metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊22D4, C14.182- (1+4), C22⋊Q8⋊8D7, C7⋊5(D4⋊6D4), C4.112(D4×D7), C4⋊C4.189D14, D14.21(C2×D4), C28.235(C2×D4), D14⋊D4⋊25C2, D28⋊C4⋊26C2, D14⋊2Q8⋊25C2, Dic7⋊5(C4○D4), (C2×Q8).126D14, C22⋊C4.16D14, C14.77(C22×D4), Dic7⋊Q8⋊14C2, D14.D4⋊25C2, D14.5D4⋊18C2, (C2×C28).503C23, (C2×C14).175C24, D14⋊C4.23C22, (C22×C4).237D14, (C2×D28).149C22, Dic7⋊C4.27C22, C4⋊Dic7.215C22, (Q8×C14).107C22, C22.196(C23×D7), C23.119(C22×D7), (C22×C28).255C22, (C22×C14).203C23, (C2×Dic7).234C23, (C4×Dic7).105C22, (C22×D7).197C23, C23.D7.116C22, C2.19(Q8.10D14), (C2×Dic14).294C22, C2.50(C2×D4×D7), (D7×C4⋊C4)⋊26C2, (C4×C7⋊D4)⋊23C2, C2.49(D7×C4○D4), (C2×C4○D28)⋊24C2, (C2×Q8⋊2D7)⋊8C2, (C7×C22⋊Q8)⋊11C2, (C2×C4×D7).95C22, C14.161(C2×C4○D4), (C2×C4).48(C22×D7), (C7×C4⋊C4).158C22, (C2×C7⋊D4).123C22, (C7×C22⋊C4).30C22, SmallGroup(448,1084)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1404 in 292 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×14], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×21], D4 [×14], Q8 [×4], C23, C23 [×3], D7 [×5], C14 [×3], C14, C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4, C22×C4 [×7], C2×D4 [×6], C2×Q8, C2×Q8, C4○D4 [×8], Dic7 [×2], Dic7 [×4], C28 [×2], C28 [×5], D14 [×4], D14 [×7], C2×C14, C2×C14 [×3], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic14 [×2], C4×D7 [×14], D28 [×4], D28 [×4], C2×Dic7 [×3], C2×Dic7 [×2], C7⋊D4 [×6], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7, C22×D7 [×2], C22×C14, D4⋊6D4, C4×Dic7, Dic7⋊C4, Dic7⋊C4 [×4], C4⋊Dic7 [×2], D14⋊C4, D14⋊C4 [×4], C23.D7, C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7, C2×C4×D7 [×6], C2×D28, C2×D28 [×2], C4○D28 [×4], Q8⋊2D7 [×4], C2×C7⋊D4, C2×C7⋊D4 [×2], C22×C28, Q8×C14, D14.D4 [×2], D14⋊D4 [×2], D7×C4⋊C4 [×2], D28⋊C4, D14.5D4 [×2], D14⋊2Q8, C4×C7⋊D4, Dic7⋊Q8, C7×C22⋊Q8, C2×C4○D28, C2×Q8⋊2D7, D28⋊22D4
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), C22×D7 [×7], D4⋊6D4, D4×D7 [×2], C23×D7, C2×D4×D7, Q8.10D14, D7×C4○D4, D28⋊22D4
Generators and relations
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=dad=a13, cbc-1=a12b, dbd=a26b, dcd=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)
(1 149)(2 148)(3 147)(4 146)(5 145)(6 144)(7 143)(8 142)(9 141)(10 168)(11 167)(12 166)(13 165)(14 164)(15 163)(16 162)(17 161)(18 160)(19 159)(20 158)(21 157)(22 156)(23 155)(24 154)(25 153)(26 152)(27 151)(28 150)(29 183)(30 182)(31 181)(32 180)(33 179)(34 178)(35 177)(36 176)(37 175)(38 174)(39 173)(40 172)(41 171)(42 170)(43 169)(44 196)(45 195)(46 194)(47 193)(48 192)(49 191)(50 190)(51 189)(52 188)(53 187)(54 186)(55 185)(56 184)(57 137)(58 136)(59 135)(60 134)(61 133)(62 132)(63 131)(64 130)(65 129)(66 128)(67 127)(68 126)(69 125)(70 124)(71 123)(72 122)(73 121)(74 120)(75 119)(76 118)(77 117)(78 116)(79 115)(80 114)(81 113)(82 140)(83 139)(84 138)(85 204)(86 203)(87 202)(88 201)(89 200)(90 199)(91 198)(92 197)(93 224)(94 223)(95 222)(96 221)(97 220)(98 219)(99 218)(100 217)(101 216)(102 215)(103 214)(104 213)(105 212)(106 211)(107 210)(108 209)(109 208)(110 207)(111 206)(112 205)
(1 92 157 205)(2 105 158 218)(3 90 159 203)(4 103 160 216)(5 88 161 201)(6 101 162 214)(7 86 163 199)(8 99 164 212)(9 112 165 197)(10 97 166 210)(11 110 167 223)(12 95 168 208)(13 108 141 221)(14 93 142 206)(15 106 143 219)(16 91 144 204)(17 104 145 217)(18 89 146 202)(19 102 147 215)(20 87 148 200)(21 100 149 213)(22 85 150 198)(23 98 151 211)(24 111 152 224)(25 96 153 209)(26 109 154 222)(27 94 155 207)(28 107 156 220)(29 69 175 121)(30 82 176 134)(31 67 177 119)(32 80 178 132)(33 65 179 117)(34 78 180 130)(35 63 181 115)(36 76 182 128)(37 61 183 113)(38 74 184 126)(39 59 185 139)(40 72 186 124)(41 57 187 137)(42 70 188 122)(43 83 189 135)(44 68 190 120)(45 81 191 133)(46 66 192 118)(47 79 193 131)(48 64 194 116)(49 77 195 129)(50 62 196 114)(51 75 169 127)(52 60 170 140)(53 73 171 125)(54 58 172 138)(55 71 173 123)(56 84 174 136)
(1 134)(2 119)(3 132)(4 117)(5 130)(6 115)(7 128)(8 113)(9 126)(10 139)(11 124)(12 137)(13 122)(14 135)(15 120)(16 133)(17 118)(18 131)(19 116)(20 129)(21 114)(22 127)(23 140)(24 125)(25 138)(26 123)(27 136)(28 121)(29 220)(30 205)(31 218)(32 203)(33 216)(34 201)(35 214)(36 199)(37 212)(38 197)(39 210)(40 223)(41 208)(42 221)(43 206)(44 219)(45 204)(46 217)(47 202)(48 215)(49 200)(50 213)(51 198)(52 211)(53 224)(54 209)(55 222)(56 207)(57 168)(58 153)(59 166)(60 151)(61 164)(62 149)(63 162)(64 147)(65 160)(66 145)(67 158)(68 143)(69 156)(70 141)(71 154)(72 167)(73 152)(74 165)(75 150)(76 163)(77 148)(78 161)(79 146)(80 159)(81 144)(82 157)(83 142)(84 155)(85 169)(86 182)(87 195)(88 180)(89 193)(90 178)(91 191)(92 176)(93 189)(94 174)(95 187)(96 172)(97 185)(98 170)(99 183)(100 196)(101 181)(102 194)(103 179)(104 192)(105 177)(106 190)(107 175)(108 188)(109 173)(110 186)(111 171)(112 184)
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), (1,149)(2,148)(3,147)(4,146)(5,145)(6,144)(7,143)(8,142)(9,141)(10,168)(11,167)(12,166)(13,165)(14,164)(15,163)(16,162)(17,161)(18,160)(19,159)(20,158)(21,157)(22,156)(23,155)(24,154)(25,153)(26,152)(27,151)(28,150)(29,183)(30,182)(31,181)(32,180)(33,179)(34,178)(35,177)(36,176)(37,175)(38,174)(39,173)(40,172)(41,171)(42,170)(43,169)(44,196)(45,195)(46,194)(47,193)(48,192)(49,191)(50,190)(51,189)(52,188)(53,187)(54,186)(55,185)(56,184)(57,137)(58,136)(59,135)(60,134)(61,133)(62,132)(63,131)(64,130)(65,129)(66,128)(67,127)(68,126)(69,125)(70,124)(71,123)(72,122)(73,121)(74,120)(75,119)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,140)(83,139)(84,138)(85,204)(86,203)(87,202)(88,201)(89,200)(90,199)(91,198)(92,197)(93,224)(94,223)(95,222)(96,221)(97,220)(98,219)(99,218)(100,217)(101,216)(102,215)(103,214)(104,213)(105,212)(106,211)(107,210)(108,209)(109,208)(110,207)(111,206)(112,205), (1,92,157,205)(2,105,158,218)(3,90,159,203)(4,103,160,216)(5,88,161,201)(6,101,162,214)(7,86,163,199)(8,99,164,212)(9,112,165,197)(10,97,166,210)(11,110,167,223)(12,95,168,208)(13,108,141,221)(14,93,142,206)(15,106,143,219)(16,91,144,204)(17,104,145,217)(18,89,146,202)(19,102,147,215)(20,87,148,200)(21,100,149,213)(22,85,150,198)(23,98,151,211)(24,111,152,224)(25,96,153,209)(26,109,154,222)(27,94,155,207)(28,107,156,220)(29,69,175,121)(30,82,176,134)(31,67,177,119)(32,80,178,132)(33,65,179,117)(34,78,180,130)(35,63,181,115)(36,76,182,128)(37,61,183,113)(38,74,184,126)(39,59,185,139)(40,72,186,124)(41,57,187,137)(42,70,188,122)(43,83,189,135)(44,68,190,120)(45,81,191,133)(46,66,192,118)(47,79,193,131)(48,64,194,116)(49,77,195,129)(50,62,196,114)(51,75,169,127)(52,60,170,140)(53,73,171,125)(54,58,172,138)(55,71,173,123)(56,84,174,136), (1,134)(2,119)(3,132)(4,117)(5,130)(6,115)(7,128)(8,113)(9,126)(10,139)(11,124)(12,137)(13,122)(14,135)(15,120)(16,133)(17,118)(18,131)(19,116)(20,129)(21,114)(22,127)(23,140)(24,125)(25,138)(26,123)(27,136)(28,121)(29,220)(30,205)(31,218)(32,203)(33,216)(34,201)(35,214)(36,199)(37,212)(38,197)(39,210)(40,223)(41,208)(42,221)(43,206)(44,219)(45,204)(46,217)(47,202)(48,215)(49,200)(50,213)(51,198)(52,211)(53,224)(54,209)(55,222)(56,207)(57,168)(58,153)(59,166)(60,151)(61,164)(62,149)(63,162)(64,147)(65,160)(66,145)(67,158)(68,143)(69,156)(70,141)(71,154)(72,167)(73,152)(74,165)(75,150)(76,163)(77,148)(78,161)(79,146)(80,159)(81,144)(82,157)(83,142)(84,155)(85,169)(86,182)(87,195)(88,180)(89,193)(90,178)(91,191)(92,176)(93,189)(94,174)(95,187)(96,172)(97,185)(98,170)(99,183)(100,196)(101,181)(102,194)(103,179)(104,192)(105,177)(106,190)(107,175)(108,188)(109,173)(110,186)(111,171)(112,184)>;
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), (1,149)(2,148)(3,147)(4,146)(5,145)(6,144)(7,143)(8,142)(9,141)(10,168)(11,167)(12,166)(13,165)(14,164)(15,163)(16,162)(17,161)(18,160)(19,159)(20,158)(21,157)(22,156)(23,155)(24,154)(25,153)(26,152)(27,151)(28,150)(29,183)(30,182)(31,181)(32,180)(33,179)(34,178)(35,177)(36,176)(37,175)(38,174)(39,173)(40,172)(41,171)(42,170)(43,169)(44,196)(45,195)(46,194)(47,193)(48,192)(49,191)(50,190)(51,189)(52,188)(53,187)(54,186)(55,185)(56,184)(57,137)(58,136)(59,135)(60,134)(61,133)(62,132)(63,131)(64,130)(65,129)(66,128)(67,127)(68,126)(69,125)(70,124)(71,123)(72,122)(73,121)(74,120)(75,119)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,140)(83,139)(84,138)(85,204)(86,203)(87,202)(88,201)(89,200)(90,199)(91,198)(92,197)(93,224)(94,223)(95,222)(96,221)(97,220)(98,219)(99,218)(100,217)(101,216)(102,215)(103,214)(104,213)(105,212)(106,211)(107,210)(108,209)(109,208)(110,207)(111,206)(112,205), (1,92,157,205)(2,105,158,218)(3,90,159,203)(4,103,160,216)(5,88,161,201)(6,101,162,214)(7,86,163,199)(8,99,164,212)(9,112,165,197)(10,97,166,210)(11,110,167,223)(12,95,168,208)(13,108,141,221)(14,93,142,206)(15,106,143,219)(16,91,144,204)(17,104,145,217)(18,89,146,202)(19,102,147,215)(20,87,148,200)(21,100,149,213)(22,85,150,198)(23,98,151,211)(24,111,152,224)(25,96,153,209)(26,109,154,222)(27,94,155,207)(28,107,156,220)(29,69,175,121)(30,82,176,134)(31,67,177,119)(32,80,178,132)(33,65,179,117)(34,78,180,130)(35,63,181,115)(36,76,182,128)(37,61,183,113)(38,74,184,126)(39,59,185,139)(40,72,186,124)(41,57,187,137)(42,70,188,122)(43,83,189,135)(44,68,190,120)(45,81,191,133)(46,66,192,118)(47,79,193,131)(48,64,194,116)(49,77,195,129)(50,62,196,114)(51,75,169,127)(52,60,170,140)(53,73,171,125)(54,58,172,138)(55,71,173,123)(56,84,174,136), (1,134)(2,119)(3,132)(4,117)(5,130)(6,115)(7,128)(8,113)(9,126)(10,139)(11,124)(12,137)(13,122)(14,135)(15,120)(16,133)(17,118)(18,131)(19,116)(20,129)(21,114)(22,127)(23,140)(24,125)(25,138)(26,123)(27,136)(28,121)(29,220)(30,205)(31,218)(32,203)(33,216)(34,201)(35,214)(36,199)(37,212)(38,197)(39,210)(40,223)(41,208)(42,221)(43,206)(44,219)(45,204)(46,217)(47,202)(48,215)(49,200)(50,213)(51,198)(52,211)(53,224)(54,209)(55,222)(56,207)(57,168)(58,153)(59,166)(60,151)(61,164)(62,149)(63,162)(64,147)(65,160)(66,145)(67,158)(68,143)(69,156)(70,141)(71,154)(72,167)(73,152)(74,165)(75,150)(76,163)(77,148)(78,161)(79,146)(80,159)(81,144)(82,157)(83,142)(84,155)(85,169)(86,182)(87,195)(88,180)(89,193)(90,178)(91,191)(92,176)(93,189)(94,174)(95,187)(96,172)(97,185)(98,170)(99,183)(100,196)(101,181)(102,194)(103,179)(104,192)(105,177)(106,190)(107,175)(108,188)(109,173)(110,186)(111,171)(112,184) );
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)], [(1,149),(2,148),(3,147),(4,146),(5,145),(6,144),(7,143),(8,142),(9,141),(10,168),(11,167),(12,166),(13,165),(14,164),(15,163),(16,162),(17,161),(18,160),(19,159),(20,158),(21,157),(22,156),(23,155),(24,154),(25,153),(26,152),(27,151),(28,150),(29,183),(30,182),(31,181),(32,180),(33,179),(34,178),(35,177),(36,176),(37,175),(38,174),(39,173),(40,172),(41,171),(42,170),(43,169),(44,196),(45,195),(46,194),(47,193),(48,192),(49,191),(50,190),(51,189),(52,188),(53,187),(54,186),(55,185),(56,184),(57,137),(58,136),(59,135),(60,134),(61,133),(62,132),(63,131),(64,130),(65,129),(66,128),(67,127),(68,126),(69,125),(70,124),(71,123),(72,122),(73,121),(74,120),(75,119),(76,118),(77,117),(78,116),(79,115),(80,114),(81,113),(82,140),(83,139),(84,138),(85,204),(86,203),(87,202),(88,201),(89,200),(90,199),(91,198),(92,197),(93,224),(94,223),(95,222),(96,221),(97,220),(98,219),(99,218),(100,217),(101,216),(102,215),(103,214),(104,213),(105,212),(106,211),(107,210),(108,209),(109,208),(110,207),(111,206),(112,205)], [(1,92,157,205),(2,105,158,218),(3,90,159,203),(4,103,160,216),(5,88,161,201),(6,101,162,214),(7,86,163,199),(8,99,164,212),(9,112,165,197),(10,97,166,210),(11,110,167,223),(12,95,168,208),(13,108,141,221),(14,93,142,206),(15,106,143,219),(16,91,144,204),(17,104,145,217),(18,89,146,202),(19,102,147,215),(20,87,148,200),(21,100,149,213),(22,85,150,198),(23,98,151,211),(24,111,152,224),(25,96,153,209),(26,109,154,222),(27,94,155,207),(28,107,156,220),(29,69,175,121),(30,82,176,134),(31,67,177,119),(32,80,178,132),(33,65,179,117),(34,78,180,130),(35,63,181,115),(36,76,182,128),(37,61,183,113),(38,74,184,126),(39,59,185,139),(40,72,186,124),(41,57,187,137),(42,70,188,122),(43,83,189,135),(44,68,190,120),(45,81,191,133),(46,66,192,118),(47,79,193,131),(48,64,194,116),(49,77,195,129),(50,62,196,114),(51,75,169,127),(52,60,170,140),(53,73,171,125),(54,58,172,138),(55,71,173,123),(56,84,174,136)], [(1,134),(2,119),(3,132),(4,117),(5,130),(6,115),(7,128),(8,113),(9,126),(10,139),(11,124),(12,137),(13,122),(14,135),(15,120),(16,133),(17,118),(18,131),(19,116),(20,129),(21,114),(22,127),(23,140),(24,125),(25,138),(26,123),(27,136),(28,121),(29,220),(30,205),(31,218),(32,203),(33,216),(34,201),(35,214),(36,199),(37,212),(38,197),(39,210),(40,223),(41,208),(42,221),(43,206),(44,219),(45,204),(46,217),(47,202),(48,215),(49,200),(50,213),(51,198),(52,211),(53,224),(54,209),(55,222),(56,207),(57,168),(58,153),(59,166),(60,151),(61,164),(62,149),(63,162),(64,147),(65,160),(66,145),(67,158),(68,143),(69,156),(70,141),(71,154),(72,167),(73,152),(74,165),(75,150),(76,163),(77,148),(78,161),(79,146),(80,159),(81,144),(82,157),(83,142),(84,155),(85,169),(86,182),(87,195),(88,180),(89,193),(90,178),(91,191),(92,176),(93,189),(94,174),(95,187),(96,172),(97,185),(98,170),(99,183),(100,196),(101,181),(102,194),(103,179),(104,192),(105,177),(106,190),(107,175),(108,188),(109,173),(110,186),(111,171),(112,184)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 3 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 9 |
| 0 | 0 | 0 | 0 | 20 | 15 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 15 |
| 0 | 0 | 0 | 0 | 14 | 9 |
G:=sub<GL(6,GF(29))| [0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,3,0,0,0,0,26,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,8,21,0,0,0,0,26,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,0,14,20,0,0,0,0,9,15],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,0,20,14,0,0,0,0,15,9] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2- (1+4) | D4×D7 | Q8.10D14 | D7×C4○D4 |
| kernel | D28⋊22D4 | D14.D4 | D14⋊D4 | D7×C4⋊C4 | D28⋊C4 | D14.5D4 | D14⋊2Q8 | C4×C7⋊D4 | Dic7⋊Q8 | C7×C22⋊Q8 | C2×C4○D28 | C2×Q8⋊2D7 | D28 | C22⋊Q8 | Dic7 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
D_{28}\rtimes_{22}D_4 % in TeX
G:=Group("D28:22D4"); // GroupNames label
G:=SmallGroup(448,1084);
// by ID
G=gap.SmallGroup(448,1084);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,1571,570,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^13,c*b*c^-1=a^12*b,d*b*d=a^26*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