72 123)(73 124)(74 125)(75 126)(76 127)(77 128)(78 129)(79 130)(80 131)(81 132)(82 133)(83 134)(84 135)(85 191)(86 192)(87 193)(88 194)(89 195)(90 196)(91 169)(92 170)(93 171)(94 172)(95 173)(96 174)(97 175)(98 176)(99 177)(100 178)(101 179)(102 180)(103 181)(104 182)(105 183)(106 184)(107 185)(108 186)(109 187)(110 188)(111 189)(112 190)
G:=sub<Sym(224)| (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,129)(28,130)(29,202)(30,203)(31,204)(32,205)(33,206)(34,207)(35,208)(36,209)(37,210)(38,211)(39,212)(40,213)(41,214)(42,215)(43,216)(44,217)(45,218)(46,219)(47,220)(48,221)(49,222)(50,223)(51,224)(52,197)(53,198)(54,199)(55,200)(56,201)(57,155)(58,156)(59,157)(60,158)(61,159)(62,160)(63,161)(64,162)(65,163)(66,164)(67,165)(68,166)(69,167)(70,168)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,177)(86,178)(87,179)(88,180)(89,181)(90,182)(91,183)(92,184)(93,185)(94,186)(95,187)(96,188)(97,189)(98,190)(99,191)(100,192)(101,193)(102,194)(103,195)(104,196)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176), (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), (1,218,66,175)(2,219,67,176)(3,220,68,177)(4,221,69,178)(5,222,70,179)(6,223,71,180)(7,224,72,181)(8,197,73,182)(9,198,74,183)(10,199,75,184)(11,200,76,185)(12,201,77,186)(13,202,78,187)(14,203,79,188)(15,204,80,189)(16,205,81,190)(17,206,82,191)(18,207,83,192)(19,208,84,193)(20,209,57,194)(21,210,58,195)(22,211,59,196)(23,212,60,169)(24,213,61,170)(25,214,62,171)(26,215,63,172)(27,216,64,173)(28,217,65,174)(29,148,95,115)(30,149,96,116)(31,150,97,117)(32,151,98,118)(33,152,99,119)(34,153,100,120)(35,154,101,121)(36,155,102,122)(37,156,103,123)(38,157,104,124)(39,158,105,125)(40,159,106,126)(41,160,107,127)(42,161,108,128)(43,162,109,129)(44,163,110,130)(45,164,111,131)(46,165,112,132)(47,166,85,133)(48,167,86,134)(49,168,87,135)(50,141,88,136)(51,142,89,137)(52,143,90,138)(53,144,91,139)(54,145,92,140)(55,146,93,113)(56,147,94,114), (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,161)(13,162)(14,163)(15,164)(16,165)(17,166)(18,167)(19,168)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,216)(30,217)(31,218)(32,219)(33,220)(34,221)(35,222)(36,223)(37,224)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,211)(53,212)(54,213)(55,214)(56,215)(57,136)(58,137)(59,138)(60,139)(61,140)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,129)(79,130)(80,131)(81,132)(82,133)(83,134)(84,135)(85,191)(86,192)(87,193)(88,194)(89,195)(90,196)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,179)(102,180)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)>;
G:=Group( (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,129)(28,130)(29,202)(30,203)(31,204)(32,205)(33,206)(34,207)(35,208)(36,209)(37,210)(38,211)(39,212)(40,213)(41,214)(42,215)(43,216)(44,217)(45,218)(46,219)(47,220)(48,221)(49,222)(50,223)(51,224)(52,197)(53,198)(54,199)(55,200)(56,201)(57,155)(58,156)(59,157)(60,158)(61,159)(62,160)(63,161)(64,162)(65,163)(66,164)(67,165)(68,166)(69,167)(70,168)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,177)(86,178)(87,179)(88,180)(89,181)(90,182)(91,183)(92,184)(93,185)(94,186)(95,187)(96,188)(97,189)(98,190)(99,191)(100,192)(101,193)(102,194)(103,195)(104,196)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176), (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), (1,218,66,175)(2,219,67,176)(3,220,68,177)(4,221,69,178)(5,222,70,179)(6,223,71,180)(7,224,72,181)(8,197,73,182)(9,198,74,183)(10,199,75,184)(11,200,76,185)(12,201,77,186)(13,202,78,187)(14,203,79,188)(15,204,80,189)(16,205,81,190)(17,206,82,191)(18,207,83,192)(19,208,84,193)(20,209,57,194)(21,210,58,195)(22,211,59,196)(23,212,60,169)(24,213,61,170)(25,214,62,171)(26,215,63,172)(27,216,64,173)(28,217,65,174)(29,148,95,115)(30,149,96,116)(31,150,97,117)(32,151,98,118)(33,152,99,119)(34,153,100,120)(35,154,101,121)(36,155,102,122)(37,156,103,123)(38,157,104,124)(39,158,105,125)(40,159,106,126)(41,160,107,127)(42,161,108,128)(43,162,109,129)(44,163,110,130)(45,164,111,131)(46,165,112,132)(47,166,85,133)(48,167,86,134)(49,168,87,135)(50,141,88,136)(51,142,89,137)(52,143,90,138)(53,144,91,139)(54,145,92,140)(55,146,93,113)(56,147,94,114), (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,161)(13,162)(14,163)(15,164)(16,165)(17,166)(18,167)(19,168)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,216)(30,217)(31,218)(32,219)(33,220)(34,221)(35,222)(36,223)(37,224)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,211)(53,212)(54,213)(55,214)(56,215)(57,136)(58,137)(59,138)(60,139)(61,140)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,129)(79,130)(80,131)(81,132)(82,133)(83,134)(84,135)(85,191)(86,192)(87,193)(88,194)(89,195)(90,196)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,179)(102,180)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190) );
G=PermutationGroup([[(1,131),(2,132),(3,133),(4,134),(5,135),(6,136),(7,137),(8,138),(9,139),(10,140),(11,113),(12,114),(13,115),(14,116),(15,117),(16,118),(17,119),(18,120),(19,121),(20,122),(21,123),(22,124),(23,125),(24,126),(25,127),(26,128),(27,129),(28,130),(29,202),(30,203),(31,204),(32,205),(33,206),(34,207),(35,208),(36,209),(37,210),(38,211),(39,212),(40,213),(41,214),(42,215),(43,216),(44,217),(45,218),(46,219),(47,220),(48,221),(49,222),(50,223),(51,224),(52,197),(53,198),(54,199),(55,200),(56,201),(57,155),(58,156),(59,157),(60,158),(61,159),(62,160),(63,161),(64,162),(65,163),(66,164),(67,165),(68,166),(69,167),(70,168),(71,141),(72,142),(73,143),(74,144),(75,145),(76,146),(77,147),(78,148),(79,149),(80,150),(81,151),(82,152),(83,153),(84,154),(85,177),(86,178),(87,179),(88,180),(89,181),(90,182),(91,183),(92,184),(93,185),(94,186),(95,187),(96,188),(97,189),(98,190),(99,191),(100,192),(101,193),(102,194),(103,195),(104,196),(105,169),(106,170),(107,171),(108,172),(109,173),(110,174),(111,175),(112,176)], [(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)], [(1,218,66,175),(2,219,67,176),(3,220,68,177),(4,221,69,178),(5,222,70,179),(6,223,71,180),(7,224,72,181),(8,197,73,182),(9,198,74,183),(10,199,75,184),(11,200,76,185),(12,201,77,186),(13,202,78,187),(14,203,79,188),(15,204,80,189),(16,205,81,190),(17,206,82,191),(18,207,83,192),(19,208,84,193),(20,209,57,194),(21,210,58,195),(22,211,59,196),(23,212,60,169),(24,213,61,170),(25,214,62,171),(26,215,63,172),(27,216,64,173),(28,217,65,174),(29,148,95,115),(30,149,96,116),(31,150,97,117),(32,151,98,118),(33,152,99,119),(34,153,100,120),(35,154,101,121),(36,155,102,122),(37,156,103,123),(38,157,104,124),(39,158,105,125),(40,159,106,126),(41,160,107,127),(42,161,108,128),(43,162,109,129),(44,163,110,130),(45,164,111,131),(46,165,112,132),(47,166,85,133),(48,167,86,134),(49,168,87,135),(50,141,88,136),(51,142,89,137),(52,143,90,138),(53,144,91,139),(54,145,92,140),(55,146,93,113),(56,147,94,114)], [(1,150),(2,151),(3,152),(4,153),(5,154),(6,155),(7,156),(8,157),(9,158),(10,159),(11,160),(12,161),(13,162),(14,163),(15,164),(16,165),(17,166),(18,167),(19,168),(20,141),(21,142),(22,143),(23,144),(24,145),(25,146),(26,147),(27,148),(28,149),(29,216),(30,217),(31,218),(32,219),(33,220),(34,221),(35,222),(36,223),(37,224),(38,197),(39,198),(40,199),(41,200),(42,201),(43,202),(44,203),(45,204),(46,205),(47,206),(48,207),(49,208),(50,209),(51,210),(52,211),(53,212),(54,213),(55,214),(56,215),(57,136),(58,137),(59,138),(60,139),(61,140),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119),(69,120),(70,121),(71,122),(72,123),(73,124),(74,125),(75,126),(76,127),(77,128),(78,129),(79,130),(80,131),(81,132),(82,133),(83,134),(84,135),(85,191),(86,192),(87,193),(88,194),(89,195),(90,196),(91,169),(92,170),(93,171),(94,172),(95,173),(96,174),(97,175),(98,176),(99,177),(100,178),(101,179),(102,180),(103,181),(104,182),(105,183),(106,184),(107,185),(108,186),(109,187),(110,188),(111,189),(112,190)]])
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 |
Matrix representation of D4×C2×C28 ►in 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] >;
D4×C2×C28 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