direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×SD16⋊C4, SD16⋊1C28, C8⋊2(C2×C28), C56⋊20(C2×C4), Q8⋊2(C2×C28), (C4×Q8)⋊2C14, C8⋊C4⋊1C14, (Q8×C28)⋊22C2, (C7×SD16)⋊5C4, (C4×D4).5C14, D4.2(C2×C28), C2.D8⋊11C14, C2.15(D4×C28), (D4×C28).20C2, C14.117(C4×D4), (C2×C28).456D4, C42.8(C2×C14), Q8⋊C4⋊16C14, D4⋊C4.6C14, C4.12(C22×C28), (C2×SD16).1C14, (C14×SD16).4C2, C22.54(D4×C14), C28.259(C4○D4), C28.157(C22×C4), (C2×C56).329C22, (C2×C28).907C23, (C4×C28).249C22, C14.129(C8⋊C22), (D4×C14).292C22, (Q8×C14).256C22, C14.129(C8.C22), C4.4(C7×C4○D4), (C7×Q8)⋊14(C2×C4), (C7×C2.D8)⋊26C2, (C7×C8⋊C4)⋊10C2, C2.4(C7×C8⋊C22), C4⋊C4.48(C2×C14), (C2×C8).18(C2×C14), (C7×D4).19(C2×C4), (C2×C4).102(C7×D4), C2.4(C7×C8.C22), (C7×Q8⋊C4)⋊39C2, (C2×D4).50(C2×C14), (C2×C14).630(C2×D4), (C2×Q8).41(C2×C14), (C7×D4⋊C4).15C2, (C7×C4⋊C4).369C22, (C2×C4).82(C22×C14), SmallGroup(448,848)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×SD16⋊C4
G = < a,b,c,d | a7=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=b5, cd=dc >
Subgroups: 202 in 120 conjugacy classes, 70 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, SD16⋊C4, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×SD16, C22×C28, D4×C14, Q8×C14, C7×C8⋊C4, C7×D4⋊C4, C7×Q8⋊C4, C7×C2.D8, D4×C28, Q8×C28, C14×SD16, C7×SD16⋊C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C8⋊C22, C8.C22, C2×C28, C7×D4, C22×C14, SD16⋊C4, C22×C28, D4×C14, C7×C4○D4, D4×C28, C7×C8⋊C22, C7×C8.C22, C7×SD16⋊C4
►On 224 points
Generators in S224
(1 62 129 78 121 70 113)(2 63 130 79 122 71 114)(3 64 131 80 123 72 115)(4 57 132 73 124 65 116)(5 58 133 74 125 66 117)(6 59 134 75 126 67 118)(7 60 135 76 127 68 119)(8 61 136 77 128 69 120)(9 162 194 25 186 17 178)(10 163 195 26 187 18 179)(11 164 196 27 188 19 180)(12 165 197 28 189 20 181)(13 166 198 29 190 21 182)(14 167 199 30 191 22 183)(15 168 200 31 192 23 184)(16 161 193 32 185 24 177)(33 175 218 49 210 41 202)(34 176 219 50 211 42 203)(35 169 220 51 212 43 204)(36 170 221 52 213 44 205)(37 171 222 53 214 45 206)(38 172 223 54 215 46 207)(39 173 224 55 216 47 208)(40 174 217 56 209 48 201)(81 105 156 97 148 89 140)(82 106 157 98 149 90 141)(83 107 158 99 150 91 142)(84 108 159 100 151 92 143)(85 109 160 101 152 93 144)(86 110 153 102 145 94 137)(87 111 154 103 146 95 138)(88 112 155 104 147 96 139)
(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 34)(2 37)(3 40)(4 35)(5 38)(6 33)(7 36)(8 39)(9 139)(10 142)(11 137)(12 140)(13 143)(14 138)(15 141)(16 144)(17 147)(18 150)(19 145)(20 148)(21 151)(22 146)(23 149)(24 152)(25 155)(26 158)(27 153)(28 156)(29 159)(30 154)(31 157)(32 160)(41 67)(42 70)(43 65)(44 68)(45 71)(46 66)(47 69)(48 72)(49 75)(50 78)(51 73)(52 76)(53 79)(54 74)(55 77)(56 80)(57 169)(58 172)(59 175)(60 170)(61 173)(62 176)(63 171)(64 174)(81 165)(82 168)(83 163)(84 166)(85 161)(86 164)(87 167)(88 162)(89 181)(90 184)(91 179)(92 182)(93 177)(94 180)(95 183)(96 178)(97 189)(98 192)(99 187)(100 190)(101 185)(102 188)(103 191)(104 186)(105 197)(106 200)(107 195)(108 198)(109 193)(110 196)(111 199)(112 194)(113 203)(114 206)(115 201)(116 204)(117 207)(118 202)(119 205)(120 208)(121 211)(122 214)(123 209)(124 212)(125 215)(126 210)(127 213)(128 216)(129 219)(130 222)(131 217)(132 220)(133 223)(134 218)(135 221)(136 224)
(1 83 34 163)(2 88 35 168)(3 85 36 165)(4 82 37 162)(5 87 38 167)(6 84 39 164)(7 81 40 161)(8 86 33 166)(9 116 141 206)(10 113 142 203)(11 118 143 208)(12 115 144 205)(13 120 137 202)(14 117 138 207)(15 114 139 204)(16 119 140 201)(17 124 149 214)(18 121 150 211)(19 126 151 216)(20 123 152 213)(21 128 145 210)(22 125 146 215)(23 122 147 212)(24 127 148 209)(25 132 157 222)(26 129 158 219)(27 134 159 224)(28 131 160 221)(29 136 153 218)(30 133 154 223)(31 130 155 220)(32 135 156 217)(41 182 69 94)(42 179 70 91)(43 184 71 96)(44 181 72 93)(45 178 65 90)(46 183 66 95)(47 180 67 92)(48 177 68 89)(49 190 77 102)(50 187 78 99)(51 192 79 104)(52 189 80 101)(53 186 73 98)(54 191 74 103)(55 188 75 100)(56 185 76 97)(57 106 171 194)(58 111 172 199)(59 108 173 196)(60 105 174 193)(61 110 175 198)(62 107 176 195)(63 112 169 200)(64 109 170 197)
G:=sub<Sym(224)| (1,62,129,78,121,70,113)(2,63,130,79,122,71,114)(3,64,131,80,123,72,115)(4,57,132,73,124,65,116)(5,58,133,74,125,66,117)(6,59,134,75,126,67,118)(7,60,135,76,127,68,119)(8,61,136,77,128,69,120)(9,162,194,25,186,17,178)(10,163,195,26,187,18,179)(11,164,196,27,188,19,180)(12,165,197,28,189,20,181)(13,166,198,29,190,21,182)(14,167,199,30,191,22,183)(15,168,200,31,192,23,184)(16,161,193,32,185,24,177)(33,175,218,49,210,41,202)(34,176,219,50,211,42,203)(35,169,220,51,212,43,204)(36,170,221,52,213,44,205)(37,171,222,53,214,45,206)(38,172,223,54,215,46,207)(39,173,224,55,216,47,208)(40,174,217,56,209,48,201)(81,105,156,97,148,89,140)(82,106,157,98,149,90,141)(83,107,158,99,150,91,142)(84,108,159,100,151,92,143)(85,109,160,101,152,93,144)(86,110,153,102,145,94,137)(87,111,154,103,146,95,138)(88,112,155,104,147,96,139), (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), (1,34)(2,37)(3,40)(4,35)(5,38)(6,33)(7,36)(8,39)(9,139)(10,142)(11,137)(12,140)(13,143)(14,138)(15,141)(16,144)(17,147)(18,150)(19,145)(20,148)(21,151)(22,146)(23,149)(24,152)(25,155)(26,158)(27,153)(28,156)(29,159)(30,154)(31,157)(32,160)(41,67)(42,70)(43,65)(44,68)(45,71)(46,66)(47,69)(48,72)(49,75)(50,78)(51,73)(52,76)(53,79)(54,74)(55,77)(56,80)(57,169)(58,172)(59,175)(60,170)(61,173)(62,176)(63,171)(64,174)(81,165)(82,168)(83,163)(84,166)(85,161)(86,164)(87,167)(88,162)(89,181)(90,184)(91,179)(92,182)(93,177)(94,180)(95,183)(96,178)(97,189)(98,192)(99,187)(100,190)(101,185)(102,188)(103,191)(104,186)(105,197)(106,200)(107,195)(108,198)(109,193)(110,196)(111,199)(112,194)(113,203)(114,206)(115,201)(116,204)(117,207)(118,202)(119,205)(120,208)(121,211)(122,214)(123,209)(124,212)(125,215)(126,210)(127,213)(128,216)(129,219)(130,222)(131,217)(132,220)(133,223)(134,218)(135,221)(136,224), (1,83,34,163)(2,88,35,168)(3,85,36,165)(4,82,37,162)(5,87,38,167)(6,84,39,164)(7,81,40,161)(8,86,33,166)(9,116,141,206)(10,113,142,203)(11,118,143,208)(12,115,144,205)(13,120,137,202)(14,117,138,207)(15,114,139,204)(16,119,140,201)(17,124,149,214)(18,121,150,211)(19,126,151,216)(20,123,152,213)(21,128,145,210)(22,125,146,215)(23,122,147,212)(24,127,148,209)(25,132,157,222)(26,129,158,219)(27,134,159,224)(28,131,160,221)(29,136,153,218)(30,133,154,223)(31,130,155,220)(32,135,156,217)(41,182,69,94)(42,179,70,91)(43,184,71,96)(44,181,72,93)(45,178,65,90)(46,183,66,95)(47,180,67,92)(48,177,68,89)(49,190,77,102)(50,187,78,99)(51,192,79,104)(52,189,80,101)(53,186,73,98)(54,191,74,103)(55,188,75,100)(56,185,76,97)(57,106,171,194)(58,111,172,199)(59,108,173,196)(60,105,174,193)(61,110,175,198)(62,107,176,195)(63,112,169,200)(64,109,170,197)>;
G:=Group( (1,62,129,78,121,70,113)(2,63,130,79,122,71,114)(3,64,131,80,123,72,115)(4,57,132,73,124,65,116)(5,58,133,74,125,66,117)(6,59,134,75,126,67,118)(7,60,135,76,127,68,119)(8,61,136,77,128,69,120)(9,162,194,25,186,17,178)(10,163,195,26,187,18,179)(11,164,196,27,188,19,180)(12,165,197,28,189,20,181)(13,166,198,29,190,21,182)(14,167,199,30,191,22,183)(15,168,200,31,192,23,184)(16,161,193,32,185,24,177)(33,175,218,49,210,41,202)(34,176,219,50,211,42,203)(35,169,220,51,212,43,204)(36,170,221,52,213,44,205)(37,171,222,53,214,45,206)(38,172,223,54,215,46,207)(39,173,224,55,216,47,208)(40,174,217,56,209,48,201)(81,105,156,97,148,89,140)(82,106,157,98,149,90,141)(83,107,158,99,150,91,142)(84,108,159,100,151,92,143)(85,109,160,101,152,93,144)(86,110,153,102,145,94,137)(87,111,154,103,146,95,138)(88,112,155,104,147,96,139), (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), (1,34)(2,37)(3,40)(4,35)(5,38)(6,33)(7,36)(8,39)(9,139)(10,142)(11,137)(12,140)(13,143)(14,138)(15,141)(16,144)(17,147)(18,150)(19,145)(20,148)(21,151)(22,146)(23,149)(24,152)(25,155)(26,158)(27,153)(28,156)(29,159)(30,154)(31,157)(32,160)(41,67)(42,70)(43,65)(44,68)(45,71)(46,66)(47,69)(48,72)(49,75)(50,78)(51,73)(52,76)(53,79)(54,74)(55,77)(56,80)(57,169)(58,172)(59,175)(60,170)(61,173)(62,176)(63,171)(64,174)(81,165)(82,168)(83,163)(84,166)(85,161)(86,164)(87,167)(88,162)(89,181)(90,184)(91,179)(92,182)(93,177)(94,180)(95,183)(96,178)(97,189)(98,192)(99,187)(100,190)(101,185)(102,188)(103,191)(104,186)(105,197)(106,200)(107,195)(108,198)(109,193)(110,196)(111,199)(112,194)(113,203)(114,206)(115,201)(116,204)(117,207)(118,202)(119,205)(120,208)(121,211)(122,214)(123,209)(124,212)(125,215)(126,210)(127,213)(128,216)(129,219)(130,222)(131,217)(132,220)(133,223)(134,218)(135,221)(136,224), (1,83,34,163)(2,88,35,168)(3,85,36,165)(4,82,37,162)(5,87,38,167)(6,84,39,164)(7,81,40,161)(8,86,33,166)(9,116,141,206)(10,113,142,203)(11,118,143,208)(12,115,144,205)(13,120,137,202)(14,117,138,207)(15,114,139,204)(16,119,140,201)(17,124,149,214)(18,121,150,211)(19,126,151,216)(20,123,152,213)(21,128,145,210)(22,125,146,215)(23,122,147,212)(24,127,148,209)(25,132,157,222)(26,129,158,219)(27,134,159,224)(28,131,160,221)(29,136,153,218)(30,133,154,223)(31,130,155,220)(32,135,156,217)(41,182,69,94)(42,179,70,91)(43,184,71,96)(44,181,72,93)(45,178,65,90)(46,183,66,95)(47,180,67,92)(48,177,68,89)(49,190,77,102)(50,187,78,99)(51,192,79,104)(52,189,80,101)(53,186,73,98)(54,191,74,103)(55,188,75,100)(56,185,76,97)(57,106,171,194)(58,111,172,199)(59,108,173,196)(60,105,174,193)(61,110,175,198)(62,107,176,195)(63,112,169,200)(64,109,170,197) );
G=PermutationGroup([[(1,62,129,78,121,70,113),(2,63,130,79,122,71,114),(3,64,131,80,123,72,115),(4,57,132,73,124,65,116),(5,58,133,74,125,66,117),(6,59,134,75,126,67,118),(7,60,135,76,127,68,119),(8,61,136,77,128,69,120),(9,162,194,25,186,17,178),(10,163,195,26,187,18,179),(11,164,196,27,188,19,180),(12,165,197,28,189,20,181),(13,166,198,29,190,21,182),(14,167,199,30,191,22,183),(15,168,200,31,192,23,184),(16,161,193,32,185,24,177),(33,175,218,49,210,41,202),(34,176,219,50,211,42,203),(35,169,220,51,212,43,204),(36,170,221,52,213,44,205),(37,171,222,53,214,45,206),(38,172,223,54,215,46,207),(39,173,224,55,216,47,208),(40,174,217,56,209,48,201),(81,105,156,97,148,89,140),(82,106,157,98,149,90,141),(83,107,158,99,150,91,142),(84,108,159,100,151,92,143),(85,109,160,101,152,93,144),(86,110,153,102,145,94,137),(87,111,154,103,146,95,138),(88,112,155,104,147,96,139)], [(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)], [(1,34),(2,37),(3,40),(4,35),(5,38),(6,33),(7,36),(8,39),(9,139),(10,142),(11,137),(12,140),(13,143),(14,138),(15,141),(16,144),(17,147),(18,150),(19,145),(20,148),(21,151),(22,146),(23,149),(24,152),(25,155),(26,158),(27,153),(28,156),(29,159),(30,154),(31,157),(32,160),(41,67),(42,70),(43,65),(44,68),(45,71),(46,66),(47,69),(48,72),(49,75),(50,78),(51,73),(52,76),(53,79),(54,74),(55,77),(56,80),(57,169),(58,172),(59,175),(60,170),(61,173),(62,176),(63,171),(64,174),(81,165),(82,168),(83,163),(84,166),(85,161),(86,164),(87,167),(88,162),(89,181),(90,184),(91,179),(92,182),(93,177),(94,180),(95,183),(96,178),(97,189),(98,192),(99,187),(100,190),(101,185),(102,188),(103,191),(104,186),(105,197),(106,200),(107,195),(108,198),(109,193),(110,196),(111,199),(112,194),(113,203),(114,206),(115,201),(116,204),(117,207),(118,202),(119,205),(120,208),(121,211),(122,214),(123,209),(124,212),(125,215),(126,210),(127,213),(128,216),(129,219),(130,222),(131,217),(132,220),(133,223),(134,218),(135,221),(136,224)], [(1,83,34,163),(2,88,35,168),(3,85,36,165),(4,82,37,162),(5,87,38,167),(6,84,39,164),(7,81,40,161),(8,86,33,166),(9,116,141,206),(10,113,142,203),(11,118,143,208),(12,115,144,205),(13,120,137,202),(14,117,138,207),(15,114,139,204),(16,119,140,201),(17,124,149,214),(18,121,150,211),(19,126,151,216),(20,123,152,213),(21,128,145,210),(22,125,146,215),(23,122,147,212),(24,127,148,209),(25,132,157,222),(26,129,158,219),(27,134,159,224),(28,131,160,221),(29,136,153,218),(30,133,154,223),(31,130,155,220),(32,135,156,217),(41,182,69,94),(42,179,70,91),(43,184,71,96),(44,181,72,93),(45,178,65,90),(46,183,66,95),(47,180,67,92),(48,177,68,89),(49,190,77,102),(50,187,78,99),(51,192,79,104),(52,189,80,101),(53,186,73,98),(54,191,74,103),(55,188,75,100),(56,185,76,97),(57,106,171,194),(58,111,172,199),(59,108,173,196),(60,105,174,193),(61,110,175,198),(62,107,176,195),(63,112,169,200),(64,109,170,197)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4L | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28AJ | 28AK | ··· | 28BT | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | C4○D4 | C7×D4 | C7×C4○D4 | C8⋊C22 | C8.C22 | C7×C8⋊C22 | C7×C8.C22 |
| kernel | C7×SD16⋊C4 | C7×C8⋊C4 | C7×D4⋊C4 | C7×Q8⋊C4 | C7×C2.D8 | D4×C28 | Q8×C28 | C14×SD16 | C7×SD16 | SD16⋊C4 | C8⋊C4 | D4⋊C4 | Q8⋊C4 | C2.D8 | C4×D4 | C4×Q8 | C2×SD16 | SD16 | C2×C28 | C28 | C2×C4 | C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 48 | 2 | 2 | 12 | 12 | 1 | 1 | 6 | 6 |
Matrix representation of C7×SD16⋊C4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 109 | 0 | 0 | 0 |
| 0 | 0 | 0 | 109 | 0 | 0 |
| 0 | 0 | 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 0 | 0 | 109 |
| 52 | 60 | 0 | 0 | 0 | 0 |
| 2 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 29 | 74 | 39 |
| 0 | 0 | 68 | 29 | 0 | 39 |
| 0 | 0 | 0 | 19 | 68 | 45 |
| 0 | 0 | 19 | 93 | 68 | 16 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 62 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 74 | 0 | 0 | 112 |
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 74 | 0 | 111 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 82 | 0 | 39 | 0 |
| 0 | 0 | 82 | 1 | 39 | 0 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,109,0,0,0,0,0,0,109,0,0,0,0,0,0,109,0,0,0,0,0,0,109],[52,2,0,0,0,0,60,61,0,0,0,0,0,0,0,68,0,19,0,0,29,29,19,93,0,0,74,0,68,68,0,0,39,39,45,16],[112,62,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,74,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,74,0,82,82,0,0,0,0,0,1,0,0,111,112,39,39,0,0,0,1,0,0] >;
C7×SD16⋊C4 in GAP, Magma, Sage, TeX
C_7\times {\rm SD}_{16}\rtimes C_4 % in TeX
G:=Group("C7xSD16:C4"); // GroupNames label
G:=SmallGroup(448,848);
// by ID
G=gap.SmallGroup(448,848);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,4790,604,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d^-1=b^5,c*d=d*c>;
// generators/relations