metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊4D8, C7⋊2(C4×D8), D4⋊D7⋊1C4, D4⋊1(C4×D7), C2.1(D7×D8), D28⋊1(C2×C4), (D4×Dic7)⋊1C2, C14.18(C2×D8), C14.30(C4×D4), D28⋊C4⋊1C2, D4⋊C4⋊21D7, C4⋊C4.128D14, (C8×Dic7)⋊17C2, (C2×C8).197D14, C28.Q8⋊1C2, C28.1(C22×C4), C2.D56⋊14C2, C22.65(D4×D7), (C2×D4).123D14, C14.36(C4○D8), C28.142(C4○D4), C4.43(D4⋊2D7), (C2×C56).175C22, (C2×C28).196C23, (C2×Dic7).199D4, (D4×C14).17C22, (C2×D28).43C22, C4⋊Dic7.56C22, C2.1(SD16⋊3D7), C2.14(Dic7⋊4D4), (C4×Dic7).220C22, C4.1(C2×C4×D7), C7⋊C8⋊11(C2×C4), (C7×D4)⋊1(C2×C4), (C2×D4⋊D7).1C2, (C7×C4⋊C4).1C22, (C7×D4⋊C4)⋊15C2, (C2×C14).209(C2×D4), (C2×C7⋊C8).207C22, (C2×C4).303(C22×D7), SmallGroup(448,290)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊4D8
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 724 in 134 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C14, C4×C8, D4⋊C4, D4⋊C4, C2.D8, C4×D4, C2×D8, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C4×D8, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D14⋊C4, D4⋊D7, C23.D7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C22×Dic7, D4×C14, C28.Q8, C8×Dic7, C2.D56, C7×D4⋊C4, D28⋊C4, C2×D4⋊D7, D4×Dic7, Dic7⋊4D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, D8, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×D8, C4○D8, C4×D7, C22×D7, C4×D8, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4, D7×D8, SD16⋊3D7, Dic7⋊4D8
►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 44 8 51)(2 43 9 50)(3 56 10 49)(4 55 11 48)(5 54 12 47)(6 53 13 46)(7 52 14 45)(15 188 22 195)(16 187 23 194)(17 186 24 193)(18 185 25 192)(19 184 26 191)(20 183 27 190)(21 196 28 189)(29 151 36 144)(30 150 37 143)(31 149 38 142)(32 148 39 141)(33 147 40 154)(34 146 41 153)(35 145 42 152)(57 72 64 79)(58 71 65 78)(59 84 66 77)(60 83 67 76)(61 82 68 75)(62 81 69 74)(63 80 70 73)(85 164 92 157)(86 163 93 156)(87 162 94 155)(88 161 95 168)(89 160 96 167)(90 159 97 166)(91 158 98 165)(99 121 106 114)(100 120 107 113)(101 119 108 126)(102 118 109 125)(103 117 110 124)(104 116 111 123)(105 115 112 122)(127 216 134 223)(128 215 135 222)(129 214 136 221)(130 213 137 220)(131 212 138 219)(132 211 139 218)(133 224 140 217)(169 199 176 206)(170 198 177 205)(171 197 178 204)(172 210 179 203)(173 209 180 202)(174 208 181 201)(175 207 182 200)
(1 75 121 17 199 90 34 131)(2 74 122 16 200 89 35 130)(3 73 123 15 201 88 36 129)(4 72 124 28 202 87 37 128)(5 71 125 27 203 86 38 127)(6 84 126 26 204 85 39 140)(7 83 113 25 205 98 40 139)(8 82 114 24 206 97 41 138)(9 81 115 23 207 96 42 137)(10 80 116 22 208 95 29 136)(11 79 117 21 209 94 30 135)(12 78 118 20 210 93 31 134)(13 77 119 19 197 92 32 133)(14 76 120 18 198 91 33 132)(43 62 105 187 175 160 145 213)(44 61 106 186 176 159 146 212)(45 60 107 185 177 158 147 211)(46 59 108 184 178 157 148 224)(47 58 109 183 179 156 149 223)(48 57 110 196 180 155 150 222)(49 70 111 195 181 168 151 221)(50 69 112 194 182 167 152 220)(51 68 99 193 169 166 153 219)(52 67 100 192 170 165 154 218)(53 66 101 191 171 164 141 217)(54 65 102 190 172 163 142 216)(55 64 103 189 173 162 143 215)(56 63 104 188 174 161 144 214)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 95)(16 96)(17 97)(18 98)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 123)(30 124)(31 125)(32 126)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 215)(58 216)(59 217)(60 218)(61 219)(62 220)(63 221)(64 222)(65 223)(66 224)(67 211)(68 212)(69 213)(70 214)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 127)(79 128)(80 129)(81 130)(82 131)(83 132)(84 133)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 153)(107 154)(108 141)(109 142)(110 143)(111 144)(112 145)(155 189)(156 190)(157 191)(158 192)(159 193)(160 194)(161 195)(162 196)(163 183)(164 184)(165 185)(166 186)(167 187)(168 188)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)
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), (1,44,8,51)(2,43,9,50)(3,56,10,49)(4,55,11,48)(5,54,12,47)(6,53,13,46)(7,52,14,45)(15,188,22,195)(16,187,23,194)(17,186,24,193)(18,185,25,192)(19,184,26,191)(20,183,27,190)(21,196,28,189)(29,151,36,144)(30,150,37,143)(31,149,38,142)(32,148,39,141)(33,147,40,154)(34,146,41,153)(35,145,42,152)(57,72,64,79)(58,71,65,78)(59,84,66,77)(60,83,67,76)(61,82,68,75)(62,81,69,74)(63,80,70,73)(85,164,92,157)(86,163,93,156)(87,162,94,155)(88,161,95,168)(89,160,96,167)(90,159,97,166)(91,158,98,165)(99,121,106,114)(100,120,107,113)(101,119,108,126)(102,118,109,125)(103,117,110,124)(104,116,111,123)(105,115,112,122)(127,216,134,223)(128,215,135,222)(129,214,136,221)(130,213,137,220)(131,212,138,219)(132,211,139,218)(133,224,140,217)(169,199,176,206)(170,198,177,205)(171,197,178,204)(172,210,179,203)(173,209,180,202)(174,208,181,201)(175,207,182,200), (1,75,121,17,199,90,34,131)(2,74,122,16,200,89,35,130)(3,73,123,15,201,88,36,129)(4,72,124,28,202,87,37,128)(5,71,125,27,203,86,38,127)(6,84,126,26,204,85,39,140)(7,83,113,25,205,98,40,139)(8,82,114,24,206,97,41,138)(9,81,115,23,207,96,42,137)(10,80,116,22,208,95,29,136)(11,79,117,21,209,94,30,135)(12,78,118,20,210,93,31,134)(13,77,119,19,197,92,32,133)(14,76,120,18,198,91,33,132)(43,62,105,187,175,160,145,213)(44,61,106,186,176,159,146,212)(45,60,107,185,177,158,147,211)(46,59,108,184,178,157,148,224)(47,58,109,183,179,156,149,223)(48,57,110,196,180,155,150,222)(49,70,111,195,181,168,151,221)(50,69,112,194,182,167,152,220)(51,68,99,193,169,166,153,219)(52,67,100,192,170,165,154,218)(53,66,101,191,171,164,141,217)(54,65,102,190,172,163,142,216)(55,64,103,189,173,162,143,215)(56,63,104,188,174,161,144,214), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,95)(16,96)(17,97)(18,98)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(29,123)(30,124)(31,125)(32,126)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,215)(58,216)(59,217)(60,218)(61,219)(62,220)(63,221)(64,222)(65,223)(66,224)(67,211)(68,212)(69,213)(70,214)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,141)(109,142)(110,143)(111,144)(112,145)(155,189)(156,190)(157,191)(158,192)(159,193)(160,194)(161,195)(162,196)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)>;
