metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.7D28, Dic14.9D4, D14⋊C8⋊6C2, (C7×D4).2D4, C4.87(D4×D7), C4.4(C2×D28), C28.3(C2×D4), D4⋊C4⋊6D7, C4⋊C4.14D14, (C2×C8).10D14, D14⋊2Q8⋊4C2, (C2×Dic28)⋊5C2, C14.Q16⋊7C2, C7⋊2(D4.7D4), C14.22C22≀C2, (C2×D4).137D14, C14.24(C4○D8), (C2×C56).10C22, (C22×D7).15D4, C22.181(D4×D7), C2.10(D8⋊3D7), (C2×C28).223C23, (C2×Dic7).143D4, (D4×C14).44C22, C2.25(C22⋊D28), C2.13(SD16⋊D7), C14.31(C8.C22), (C2×Dic14).60C22, (C2×D4.D7)⋊6C2, (C7×D4⋊C4)⋊6C2, (C2×C7⋊C8).21C22, (C2×C4×D7).15C22, (C2×D4⋊2D7).5C2, (C2×C14).236(C2×D4), (C7×C4⋊C4).24C22, (C2×C4).330(C22×D7), SmallGroup(448,317)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D28
G = < a,b,c,d | a4=b2=c28=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 820 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, D4.7D4, Dic28, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, D4.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, C14.Q16, D14⋊C8, C7×D4⋊C4, D14⋊2Q8, C2×Dic28, C2×D4.D7, C2×D4⋊2D7, D4.D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C4○D8, C8.C22, D28, C22×D7, D4.7D4, C2×D28, D4×D7, C22⋊D28, D8⋊3D7, SD16⋊D7, D4.D28
►On 224 points
Generators in S224
(1 50 151 126)(2 127 152 51)(3 52 153 128)(4 129 154 53)(5 54 155 130)(6 131 156 55)(7 56 157 132)(8 133 158 29)(9 30 159 134)(10 135 160 31)(11 32 161 136)(12 137 162 33)(13 34 163 138)(14 139 164 35)(15 36 165 140)(16 113 166 37)(17 38 167 114)(18 115 168 39)(19 40 141 116)(20 117 142 41)(21 42 143 118)(22 119 144 43)(23 44 145 120)(24 121 146 45)(25 46 147 122)(26 123 148 47)(27 48 149 124)(28 125 150 49)(57 87 215 193)(58 194 216 88)(59 89 217 195)(60 196 218 90)(61 91 219 169)(62 170 220 92)(63 93 221 171)(64 172 222 94)(65 95 223 173)(66 174 224 96)(67 97 197 175)(68 176 198 98)(69 99 199 177)(70 178 200 100)(71 101 201 179)(72 180 202 102)(73 103 203 181)(74 182 204 104)(75 105 205 183)(76 184 206 106)(77 107 207 185)(78 186 208 108)(79 109 209 187)(80 188 210 110)(81 111 211 189)(82 190 212 112)(83 85 213 191)(84 192 214 86)
(1 212)(2 85)(3 214)(4 87)(5 216)(6 89)(7 218)(8 91)(9 220)(10 93)(11 222)(12 95)(13 224)(14 97)(15 198)(16 99)(17 200)(18 101)(19 202)(20 103)(21 204)(22 105)(23 206)(24 107)(25 208)(26 109)(27 210)(28 111)(29 219)(30 170)(31 221)(32 172)(33 223)(34 174)(35 197)(36 176)(37 199)(38 178)(39 201)(40 180)(41 203)(42 182)(43 205)(44 184)(45 207)(46 186)(47 209)(48 188)(49 211)(50 190)(51 213)(52 192)(53 215)(54 194)(55 217)(56 196)(57 129)(58 155)(59 131)(60 157)(61 133)(62 159)(63 135)(64 161)(65 137)(66 163)(67 139)(68 165)(69 113)(70 167)(71 115)(72 141)(73 117)(74 143)(75 119)(76 145)(77 121)(78 147)(79 123)(80 149)(81 125)(82 151)(83 127)(84 153)(86 128)(88 130)(90 132)(92 134)(94 136)(96 138)(98 140)(100 114)(102 116)(104 118)(106 120)(108 122)(110 124)(112 126)(142 181)(144 183)(146 185)(148 187)(150 189)(152 191)(154 193)(156 195)(158 169)(160 171)(162 173)(164 175)(166 177)(168 179)
(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)
(1 150 151 28)(2 27 152 149)(3 148 153 26)(4 25 154 147)(5 146 155 24)(6 23 156 145)(7 144 157 22)(8 21 158 143)(9 142 159 20)(10 19 160 141)(11 168 161 18)(12 17 162 167)(13 166 163 16)(14 15 164 165)(29 42 133 118)(30 117 134 41)(31 40 135 116)(32 115 136 39)(33 38 137 114)(34 113 138 37)(35 36 139 140)(43 56 119 132)(44 131 120 55)(45 54 121 130)(46 129 122 53)(47 52 123 128)(48 127 124 51)(49 50 125 126)(57 208 215 78)(58 77 216 207)(59 206 217 76)(60 75 218 205)(61 204 219 74)(62 73 220 203)(63 202 221 72)(64 71 222 201)(65 200 223 70)(66 69 224 199)(67 198 197 68)(79 214 209 84)(80 83 210 213)(81 212 211 82)(85 188 191 110)(86 109 192 187)(87 186 193 108)(88 107 194 185)(89 184 195 106)(90 105 196 183)(91 182 169 104)(92 103 170 181)(93 180 171 102)(94 101 172 179)(95 178 173 100)(96 99 174 177)(97 176 175 98)(111 190 189 112)
G:=sub<Sym(224)| (1,50,151,126)(2,127,152,51)(3,52,153,128)(4,129,154,53)(5,54,155,130)(6,131,156,55)(7,56,157,132)(8,133,158,29)(9,30,159,134)(10,135,160,31)(11,32,161,136)(12,137,162,33)(13,34,163,138)(14,139,164,35)(15,36,165,140)(16,113,166,37)(17,38,167,114)(18,115,168,39)(19,40,141,116)(20,117,142,41)(21,42,143,118)(22,119,144,43)(23,44,145,120)(24,121,146,45)(25,46,147,122)(26,123,148,47)(27,48,149,124)(28,125,150,49)(57,87,215,193)(58,194,216,88)(59,89,217,195)(60,196,218,90)(61,91,219,169)(62,170,220,92)(63,93,221,171)(64,172,222,94)(65,95,223,173)(66,174,224,96)(67,97,197,175)(68,176,198,98)(69,99,199,177)(70,178,200,100)(71,101,201,179)(72,180,202,102)(73,103,203,181)(74,182,204,104)(75,105,205,183)(76,184,206,106)(77,107,207,185)(78,186,208,108)(79,109,209,187)(80,188,210,110)(81,111,211,189)(82,190,212,112)(83,85,213,191)(84,192,214,86), (1,212)(2,85)(3,214)(4,87)(5,216)(6,89)(7,218)(8,91)(9,220)(10,93)(11,222)(12,95)(13,224)(14,97)(15,198)(16,99)(17,200)(18,101)(19,202)(20,103)(21,204)(22,105)(23,206)(24,107)(25,208)(26,109)(27,210)(28,111)(29,219)(30,170)(31,221)(32,172)(33,223)(34,174)(35,197)(36,176)(37,199)(38,178)(39,201)(40,180)(41,203)(42,182)(43,205)(44,184)(45,207)(46,186)(47,209)(48,188)(49,211)(50,190)(51,213)(52,192)(53,215)(54,194)(55,217)(56,196)(57,129)(58,155)(59,131)(60,157)(61,133)(62,159)(63,135)(64,161)(65,137)(66,163)(67,139)(68,165)(69,113)(70,167)(71,115)(72,141)(73,117)(74,143)(75,119)(76,145)(77,121)(78,147)(79,123)(80,149)(81,125)(82,151)(83,127)(84,153)(86,128)(88,130)(90,132)(92,134)(94,136)(96,138)(98,140)(100,114)(102,116)(104,118)(106,120)(108,122)(110,124)(112,126)(142,181)(144,183)(146,185)(148,187)(150,189)(152,191)(154,193)(156,195)(158,169)(160,171)(162,173)(164,175)(166,177)(168,179), (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), (1,150,151,28)(2,27,152,149)(3,148,153,26)(4,25,154,147)(5,146,155,24)(6,23,156,145)(7,144,157,22)(8,21,158,143)(9,142,159,20)(10,19,160,141)(11,168,161,18)(12,17,162,167)(13,166,163,16)(14,15,164,165)(29,42,133,118)(30,117,134,41)(31,40,135,116)(32,115,136,39)(33,38,137,114)(34,113,138,37)(35,36,139,140)(43,56,119,132)(44,131,120,55)(45,54,121,130)(46,129,122,53)(47,52,123,128)(48,127,124,51)(49,50,125,126)(57,208,215,78)(58,77,216,207)(59,206,217,76)(60,75,218,205)(61,204,219,74)(62,73,220,203)(63,202,221,72)(64,71,222,201)(65,200,223,70)(66,69,224,199)(67,198,197,68)(79,214,209,84)(80,83,210,213)(81,212,211,82)(85,188,191,110)(86,109,192,187)(87,186,193,108)(88,107,194,185)(89,184,195,106)(90,105,196,183)(91,182,169,104)(92,103,170,181)(93,180,171,102)(94,101,172,179)(95,178,173,100)(96,99,174,177)(97,176,175,98)(111,190,189,112)>;
G:=Group( (1,50,151,126)(2,127,152,51)(3,52,153,128)(4,129,154,53)(5,54,155,130)(6,131,156,55)(7,56,157,132)(8,133,158,29)(9,30,159,134)(10,135,160,31)(11,32,161,136)(12,137,162,33)(13,34,163,138)(14,139,164,35)(15,36,165,140)(16,113,166,37)(17,38,167,114)(18,115,168,39)(19,40,141,116)(20,117,142,41)(21,42,143,118)(22,119,144,43)(23,44,145,120)(24,121,146,45)(25,46,147,122)(26,123,148,47)(27,48,149,124)(28,125,150,49)(57,87,215,193)(58,194,216,88)(59,89,217,195)(60,196,218,90)(61,91,219,169)(62,170,220,92)(63,93,221,171)(64,172,222,94)(65,95,223,173)(66,174,224,96)(67,97,197,175)(68,176,198,98)(69,99,199,177)(70,178,200,100)(71,101,201,179)(72,180,202,102)(73,103,203,181)(74,182,204,104)(75,105,205,183)(76,184,206,106)(77,107,207,185)(78,186,208,108)(79,109,209,187)(80,188,210,110)(81,111,211,189)(82,190,212,112)(83,85,213,191)(84,192,214,86), (1,212)(2,85)(3,214)(4,87)(5,216)(6,89)(7,218)(8,91)(9,220)(10,93)(11,222)(12,95)(13,224)(14,97)(15,198)(16,99)(17,200)(18,101)(19,202)(20,103)(21,204)(22,105)(23,206)(24,107)(25,208)(26,109)(27,210)(28,111)(29,219)(30,170)(31,221)(32,172)(33,223)(34,174)(35,197)(36,176)(37,199)(38,178)(39,201)(40,180)(41,203)(42,182)(43,205)(44,184)(45,207)(46,186)(47,209)(48,188)(49,211)(50,190)(51,213)(52,192)(53,215)(54,194)(55,217)(56,196)(57,129)(58,155)(59,131)(60,157)(61,133)(62,159)(63,135)(64,161)(65,137)(66,163)(67,139)(68,165)(69,113)(70,167)(71,115)(72,141)(73,117)(74,143)(75,119)(76,145)(77,121)(78,147)(79,123)(80,149)(81,125)(82,151)(83,127)(84,153)(86,128)(88,130)(90,132)(92,134)(94,136)(96,138)(98,140)(100,114)(102,116)(104,118)(106,120)(108,122)(110,124)(112,126)(142,181)(144,183)(146,185)(148,187)(150,189)(152,191)(154,193)(156,195)(158,169)(160,171)(162,173)(164,175)(166,177)(168,179), (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), (1,150,151,28)(2,27,152,149)(3,148,153,26)(4,25,154,147)(5,146,155,24)(6,23,156,145)(7,144,157,22)(8,21,158,143)(9,142,159,20)(10,19,160,141)(11,168,161,18)(12,17,162,167)(13,166,163,16)(14,15,164,165)(29,42,133,118)(30,117,134,41)(31,40,135,116)(32,115,136,39)(33,38,137,114)(34,113,138,37)(35,36,139,140)(43,56,119,132)(44,131,120,55)(45,54,121,130)(46,129,122,53)(47,52,123,128)(48,127,124,51)(49,50,125,126)(57,208,215,78)(58,77,216,207)(59,206,217,76)(60,75,218,205)(61,204,219,74)(62,73,220,203)(63,202,221,72)(64,71,222,201)(65,200,223,70)(66,69,224,199)(67,198,197,68)(79,214,209,84)(80,83,210,213)(81,212,211,82)(85,188,191,110)(86,109,192,187)(87,186,193,108)(88,107,194,185)(89,184,195,106)(90,105,196,183)(91,182,169,104)(92,103,170,181)(93,180,171,102)(94,101,172,179)(95,178,173,100)(96,99,174,177)(97,176,175,98)(111,190,189,112) );
G=PermutationGroup([[(1,50,151,126),(2,127,152,51),(3,52,153,128),(4,129,154,53),(5,54,155,130),(6,131,156,55),(7,56,157,132),(8,133,158,29),(9,30,159,134),(10,135,160,31),(11,32,161,136),(12,137,162,33),(13,34,163,138),(14,139,164,35),(15,36,165,140),(16,113,166,37),(17,38,167,114),(18,115,168,39),(19,40,141,116),(20,117,142,41),(21,42,143,118),(22,119,144,43),(23,44,145,120),(24,121,146,45),(25,46,147,122),(26,123,148,47),(27,48,149,124),(28,125,150,49),(57,87,215,193),(58,194,216,88),(59,89,217,195),(60,196,218,90),(61,91,219,169),(62,170,220,92),(63,93,221,171),(64,172,222,94),(65,95,223,173),(66,174,224,96),(67,97,197,175),(68,176,198,98),(69,99,199,177),(70,178,200,100),(71,101,201,179),(72,180,202,102),(73,103,203,181),(74,182,204,104),(75,105,205,183),(76,184,206,106),(77,107,207,185),(78,186,208,108),(79,109,209,187),(80,188,210,110),(81,111,211,189),(82,190,212,112),(83,85,213,191),(84,192,214,86)], [(1,212),(2,85),(3,214),(4,87),(5,216),(6,89),(7,218),(8,91),(9,220),(10,93),(11,222),(12,95),(13,224),(14,97),(15,198),(16,99),(17,200),(18,101),(19,202),(20,103),(21,204),(22,105),(23,206),(24,107),(25,208),(26,109),(27,210),(28,111),(29,219),(30,170),(31,221),(32,172),(33,223),(34,174),(35,197),(36,176),(37,199),(38,178),(39,201),(40,180),(41,203),(42,182),(43,205),(44,184),(45,207),(46,186),(47,209),(48,188),(49,211),(50,190),(51,213),(52,192),(53,215),(54,194),(55,217),(56,196),(57,129),(58,155),(59,131),(60,157),(61,133),(62,159),(63,135),(64,161),(65,137),(66,163),(67,139),(68,165),(69,113),(70,167),(71,115),(72,141),(73,117),(74,143),(75,119),(76,145),(77,121),(78,147),(79,123),(80,149),(81,125),(82,151),(83,127),(84,153),(86,128),(88,130),(90,132),(92,134),(94,136),(96,138),(98,140),(100,114),(102,116),(104,118),(106,120),(108,122),(110,124),(112,126),(142,181),(144,183),(146,185),(148,187),(150,189),(152,191),(154,193),(156,195),(158,169),(160,171),(162,173),(164,175),(166,177),(168,179)], [(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)], [(1,150,151,28),(2,27,152,149),(3,148,153,26),(4,25,154,147),(5,146,155,24),(6,23,156,145),(7,144,157,22),(8,21,158,143),(9,142,159,20),(10,19,160,141),(11,168,161,18),(12,17,162,167),(13,166,163,16),(14,15,164,165),(29,42,133,118),(30,117,134,41),(31,40,135,116),(32,115,136,39),(33,38,137,114),(34,113,138,37),(35,36,139,140),(43,56,119,132),(44,131,120,55),(45,54,121,130),(46,129,122,53),(47,52,123,128),(48,127,124,51),(49,50,125,126),(57,208,215,78),(58,77,216,207),(59,206,217,76),(60,75,218,205),(61,204,219,74),(62,73,220,203),(63,202,221,72),(64,71,222,201),(65,200,223,70),(66,69,224,199),(67,198,197,68),(79,214,209,84),(80,83,210,213),(81,212,211,82),(85,188,191,110),(86,109,192,187),(87,186,193,108),(88,107,194,185),(89,184,195,106),(90,105,196,183),(91,182,169,104),(92,103,170,181),(93,180,171,102),(94,101,172,179),(95,178,173,100),(96,99,174,177),(97,176,175,98),(111,190,189,112)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 4 | 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 | 4 | 4 | 28 | 2 | 2 | 8 | 14 | 14 | 28 | 28 | 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 | D28 | C8.C22 | D4×D7 | D4×D7 | D8⋊3D7 | SD16⋊D7 |
| kernel | D4.D28 | C14.Q16 | D14⋊C8 | C7×D4⋊C4 | D14⋊2Q8 | C2×Dic28 | C2×D4.D7 | C2×D4⋊2D7 | Dic14 | C2×Dic7 | C7×D4 | C22×D7 | D4⋊C4 | C4⋊C4 | C2×C8 | C2×D4 | C14 | D4 | 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 D4.D28 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 84 | 15 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 21 | 64 |
| 0 | 0 | 92 | 92 |
| 94 | 13 | 0 | 0 |
| 100 | 104 | 0 | 0 |
| 0 | 0 | 74 | 91 |
| 0 | 0 | 28 | 39 |
| 94 | 13 | 0 | 0 |
| 94 | 19 | 0 | 0 |
| 0 | 0 | 74 | 91 |
| 0 | 0 | 100 | 39 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,98,84,0,0,0,15],[112,0,0,0,0,112,0,0,0,0,21,92,0,0,64,92],[94,100,0,0,13,104,0,0,0,0,74,28,0,0,91,39],[94,94,0,0,13,19,0,0,0,0,74,100,0,0,91,39] >;
D4.D28 in GAP, Magma, Sage, TeX
D_4.D_{28} % in TeX
G:=Group("D4.D28"); // GroupNames label
G:=SmallGroup(448,317);
// by ID
G=gap.SmallGroup(448,317);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,254,219,226,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^28=1,d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations