metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊7D4, (C7×Q8)⋊6D4, C4.64(D4×D7), D14⋊C8⋊34C2, C28⋊2D4⋊7C2, C7⋊6(D4⋊D4), Q8⋊3(C7⋊D4), C28.49(C2×D4), (C2×SD16)⋊13D7, (C2×D4).74D14, (C2×C8).148D14, C2.D56⋊36C2, C14.59C22≀C2, (C14×SD16)⋊22C2, C14.64(C4○D8), Q8⋊Dic7⋊30C2, (C2×Q8).117D14, (C22×D7).37D4, C22.269(D4×D7), C2.29(D56⋊C2), C14.79(C8⋊C22), (C2×C28).449C23, (C2×C56).295C22, (C2×Dic7).184D4, (D4×C14).98C22, (Q8×C14).78C22, C2.27(C23⋊D14), (C2×D28).121C22, C4⋊Dic7.176C22, C2.30(SD16⋊3D7), (C2×D4⋊D7)⋊20C2, C4.44(C2×C7⋊D4), (C2×Q8⋊2D7)⋊2C2, (C2×C4×D7).49C22, (C2×C14).361(C2×D4), (C2×C7⋊C8).158C22, (C2×C4).538(C22×D7), SmallGroup(448,706)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊7D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=cac-1=a-1, dad=a15, cbc-1=a19b, dbd=a21b, dcd=c-1 >
Subgroups: 964 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×D7, C22×C14, D4⋊D4, C2×C7⋊C8, C4⋊Dic7, D4⋊D7, C23.D7, C2×C56, C7×SD16, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, Q8⋊2D7, C2×C7⋊D4, D4×C14, Q8×C14, D14⋊C8, C2.D56, Q8⋊Dic7, C2×D4⋊D7, C28⋊2D4, C14×SD16, C2×Q8⋊2D7, D28⋊7D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C4○D8, C8⋊C22, C7⋊D4, C22×D7, D4⋊D4, D4×D7, C2×C7⋊D4, D56⋊C2, SD16⋊3D7, C23⋊D14, D28⋊7D4
►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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(55 56)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(85 87)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(113 118)(114 117)(115 116)(119 140)(120 139)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)(169 170)(171 196)(172 195)(173 194)(174 193)(175 192)(176 191)(177 190)(178 189)(179 188)(180 187)(181 186)(182 185)(183 184)(197 221)(198 220)(199 219)(200 218)(201 217)(202 216)(203 215)(204 214)(205 213)(206 212)(207 211)(208 210)(222 224)
(1 199 184 68)(2 198 185 67)(3 197 186 66)(4 224 187 65)(5 223 188 64)(6 222 189 63)(7 221 190 62)(8 220 191 61)(9 219 192 60)(10 218 193 59)(11 217 194 58)(12 216 195 57)(13 215 196 84)(14 214 169 83)(15 213 170 82)(16 212 171 81)(17 211 172 80)(18 210 173 79)(19 209 174 78)(20 208 175 77)(21 207 176 76)(22 206 177 75)(23 205 178 74)(24 204 179 73)(25 203 180 72)(26 202 181 71)(27 201 182 70)(28 200 183 69)(29 150 117 103)(30 149 118 102)(31 148 119 101)(32 147 120 100)(33 146 121 99)(34 145 122 98)(35 144 123 97)(36 143 124 96)(37 142 125 95)(38 141 126 94)(39 168 127 93)(40 167 128 92)(41 166 129 91)(42 165 130 90)(43 164 131 89)(44 163 132 88)(45 162 133 87)(46 161 134 86)(47 160 135 85)(48 159 136 112)(49 158 137 111)(50 157 138 110)(51 156 139 109)(52 155 140 108)(53 154 113 107)(54 153 114 106)(55 152 115 105)(56 151 116 104)
(1 144)(2 159)(3 146)(4 161)(5 148)(6 163)(7 150)(8 165)(9 152)(10 167)(11 154)(12 141)(13 156)(14 143)(15 158)(16 145)(17 160)(18 147)(19 162)(20 149)(21 164)(22 151)(23 166)(24 153)(25 168)(26 155)(27 142)(28 157)(29 221)(30 208)(31 223)(32 210)(33 197)(34 212)(35 199)(36 214)(37 201)(38 216)(39 203)(40 218)(41 205)(42 220)(43 207)(44 222)(45 209)(46 224)(47 211)(48 198)(49 213)(50 200)(51 215)(52 202)(53 217)(54 204)(55 219)(56 206)(57 126)(58 113)(59 128)(60 115)(61 130)(62 117)(63 132)(64 119)(65 134)(66 121)(67 136)(68 123)(69 138)(70 125)(71 140)(72 127)(73 114)(74 129)(75 116)(76 131)(77 118)(78 133)(79 120)(80 135)(81 122)(82 137)(83 124)(84 139)(85 172)(86 187)(87 174)(88 189)(89 176)(90 191)(91 178)(92 193)(93 180)(94 195)(95 182)(96 169)(97 184)(98 171)(99 186)(100 173)(101 188)(102 175)(103 190)(104 177)(105 192)(106 179)(107 194)(108 181)(109 196)(110 183)(111 170)(112 185)
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(55,56)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,87)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(113,118)(114,117)(115,116)(119,140)(120,139)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(169,170)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184)(197,221)(198,220)(199,219)(200,218)(201,217)(202,216)(203,215)(204,214)(205,213)(206,212)(207,211)(208,210)(222,224), (1,199,184,68)(2,198,185,67)(3,197,186,66)(4,224,187,65)(5,223,188,64)(6,222,189,63)(7,221,190,62)(8,220,191,61)(9,219,192,60)(10,218,193,59)(11,217,194,58)(12,216,195,57)(13,215,196,84)(14,214,169,83)(15,213,170,82)(16,212,171,81)(17,211,172,80)(18,210,173,79)(19,209,174,78)(20,208,175,77)(21,207,176,76)(22,206,177,75)(23,205,178,74)(24,204,179,73)(25,203,180,72)(26,202,181,71)(27,201,182,70)(28,200,183,69)(29,150,117,103)(30,149,118,102)(31,148,119,101)(32,147,120,100)(33,146,121,99)(34,145,122,98)(35,144,123,97)(36,143,124,96)(37,142,125,95)(38,141,126,94)(39,168,127,93)(40,167,128,92)(41,166,129,91)(42,165,130,90)(43,164,131,89)(44,163,132,88)(45,162,133,87)(46,161,134,86)(47,160,135,85)(48,159,136,112)(49,158,137,111)(50,157,138,110)(51,156,139,109)(52,155,140,108)(53,154,113,107)(54,153,114,106)(55,152,115,105)(56,151,116,104), (1,144)(2,159)(3,146)(4,161)(5,148)(6,163)(7,150)(8,165)(9,152)(10,167)(11,154)(12,141)(13,156)(14,143)(15,158)(16,145)(17,160)(18,147)(19,162)(20,149)(21,164)(22,151)(23,166)(24,153)(25,168)(26,155)(27,142)(28,157)(29,221)(30,208)(31,223)(32,210)(33,197)(34,212)(35,199)(36,214)(37,201)(38,216)(39,203)(40,218)(41,205)(42,220)(43,207)(44,222)(45,209)(46,224)(47,211)(48,198)(49,213)(50,200)(51,215)(52,202)(53,217)(54,204)(55,219)(56,206)(57,126)(58,113)(59,128)(60,115)(61,130)(62,117)(63,132)(64,119)(65,134)(66,121)(67,136)(68,123)(69,138)(70,125)(71,140)(72,127)(73,114)(74,129)(75,116)(76,131)(77,118)(78,133)(79,120)(80,135)(81,122)(82,137)(83,124)(84,139)(85,172)(86,187)(87,174)(88,189)(89,176)(90,191)(91,178)(92,193)(93,180)(94,195)(95,182)(96,169)(97,184)(98,171)(99,186)(100,173)(101,188)(102,175)(103,190)(104,177)(105,192)(106,179)(107,194)(108,181)(109,196)(110,183)(111,170)(112,185)>;
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(55,56)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,87)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(113,118)(114,117)(115,116)(119,140)(120,139)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(169,170)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184)(197,221)(198,220)(199,219)(200,218)(201,217)(202,216)(203,215)(204,214)(205,213)(206,212)(207,211)(208,210)(222,224), (1,199,184,68)(2,198,185,67)(3,197,186,66)(4,224,187,65)(5,223,188,64)(6,222,189,63)(7,221,190,62)(8,220,191,61)(9,219,192,60)(10,218,193,59)(11,217,194,58)(12,216,195,57)(13,215,196,84)(14,214,169,83)(15,213,170,82)(16,212,171,81)(17,211,172,80)(18,210,173,79)(19,209,174,78)(20,208,175,77)(21,207,176,76)(22,206,177,75)(23,205,178,74)(24,204,179,73)(25,203,180,72)(26,202,181,71)(27,201,182,70)(28,200,183,69)(29,150,117,103)(30,149,118,102)(31,148,119,101)(32,147,120,100)(33,146,121,99)(34,145,122,98)(35,144,123,97)(36,143,124,96)(37,142,125,95)(38,141,126,94)(39,168,127,93)(40,167,128,92)(41,166,129,91)(42,165,130,90)(43,164,131,89)(44,163,132,88)(45,162,133,87)(46,161,134,86)(47,160,135,85)(48,159,136,112)(49,158,137,111)(50,157,138,110)(51,156,139,109)(52,155,140,108)(53,154,113,107)(54,153,114,106)(55,152,115,105)(56,151,116,104), (1,144)(2,159)(3,146)(4,161)(5,148)(6,163)(7,150)(8,165)(9,152)(10,167)(11,154)(12,141)(13,156)(14,143)(15,158)(16,145)(17,160)(18,147)(19,162)(20,149)(21,164)(22,151)(23,166)(24,153)(25,168)(26,155)(27,142)(28,157)(29,221)(30,208)(31,223)(32,210)(33,197)(34,212)(35,199)(36,214)(37,201)(38,216)(39,203)(40,218)(41,205)(42,220)(43,207)(44,222)(45,209)(46,224)(47,211)(48,198)(49,213)(50,200)(51,215)(52,202)(53,217)(54,204)(55,219)(56,206)(57,126)(58,113)(59,128)(60,115)(61,130)(62,117)(63,132)(64,119)(65,134)(66,121)(67,136)(68,123)(69,138)(70,125)(71,140)(72,127)(73,114)(74,129)(75,116)(76,131)(77,118)(78,133)(79,120)(80,135)(81,122)(82,137)(83,124)(84,139)(85,172)(86,187)(87,174)(88,189)(89,176)(90,191)(91,178)(92,193)(93,180)(94,195)(95,182)(96,169)(97,184)(98,171)(99,186)(100,173)(101,188)(102,175)(103,190)(104,177)(105,192)(106,179)(107,194)(108,181)(109,196)(110,183)(111,170)(112,185) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(55,56),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(85,87),(88,112),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101),(113,118),(114,117),(115,116),(119,140),(120,139),(121,138),(122,137),(123,136),(124,135),(125,134),(126,133),(127,132),(128,131),(129,130),(141,153),(142,152),(143,151),(144,150),(145,149),(146,148),(154,168),(155,167),(156,166),(157,165),(158,164),(159,163),(160,162),(169,170),(171,196),(172,195),(173,194),(174,193),(175,192),(176,191),(177,190),(178,189),(179,188),(180,187),(181,186),(182,185),(183,184),(197,221),(198,220),(199,219),(200,218),(201,217),(202,216),(203,215),(204,214),(205,213),(206,212),(207,211),(208,210),(222,224)], [(1,199,184,68),(2,198,185,67),(3,197,186,66),(4,224,187,65),(5,223,188,64),(6,222,189,63),(7,221,190,62),(8,220,191,61),(9,219,192,60),(10,218,193,59),(11,217,194,58),(12,216,195,57),(13,215,196,84),(14,214,169,83),(15,213,170,82),(16,212,171,81),(17,211,172,80),(18,210,173,79),(19,209,174,78),(20,208,175,77),(21,207,176,76),(22,206,177,75),(23,205,178,74),(24,204,179,73),(25,203,180,72),(26,202,181,71),(27,201,182,70),(28,200,183,69),(29,150,117,103),(30,149,118,102),(31,148,119,101),(32,147,120,100),(33,146,121,99),(34,145,122,98),(35,144,123,97),(36,143,124,96),(37,142,125,95),(38,141,126,94),(39,168,127,93),(40,167,128,92),(41,166,129,91),(42,165,130,90),(43,164,131,89),(44,163,132,88),(45,162,133,87),(46,161,134,86),(47,160,135,85),(48,159,136,112),(49,158,137,111),(50,157,138,110),(51,156,139,109),(52,155,140,108),(53,154,113,107),(54,153,114,106),(55,152,115,105),(56,151,116,104)], [(1,144),(2,159),(3,146),(4,161),(5,148),(6,163),(7,150),(8,165),(9,152),(10,167),(11,154),(12,141),(13,156),(14,143),(15,158),(16,145),(17,160),(18,147),(19,162),(20,149),(21,164),(22,151),(23,166),(24,153),(25,168),(26,155),(27,142),(28,157),(29,221),(30,208),(31,223),(32,210),(33,197),(34,212),(35,199),(36,214),(37,201),(38,216),(39,203),(40,218),(41,205),(42,220),(43,207),(44,222),(45,209),(46,224),(47,211),(48,198),(49,213),(50,200),(51,215),(52,202),(53,217),(54,204),(55,219),(56,206),(57,126),(58,113),(59,128),(60,115),(61,130),(62,117),(63,132),(64,119),(65,134),(66,121),(67,136),(68,123),(69,138),(70,125),(71,140),(72,127),(73,114),(74,129),(75,116),(76,131),(77,118),(78,133),(79,120),(80,135),(81,122),(82,137),(83,124),(84,139),(85,172),(86,187),(87,174),(88,189),(89,176),(90,191),(91,178),(92,193),(93,180),(94,195),(95,182),(96,169),(97,184),(98,171),(99,186),(100,173),(101,188),(102,175),(103,190),(104,177),(105,192),(106,179),(107,194),(108,181),(109,196),(110,183),(111,170),(112,185)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C8⋊C22 | D4×D7 | D4×D7 | D56⋊C2 | SD16⋊3D7 |
| kernel | D28⋊7D4 | D14⋊C8 | C2.D56 | Q8⋊Dic7 | C2×D4⋊D7 | C28⋊2D4 | C14×SD16 | C2×Q8⋊2D7 | D28 | C2×Dic7 | C7×Q8 | C22×D7 | C2×SD16 | C2×C8 | C2×D4 | C2×Q8 | C14 | Q8 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of D28⋊7D4 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 34 | 0 | 0 |
| 0 | 0 | 79 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 22 |
| 0 | 0 | 0 | 0 | 41 | 1 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 63 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 79 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 22 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 99 | 0 | 0 | 0 | 0 |
| 41 | 102 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 79 | 0 | 0 |
| 0 | 0 | 25 | 88 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 11 | 99 | 0 | 0 | 0 | 0 |
| 57 | 102 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 109 |
| 0 | 0 | 0 | 0 | 85 | 51 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,79,0,0,0,0,34,25,0,0,0,0,0,0,112,41,0,0,0,0,22,1],[112,63,0,0,0,0,0,1,0,0,0,0,0,0,112,79,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,22,1],[11,41,0,0,0,0,99,102,0,0,0,0,0,0,25,25,0,0,0,0,79,88,0,0,0,0,0,0,0,32,0,0,0,0,60,0],[11,57,0,0,0,0,99,102,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,62,85,0,0,0,0,109,51] >;
D28⋊7D4 in GAP, Magma, Sage, TeX
D_{28}\rtimes_7D_4 % in TeX
G:=Group("D28:7D4"); // GroupNames label
G:=SmallGroup(448,706);
// by ID
G=gap.SmallGroup(448,706);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,758,135,184,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^15,c*b*c^-1=a^19*b,d*b*d=a^21*b,d*c*d=c^-1>;
// generators/relations