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), (1,44,8,51)(2,43,9,50)(3,56,10,49)(4,55,11,48)(5,54,12,47)(6,53,13,46)(7,52,14,45)(15,188,22,195)(16,187,23,194)(17,186,24,193)(18,185,25,192)(19,184,26,191)(20,183,27,190)(21,196,28,189)(29,151,36,144)(30,150,37,143)(31,149,38,142)(32,148,39,141)(33,147,40,154)(34,146,41,153)(35,145,42,152)(57,72,64,79)(58,71,65,78)(59,84,66,77)(60,83,67,76)(61,82,68,75)(62,81,69,74)(63,80,70,73)(85,164,92,157)(86,163,93,156)(87,162,94,155)(88,161,95,168)(89,160,96,167)(90,159,97,166)(91,158,98,165)(99,121,106,114)(100,120,107,113)(101,119,108,126)(102,118,109,125)(103,117,110,124)(104,116,111,123)(105,115,112,122)(127,216,134,223)(128,215,135,222)(129,214,136,221)(130,213,137,220)(131,212,138,219)(132,211,139,218)(133,224,140,217)(169,199,176,206)(170,198,177,205)(171,197,178,204)(172,210,179,203)(173,209,180,202)(174,208,181,201)(175,207,182,200), (1,75,121,17,199,90,34,131)(2,74,122,16,200,89,35,130)(3,73,123,15,201,88,36,129)(4,72,124,28,202,87,37,128)(5,71,125,27,203,86,38,127)(6,84,126,26,204,85,39,140)(7,83,113,25,205,98,40,139)(8,82,114,24,206,97,41,138)(9,81,115,23,207,96,42,137)(10,80,116,22,208,95,29,136)(11,79,117,21,209,94,30,135)(12,78,118,20,210,93,31,134)(13,77,119,19,197,92,32,133)(14,76,120,18,198,91,33,132)(43,62,105,187,175,160,145,213)(44,61,106,186,176,159,146,212)(45,60,107,185,177,158,147,211)(46,59,108,184,178,157,148,224)(47,58,109,183,179,156,149,223)(48,57,110,196,180,155,150,222)(49,70,111,195,181,168,151,221)(50,69,112,194,182,167,152,220)(51,68,99,193,169,166,153,219)(52,67,100,192,170,165,154,218)(53,66,101,191,171,164,141,217)(54,65,102,190,172,163,142,216)(55,64,103,189,173,162,143,215)(56,63,104,188,174,161,144,214), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,95)(16,96)(17,97)(18,98)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(29,123)(30,124)(31,125)(32,126)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,215)(58,216)(59,217)(60,218)(61,219)(62,220)(63,221)(64,222)(65,223)(66,224)(67,211)(68,212)(69,213)(70,214)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,141)(109,142)(110,143)(111,144)(112,145)(155,189)(156,190)(157,191)(158,192)(159,193)(160,194)(161,195)(162,196)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210) );
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)], [(1,44,8,51),(2,43,9,50),(3,56,10,49),(4,55,11,48),(5,54,12,47),(6,53,13,46),(7,52,14,45),(15,188,22,195),(16,187,23,194),(17,186,24,193),(18,185,25,192),(19,184,26,191),(20,183,27,190),(21,196,28,189),(29,151,36,144),(30,150,37,143),(31,149,38,142),(32,148,39,141),(33,147,40,154),(34,146,41,153),(35,145,42,152),(57,72,64,79),(58,71,65,78),(59,84,66,77),(60,83,67,76),(61,82,68,75),(62,81,69,74),(63,80,70,73),(85,164,92,157),(86,163,93,156),(87,162,94,155),(88,161,95,168),(89,160,96,167),(90,159,97,166),(91,158,98,165),(99,121,106,114),(100,120,107,113),(101,119,108,126),(102,118,109,125),(103,117,110,124),(104,116,111,123),(105,115,112,122),(127,216,134,223),(128,215,135,222),(129,214,136,221),(130,213,137,220),(131,212,138,219),(132,211,139,218),(133,224,140,217),(169,199,176,206),(170,198,177,205),(171