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, (C2×C28).783C23, (C4×C28).152C22, D14⋊C4.98C22, C22⋊C4.110D14, (C22×C4).208D14, C23.95(C22×D7), (D4×C14).257C22, (C2×D28).287C22, 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, C2.46(C2×C4○D28), C14.42(C2×C4○D4), (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
Generators and relations for D28⋊24D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a14b, dcd=c-1 >
Subgroups: 1332 in 292 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, D4⋊6D4, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, D4×C14, C28⋊2Q8, C4×D28, D14.D4, D7×C4⋊C4, C28.48D4, C28⋊2D4, D4×C28, C2×C4○D28, D28⋊24D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2- 1+4, C22×D7, D4⋊6D4, C4○D28, D4×D7, C23×D7, C2×C4○D28, C2×D4×D7, D4.10D14, D28⋊24D4
►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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(57 116)(58 115)(59 114)(60 113)(61 140)(62 139)(63 138)(64 137)(65 136)(66 135)(67 134)(68 133)(69 132)(70 131)(71 130)(72 129)(73 128)(74 127)(75 126)(76 125)(77 124)(78 123)(79 122)(80 121)(81 120)(82 119)(83 118)(84 117)(85 145)(86 144)(87 143)(88 142)(89 141)(90 168)(91 167)(92 166)(93 165)(94 164)(95 163)(96 162)(97 161)(98 160)(99 159)(100 158)(101 157)(102 156)(103 155)(104 154)(105 153)(106 152)(107 151)(108 150)(109 149)(110 148)(111 147)(112 146)(169 205)(170 204)(171 203)(172 202)(173 201)(174 200)(175 199)(176 198)(177 197)(178 224)(179 223)(180 222)(181 221)(182 220)(183 219)(184 218)(185 217)(186 216)(187 215)(188 214)(189 213)(190 212)(191 211)(192 210)(193 209)(194 208)(195 207)(196 206)
(1 162 40 90)(2 163 41 91)(3 164 42 92)(4 165 43 93)(5 166 44 94)(6 167 45 95)(7 168 46 96)(8 141 47 97)(9 142 48 98)(10 143 49 99)(11 144 50 100)(12 145 51 101)(13 146 52 102)(14 147 53 103)(15 148 54 104)(16 149 55 105)(17 150 56 106)(18 151 29 107)(19 152 30 108)(20 153 31 109)(21 154 32 110)(22 155 33 111)(23 156 34 112)(24 157 35 85)(25 158 36 86)(26 159 37 87)(27 160 38 88)(28 161 39 89)(57 207 134 185)(58 208 135 186)(59 209 136 187)(60 210 137 188)(61 211 138 189)(62 212 139 190)(63 213 140 191)(64 214 113 192)(65 215 114 193)(66 216 115 194)(67 217 116 195)(68 218 117 196)(69 219 118 169)(70 220 119 170)(71 221 120 171)(72 222 121 172)(73 223 122 173)(74 224 123 174)(75 197 124 175)(76 198 125 176)(77 199 126 177)(78 200 127 178)(79 201 128 179)(80 202 129 180)(81 203 130 181)(82 204 131 182)(83 205 132 183)(84 206 133 184)
(1 216)(2 217)(3 218)(4 219)(5 220)(6 221)(7 222)(8 223)(9 224)(10 197)(11 198)(12 199)(13 200)(14 201)(15 202)(16 203)(17 204)(18 205)(19 206)(20 207)(21 208)(22 209)(23 210)(24 211)(25 212)(26 213)(27 214)(28 215)(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)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 167)(72 168)(73 141)(74 142)(75 143)(76 144)(77 145)(78 146)(79 147)(80 148)(81 149)(82 150)(83 151)(84 152)(85 138)(86 139)(87 140)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 121)(97 122)(98 123)(99 124)(100 125)(101 126)(102 127)(103 128)(104 129)(105 130)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 137)
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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(57,116)(58,115)(59,114)(60,113)(61,140)(62,139)(63,138)(64,137)(65,136)(66,135)(67,134)(68,133)(69,132)(70,131)(71,130)(72,129)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,145)(86,144)(87,143)(88,142)(89,141)(90,168)(91,167)(92,166)(93,165)(94,164)(95,163)(96,162)(97,161)(98,160)(99,159)(100,158)(101,157)(102,156)(103,155)(104,154)(105,153)(106,152)(107,151)(108,150)(109,149)(110,148)(111,147)(112,146)(169,205)(170,204)(171,203)(172,202)(173,201)(174,200)(175,199)(176,198)(177,197)(178,224)(179,223)(180,222)(181,221)(182,220)(183,219)(184,218)(185,217)(186,216)(187,215)(188,214)(189,213)(190,212)(191,211)(192,210)(193,209)(194,208)(195,207)(196,206), (1,162,40,90)(2,163,41,91)(3,164,42,92)(4,165,43,93)(5,166,44,94)(6,167,45,95)(7,168,46,96)(8,141,47,97)(9,142,48,98)(10,143,49,99)(11,144,50,100)(12,145,51,101)(13,146,52,102)(14,147,53,103)(15,148,54,104)(16,149,55,105)(17,150,56,106)(18,151,29,107)(19,152,30,108)(20,153,31,109)(21,154,32,110)(22,155,33,111)(23,156,34,112)(24,157,35,85)(25,158,36,86)(26,159,37,87)(27,160,38,88)(28,161,39,89)(57,207,134,185)(58,208,135,186)(59,209,136,187)(60,210,137,188)(61,211,138,189)(62,212,139,190)(63,213,140,191)(64,214,113,192)(65,215,114,193)(66,216,115,194)(67,217,116,195)(68,218,117,196)(69,219,118,169)(70,220,119,170)(71,221,120,171)(72,222,121,172)(73,223,122,173)(74,224,123,174)(75,197,124,175)(76,198,125,176)(77,199,126,177)(78,200,127,178)(79,201,128,179)(80,202,129,180)(81,203,130,181)(82,204,131,182)(83,205,132,183)(84,206,133,184), (1,216)(2,217)(3,218)(4,219)(5,220)(6,221)(7,222)(8,223)(9,224)(10,197)(11,198)(12,199)(13,200)(14,201)(15,202)(16,203)(17,204)(18,205)(19,206)(20,207)(21,208)(22,209)(23,210)(24,211)(25,212)(26,213)(27,214)(28,215)(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)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,167)(72,168)(73,141)(74,142)(75,143)(76,144)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,138)(86,139)(87,140)(88,113)(89,114)(90,115)(91,116)(92,117)(93,118)(94,119)(95,120)(96,121)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)>;
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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(57,116)(58,115)(59,114)(60,113)(61,140)(62,139)(63,138)(64,137)(65,136)(66,135)(67,134)(68,133)(69,132)(70,131)(71,130)(72,129)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,145)(86,144)(87,143)(88,142)(89,141)(90,168)(91,167)(92,166)(93,165)(94,164)(95,163)(96,162)(97,161)(98,160)(99,159)(100,158)(101,157)(102,156)(103,155)(104,154)(105,153)(106,152)(107,151)(108,150)(109,149)(110,148)(111,147)(112,146)(169,205)(170,204)(171,203)(172,202)(173,201)(174,200)(175,199)(176,198)(177,197)(178,224)(179,223)(180,222)(181,221)(182,220)(183,219)(184,218)(185,217)(186,216)(187,215)(188,214)(189,213)(190,212)(191,211)(192,210)(193,209)(194,208)(195,207)(196,206), (1,162,40,90)(2,163,41,91)(3,164,42,92)(4,165,43,93)(5,166,44,94)(6,167,45,95)(7,168,46,96)(8,141,47,97)(9,142,48,98)(10,143,49,99)(11,144,50,100)(12,145,51,101)(13,146,52,102)(14,147,53,103)(15,148,54,104)(16,149,55,105)(17,150,56,106)(18,151,29,107)(19,152,30,108)(20,153,31,109)(21,154,32,110)(22,155,33,111)(23,156,34,112)(24,157,35,85)(25,158,36,86)(26,159,37,87)(27,160,38,88)(28,161,39,89)(57,207,134,185)(58,208,135,186)(59,209,136,187)(60,210,137,188)(61,211,138,189)(62,212,139,190)(63,213,140,191)(64,214,113,192)(65,215,114,193)(66,216,115,194)(67,217,116,195)(68,218,117,196)(69,219,118,169)(70,220,119,170)(71,221,120,171)(72,222,121,172)(73,223,122,173)(74,224,123,174)(75,197,124,175)(76,198,125,176)(77,199,126,177)(78,200,127,178)(79,201,128,179)(80,202,129,180)(81,203,130,181)(82,204,131,182)(83,205,132,183)(84,206,133,184), (1,216)(2,217)(3,218)(4,219)(5,220)(6,221)(7,222)(8,223)(9,224)(10,197)(11,198)(12,199)(13,200)(14,201)(15,202)(16,203)(17,204)(18,205)(19,206)(20,207)(21,208)(22,209)(23,210)(24,211)(25,212)(26,213)(27,214)(28,215)(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)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,167)(72,168)(73,141)(74,142)(75,143)(76,144)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,138)(86,139)(87,140)(88,113)(89,114)(90,115)(91,116)(92,117)(93,118)(94,119)(95,120)(96,121)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137) );
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)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,56),(20,55),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,48),(28,47),(57,116),(58,115),(59,114),(60,113),(61,140),(62,139),(63,138),(64,137),(65,136),(66,135),(67,134),(68,133),(69,132),(70,131),(71,130),(72,129),(73,128),(74,127),(75,126),(76,125),(77,124),(78,123),(79,122),(80,121),(81,120),(82,119),(83,118),(84,117),(85,145),(86,144),(87,143),(88,142),(89,141),(90,168),(91,167),(92,166),(93,165),(94,164),(95,163),(96,162),(97,161),(98,160),(99,159),(100,158),(101,157),(102,156),(103,155),(104,154),(105,153),(106,152),(107,151),(108,150),(109,149),(110,148),(111,147),(112,146),(169,205),(170,204),(171,203),(172,202),(173,201),(174,200),(175,199),(176,198),(177,197),(178,224),(179,223),(180,222),(181,221),(182,220),(183,219),(184,218),(185,217),(186,216),(187,215),(188,214),(189,213),(190,212),(191,211),(192,210),(193,209),(194,208),(195,207),(196,206)], [(1,162,40,90),(2,163,41,91),(3,164,42,92),(4,165,43,93),(5,166,44,94),(6,167,45,95),(7,168,46,96),(8,141,47,97),(9,142,48,98),(10,143,49,99),(11,144,50,100),(12,145,51,101),(13,146,52,102),(14,147,53,103),(15,148,54,104),(16,149,55,105),(17,150,56,106),(18,151,29,107),(19,152,30,108),(20,153,31,109),(21,154,32,110),(22,155,33,111),(23,156,34,112),(24,157,35,85),(25,158,36,86),(26,159,37,87),(27,160,38,88),(28,161,39,89),(57,207,134,185),(58,208,135,186),(59,209,136,187),(60,210,137,188),(61,211,138,189),(62,212,139,190),(63,213,140,191),(64,214,113,192),(65,215,114,193),(66,216,115,194),(67,217,116,195),(68,218,117,196),(69,219,118,169),(70,220,119,170),(71,221,120,171),(72,222,121,172),(73,223,122,173),(74,224,123,174),(75,197,124,175),(76,198,125,176),(77,199,126,177),(78,200,127,178),(79,201,128,179),(80,202,129,180),(81,203,130,181),(82,204,131,182),(83,205,132,183),(84,206,133,184)], [(1,216),(2,217),(3,218),(4,219),(5,220),(6,221),(7,222),(8,223),(9,224),(10,197),(11,198),(12,199),(13,200),(14,201),(15,202),(16,203),(17,204),(18,205),(19,206),(20,207),(21,208),(22,209),(23,210),(24,211),(25,212),(26,213),(27,214),(28,215),(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),(57,153),(58,154),(59,155),(60,156),(61,157),(62,158),(63,159),(64,160),(65,161),(66,162),(67,163),(68,164),(69,165),(70,166),(71,167),(72,168),(73,141),(74,142),(75,143),(76,144),(77,145),(78,146),(79,147),(80,148),(81,149),(82,150),(83,151),(84,152),(85,138),(86,139),(87,140),(88,113),(89,114),(90,115),(91,116),(92,117),(93,118),(94,119),(95,120),(96,121),(97,122),(98,123),(99,124),(100,125),(101,126),(102,127),(103,128),(104,129),(105,130),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,137)]])
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 |
Matrix representation of D28⋊24D4 ►in 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] >;
D28⋊24D4 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