metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊24D4, C42.109D14, C14.592- (1+4), (C4×D4)⋊13D7, (C4×D28)⋊29C2, (D4×C28)⋊15C2, C28⋊2(C4○D4), C4⋊3(C4○D28), C28⋊2D4⋊8C2, C7⋊2(D4⋊6D4), C4.140(D4×D7), C4⋊C4.316D14, C28⋊2Q8⋊24C2, D14.15(C2×D4), C28.346(C2×D4), (C2×D4).214D14, D14.D4⋊6C2, (C2×C14).95C24, C14.50(C22×D4), C28.48D4⋊20C2, (C4×C28).152C22, (C2×C28).783C23, D14⋊C4.98C22, C22⋊C4.110D14, (C22×C4).208D14, C23.95(C22×D7), (C2×D28).287C22, (D4×C14).257C22, Dic7⋊C4.65C22, C4⋊Dic7.199C22, (C2×Dic7).41C23, C22.120(C23×D7), C23.D7.12C22, (C22×C14).165C23, (C22×C28).107C22, (C22×D7).173C23, C2.16(D4.10D14), (C2×Dic14).239C22, C2.23(C2×D4×D7), (D7×C4⋊C4)⋊15C2, (C2×C4○D28)⋊8C2, C14.42(C2×C4○D4), C2.46(C2×C4○D28), (C2×C4×D7).64C22, (C7×C4⋊C4).326C22, (C2×C4).579(C22×D7), (C2×C7⋊D4).113C22, (C7×C22⋊C4).122C22, SmallGroup(448,1004)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1332 in 292 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×14], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×14], Q8 [×4], C23 [×2], C23 [×2], D7 [×4], C14 [×3], C14 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8 [×2], C4○D4 [×8], Dic7 [×6], C28 [×4], C28 [×3], D14 [×4], D14 [×4], C2×C14, C2×C14 [×6], C2×C4⋊C4 [×2], C4×D4, C4×D4, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic14 [×4], C4×D7 [×12], D28 [×4], C2×Dic7 [×6], C7⋊D4 [×8], C2×C28 [×3], C2×C28 [×2], C2×C28 [×4], C7×D4 [×2], C22×D7 [×2], C22×C14 [×2], D4⋊6D4, Dic7⋊C4 [×4], C4⋊Dic7, C4⋊Dic7 [×4], D14⋊C4 [×2], C23.D7 [×4], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7 [×6], C2×D28, C4○D28 [×8], C2×C7⋊D4 [×4], C22×C28 [×2], D4×C14, C28⋊2Q8, C4×D28, D14.D4 [×4], D7×C4⋊C4 [×2], C28.48D4 [×2], C28⋊2D4 [×2], D4×C28, C2×C4○D28 [×2], D28⋊24D4
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, C4○D28 [×2], D4×D7 [×2], C23×D7, C2×C4○D28, C2×D4×D7, D4.10D14, D28⋊24D4
Generators and relations
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a14b, 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)
(1 221)(2 220)(3 219)(4 218)(5 217)(6 216)(7 215)(8 214)(9 213)(10 212)(11 211)(12 210)(13 209)(14 208)(15 207)(16 206)(17 205)(18 204)(19 203)(20 202)(21 201)(22 200)(23 199)(24 198)(25 197)(26 224)(27 223)(28 222)(29 149)(30 148)(31 147)(32 146)(33 145)(34 144)(35 143)(36 142)(37 141)(38 168)(39 167)(40 166)(41 165)(42 164)(43 163)(44 162)(45 161)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 187)(58 186)(59 185)(60 184)(61 183)(62 182)(63 181)(64 180)(65 179)(66 178)(67 177)(68 176)(69 175)(70 174)(71 173)(72 172)(73 171)(74 170)(75 169)(76 196)(77 195)(78 194)(79 193)(80 192)(81 191)(82 190)(83 189)(84 188)(85 133)(86 132)(87 131)(88 130)(89 129)(90 128)(91 127)(92 126)(93 125)(94 124)(95 123)(96 122)(97 121)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)(106 140)(107 139)(108 138)(109 137)(110 136)(111 135)(112 134)
(1 144 215 56)(2 145 216 29)(3 146 217 30)(4 147 218 31)(5 148 219 32)(6 149 220 33)(7 150 221 34)(8 151 222 35)(9 152 223 36)(10 153 224 37)(11 154 197 38)(12 155 198 39)(13 156 199 40)(14 157 200 41)(15 158 201 42)(16 159 202 43)(17 160 203 44)(18 161 204 45)(19 162 205 46)(20 163 206 47)(21 164 207 48)(22 165 208 49)(23 166 209 50)(24 167 210 51)(25 168 211 52)(26 141 212 53)(27 142 213 54)(28 143 214 55)(57 134 177 102)(58 135 178 103)(59 136 179 104)(60 137 180 105)(61 138 181 106)(62 139 182 107)(63 140 183 108)(64 113 184 109)(65 114 185 110)(66 115 186 111)(67 116 187 112)(68 117 188 85)(69 118 189 86)(70 119 190 87)(71 120 191 88)(72 121 192 89)(73 122 193 90)(74 123 194 91)(75 124 195 92)(76 125 196 93)(77 126 169 94)(78 127 170 95)(79 128 171 96)(80 129 172 97)(81 130 173 98)(82 131 174 99)(83 132 175 100)(84 133 176 101)
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 136)(9 137)(10 138)(11 139)(12 140)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 127)(28 128)(29 173)(30 174)(31 175)(32 176)(33 177)(34 178)(35 179)(36 180)(37 181)(38 182)(39 183)(40 184)(41 185)(42 186)(43 187)(44 188)(45 189)(46 190)(47 191)(48 192)(49 193)(50 194)(51 195)(52 196)(53 169)(54 170)(55 171)(56 172)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 141)(78 142)(79 143)(80 144)(81 145)(82 146)(83 147)(84 148)(85 203)(86 204)(87 205)(88 206)(89 207)(90 208)(91 209)(92 210)(93 211)(94 212)(95 213)(96 214)(97 215)(98 216)(99 217)(100 218)(101 219)(102 220)(103 221)(104 222)(105 223)(106 224)(107 197)(108 198)(109 199)(110 200)(111 201)(112 202)
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), (1,221)(2,220)(3,219)(4,218)(5,217)(6,216)(7,215)(8,214)(9,213)(10,212)(11,211)(12,210)(13,209)(14,208)(15,207)(16,206)(17,205)(18,204)(19,203)(20,202)(21,201)(22,200)(23,199)(24,198)(25,197)(26,224)(27,223)(28,222)(29,149)(30,148)(31,147)(32,146)(33,145)(34,144)(35,143)(36,142)(37,141)(38,168)(39,167)(40,166)(41,165)(42,164)(43,163)(44,162)(45,161)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,187)(58,186)(59,185)(60,184)(61,183)(62,182)(63,181)(64,180)(65,179)(66,178)(67,177)(68,176)(69,175)(70,174)(71,173)(72,172)(73,171)(74,170)(75,169)(76,196)(77,195)(78,194)(79,193)(80,192)(81,191)(82,190)(83,189)(84,188)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(106,140)(107,139)(108,138)(109,137)(110,136)(111,135)(112,134), (1,144,215,56)(2,145,216,29)(3,146,217,30)(4,147,218,31)(5,148,219,32)(6,149,220,33)(7,150,221,34)(8,151,222,35)(9,152,223,36)(10,153,224,37)(11,154,197,38)(12,155,198,39)(13,156,199,40)(14,157,200,41)(15,158,201,42)(16,159,202,43)(17,160,203,44)(18,161,204,45)(19,162,205,46)(20,163,206,47)(21,164,207,48)(22,165,208,49)(23,166,209,50)(24,167,210,51)(25,168,211,52)(26,141,212,53)(27,142,213,54)(28,143,214,55)(57,134,177,102)(58,135,178,103)(59,136,179,104)(60,137,180,105)(61,138,181,106)(62,139,182,107)(63,140,183,108)(64,113,184,109)(65,114,185,110)(66,115,186,111)(67,116,187,112)(68,117,188,85)(69,118,189,86)(70,119,190,87)(71,120,191,88)(72,121,192,89)(73,122,193,90)(74,123,194,91)(75,124,195,92)(76,125,196,93)(77,126,169,94)(78,127,170,95)(79,128,171,96)(80,129,172,97)(81,130,173,98)(82,131,174,99)(83,132,175,100)(84,133,176,101), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,181)(38,182)(39,183)(40,184)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,169)(54,170)(55,171)(56,172)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,141)(78,142)(79,143)(80,144)(81,145)(82,146)(83,147)(84,148)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,209)(92,210)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(99,217)(100,218)(101,219)(102,220)(103,221)(104,222)(105,223)(106,224)(107,197)(108,198)(109,199)(110,200)(111,201)(112,202)>;
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), (1,221)(2,220)(3,219)(4,218)(5,217)(6,216)(7,215)(8,214)(9,213)(10,212)(11,211)(12,210)(13,209)(14,208)(15,207)(16,206)(17,205)(18,204)(19,203)(20,202)(21,201)(22,200)(23,199)(24,198)(25,197)(26,224)(27,223)(28,222)(29,149)(30,148)(31,147)(32,146)(33,145)(34,144)(35,143)(36,142)(37,141)(38,168)(39,167)(40,166)(41,165)(42,164)(43,163)(44,162)(45,161)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,187)(58,186)(59,185)(60,184)(61,183)(62,182)(63,181)(64,180)(65,179)(66,178)(67,177)(68,176)(69,175)(70,174)(71,173)(72,172)(73,171)(74,170)(75,169)(76,196)(77,195)(78,194)(79,193)(80,192)(81,191)(82,190)(83,189)(84,188)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(106,140)(107,139)(108,138)(109,137)(110,136)(111,135)(112,134), (1,144,215,56)(2,145,216,29)(3,146,217,30)(4,147,218,31)(5,148,219,32)(6,149,220,33)(7,150,221,34)(8,151,222,35)(9,152,223,36)(10,153,224,37)(11,154,197,38)(12,155,198,39)(13,156,199,40)(14,157,200,41)(15,158,201,42)(16,159,202,43)(17,160,203,44)(18,161,204,45)(19,162,205,46)(20,163,206,47)(21,164,207,48)(22,165,208,49)(23,166,209,50)(24,167,210,51)(25,168,211,52)(26,141,212,53)(27,142,213,54)(28,143,214,55)(57,134,177,102)(58,135,178,103)(59,136,179,104)(60,137,180,105)(61,138,181,106)(62,139,182,107)(63,140,183,108)(64,113,184,109)(65,114,185,110)(66,115,186,111)(67,116,187,112)(68,117,188,85)(69,118,189,86)(70,119,190,87)(71,120,191,88)(72,121,192,89)(73,122,193,90)(74,123,194,91)(75,124,195,92)(76,125,196,93)(77,126,169,94)(78,127,170,95)(79,128,171,96)(80,129,172,97)(81,130,173,98)(82,131,174,99)(83,132,175,100)(84,133,176,101), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,181)(38,182)(39,183)(40,184)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,169)(54,170)(55,171)(56,172)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,141)(78,142)(79,143)(80,144)(81,145)(82,146)(83,147)(84,148)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,209)(92,210)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(99,217)(100,218)(101,219)(102,220)(103,221)(104,222)(105,223)(106,224)(107,197)(108,198)(109,199)(110,200)(111,201)(112,202) );
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)], [(1,221),(2,220),(3,219),(4,218),(5,217),(6,216),(7,215),(8,214),(9,213),(10,212),(11,211),(12,210),(13,209),(14,208),(15,207),(16,206),(17,205),(18,204),(19,203),(20,202),(21,201),(22,200),(23,199),(24,198),(25,197),(26,224),(27,223),(28,222),(29,149),(30,148),(31,147),(32,146),(33,145),(34,144),(35,143),(36,142),(37,141),(38,168),(39,167),(40,166),(41,165),(42,164),(43,163),(44,162),(45,161),(46,160),(47,159),(48,158),(49,157),(50,156),(51,155),(52,154),(53,153),(54,152),(55,151),(56,150),(57,187),(58,186),(59,185),(60,184),(61,183),(62,182),(63,181),(64,180),(65,179),(66,178),(67,177),(68,176),(69,175),(70,174),(71,173),(72,172),(73,171),(74,170),(75,169),(76,196),(77,195),(78,194),(79,193),(80,192),(81,191),(82,190),(83,189),(84,188),(85,133),(86,132),(87,131),(88,130),(89,129),(90,128),(91,127),(92,126),(93,125),(94,124),(95,123),(96,122),(97,121),(98,120),(99,119),(100,118),(101,117),(102,116),(103,115),(104,114),(105,113),(106,140),(107,139),(108,138),(109,137),(110,136),(111,135),(112,134)], [(1,144,215,56),(2,145,216,29),(3,146,217,30),(4,147,218,31),(5,148,219,32),(6,149,220,33),(7,150,221,34),(8,151,222,35),(9,152,223,36),(10,153,224,37),(11,154,197,38),(12,155,198,39),(13,156,199,40),(14,157,200,41),(15,158,201,42),(16,159,202,43),(17,160,203,44),(18,161,204,45),(19,162,205,46),(20,163,206,47),(21,164,207,48),(22,165,208,49),(23,166,209,50),(24,167,210,51),(25,168,211,52),(26,141,212,53),(27,142,213,54),(28,143,214,55),(57,134,177,102),(58,135,178,103),(59,136,179,104),(60,137,180,105),(61,138,181,106),(62,139,182,107),(63,140,183,108),(64,113,184,109),(65,114,185,110),(66,115,186,111),(67,116,187,112),(68,117,188,85),(69,118,189,86),(70,119,190,87),(71,120,191,88),(72,121,192,89),(73,122,193,90),(74,123,194,91),(75,124,195,92),(76,125,196,93),(77,126,169,94),(78,127,170,95),(79,128,171,96),(80,129,172,97),(81,130,173,98),(82,131,174,99),(83,132,175,100),(84,133,176,101)], [(1,129),(2,130),(3,131),(4,132),(5,133),(6,134),(7,135),(8,136),(9,137),(10,138),(11,139),(12,140),(13,113),(14,114),(15,115),(16,116),(17,117),(18,118),(19,119),(20,120),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(27,127),(28,128),(29,173),(30,174),(31,175),(32,176),(33,177),(34,178),(35,179),(36,180),(37,181),(38,182),(39,183),(40,184),(41,185),(42,186),(43,187),(44,188),(45,189),(46,190),(47,191),(48,192),(49,193),(50,194),(51,195),(52,196),(53,169),(54,170),(55,171),(56,172),(57,149),(58,150),(59,151),(60,152),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,161),(70,162),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,141),(78,142),(79,143),(80,144),(81,145),(82,146),(83,147),(84,148),(85,203),(86,204),(87,205),(88,206),(89,207),(90,208),(91,209),(92,210),(93,211),(94,212),(95,213),(96,214),(97,215),(98,216),(99,217),(100,218),(101,219),(102,220),(103,221),(104,222),(105,223),(106,224),(107,197),(108,198),(109,199),(110,200),(111,201),(112,202)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 14 | 2 | 0 | 0 | 0 | 0 |
| 3 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 11 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 14 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 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 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 0 | 0 | 0 | 0 | 5 | 28 |
| 23 | 24 | 0 | 0 | 0 | 0 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 27 |
| 0 | 0 | 0 | 0 | 3 | 6 |
G:=sub<GL(6,GF(29))| [14,3,0,0,0,0,2,15,0,0,0,0,0,0,1,11,0,0,0,0,18,25,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,14,0,0,0,0,0,1,0,0,0,0,0,0,28,18,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,5,0,0,0,0,17,28],[23,7,0,0,0,0,24,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,3,0,0,0,0,27,6] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | 4J | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4○D28 | 2- (1+4) | D4×D7 | D4.10D14 |
| kernel | D28⋊24D4 | C28⋊2Q8 | C4×D28 | D14.D4 | D7×C4⋊C4 | C28.48D4 | C28⋊2D4 | D4×C28 | C2×C4○D28 | D28 | C4×D4 | C28 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 4 | 3 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
D_{28}\rtimes_{24}D_4 % in TeX
G:=Group("D28:24D4"); // GroupNames label
G:=SmallGroup(448,1004);
// by ID
G=gap.SmallGroup(448,1004);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,100,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^14*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
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