,197,178,204),(172,210,179,203),(173,209,180,202),(174,208,181,201),(175,207,182,200)], [(1,75,121,17,199,90,34,131),(2,74,122,16,200,89,35,130),(3,73,123,15,201,88,36,129),(4,72,124,28,202,87,37,128),(5,71,125,27,203,86,38,127),(6,84,126,26,204,85,39,140),(7,83,113,25,205,98,40,139),(8,82,114,24,206,97,41,138),(9,81,115,23,207,96,42,137),(10,80,116,22,208,95,29,136),(11,79,117,21,209,94,30,135),(12,78,118,20,210,93,31,134),(13,77,119,19,197,92,32,133),(14,76,120,18,198,91,33,132),(43,62,105,187,175,160,145,213),(44,61,106,186,176,159,146,212),(45,60,107,185,177,158,147,211),(46,59,108,184,178,157,148,224),(47,58,109,183,179,156,149,223),(48,57,110,196,180,155,150,222),(49,70,111,195,181,168,151,221),(50,69,112,194,182,167,152,220),(51,68,99,193,169,166,153,219),(52,67,100,192,170,165,154,218),(53,66,101,191,171,164,141,217),(54,65,102,190,172,163,142,216),(55,64,103,189,173,162,143,215),(56,63,104,188,174,161,144,214)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,95),(16,96),(17,97),(18,98),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,91),(26,92),(27,93),(28,94),(29,123),(30,124),(31,125),(32,126),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,215),(58,216),(59,217),(60,218),(61,219),(62,220),(63,221),(64,222),(65,223),(66,224),(67,211),(68,212),(69,213),(70,214),(71,134),(72,135),(73,136),(74,137),(75,138),(76,139),(77,140),(78,127),(79,128),(80,129),(81,130),(82,131),(83,132),(84,133),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,153),(107,154),(108,141),(109,142),(110,143),(111,144),(112,145),(155,189),(156,190),(157,191),(158,192),(159,193),(160,194),(161,195),(162,196),(163,183),(164,184),(165,185),(166,186),(167,187),(168,188),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 28 | 28 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | D8 | C4○D4 | D14 | D14 | D14 | C4○D8 | C4×D7 | D4⋊2D7 | D4×D7 | D7×D8 | SD16⋊3D7 |
| kernel | Dic7⋊4D8 | C28.Q8 | C8×Dic7 | C2.D56 | C7×D4⋊C4 | D28⋊C4 | C2×D4⋊D7 | D4×Dic7 | D4⋊D7 | C2×Dic7 | D4⋊C4 | Dic7 | C28 | C4⋊C4 | C2×C8 | C2×D4 | C14 | D4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 4 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of Dic7⋊4D8 ►in GL4(𝔽113) generated by
| 79 | 112 | 0 | 0 |
| 60 | 25 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 69 | 110 | 0 | 0 |
| 43 | 44 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 18 | 68 | 0 | 0 |
| 80 | 95 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 8 | 62 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 44 | 1 |
G:=sub<GL(4,GF(113))| [79,60,0,0,112,25,0,0,0,0,112,0,0,0,0,112],[69,43,0,0,110,44,0,0,0,0,98,0,0,0,0,98],[18,80,0,0,68,95,0,0,0,0,0,8,0,0,14,62],[112,0,0,0,0,112,0,0,0,0,112,44,0,0,0,1] >;
Dic7⋊4D8 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes_4D_8 % in TeX
G:=Group("Dic7:4D8"); // GroupNames label
G:=SmallGroup(448,290);
// by ID
G=gap.SmallGroup(448,290);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,135,100,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^8=d^2=1,b^2=a^7,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations