direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C8⋊9D4, C56⋊35D4, C8⋊9(C7×D4), C4⋊C8⋊15C14, C4⋊C4.7C28, C8⋊C4⋊9C14, (C2×D4).8C28, (C4×D4).2C14, C4.80(D4×C14), C2.10(D4×C28), C22⋊C8⋊13C14, (C22×C56)⋊25C2, (C22×C8)⋊11C14, (D4×C28).17C2, (D4×C14).20C4, C14.112(C4×D4), C28.485(C2×D4), C22⋊C4.4C28, (C2×C14)⋊4M4(2), C42.7(C2×C14), C14.49(C8○D4), C23.19(C2×C28), C2.9(C14×M4(2)), C22⋊1(C7×M4(2)), C28.354(C4○D4), (C14×M4(2))⋊32C2, (C2×M4(2))⋊14C14, (C4×C28).248C22, (C2×C28).991C23, (C2×C56).446C22, C14.53(C2×M4(2)), C22.47(C22×C28), (C22×C28).417C22, (C7×C4⋊C8)⋊34C2, C2.7(C7×C8○D4), (C7×C4⋊C4).19C4, (C7×C8⋊C4)⋊23C2, C4.52(C7×C4○D4), (C7×C22⋊C8)⋊30C2, (C2×C8).52(C2×C14), (C2×C4).28(C2×C28), (C2×C28).201(C2×C4), (C7×C22⋊C4).11C4, (C22×C14).23(C2×C4), (C22×C4).96(C2×C14), (C2×C14).241(C22×C4), (C2×C4).159(C22×C14), SmallGroup(448,843)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊9D4
G = < a,b,c,d | a7=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b5, dcd=c-1 >
Subgroups: 178 in 124 conjugacy classes, 74 normal (66 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C28, C28, C2×C14, C2×C14, C2×C14, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C56, C56, C2×C28, C2×C28, C7×D4, C22×C14, C8⋊9D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C7×M4(2), C22×C28, D4×C14, C7×C8⋊C4, C7×C22⋊C8, C7×C4⋊C8, D4×C28, C22×C56, C14×M4(2), C7×C8⋊9D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, M4(2), C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C2×M4(2), C8○D4, C2×C28, C7×D4, C22×C14, C8⋊9D4, C7×M4(2), C22×C28, D4×C14, C7×C4○D4, D4×C28, C14×M4(2), C7×C8○D4, C7×C8⋊9D4
►On 224 points
Generators in S224
(1 16 71 81 25 73 17)(2 9 72 82 26 74 18)(3 10 65 83 27 75 19)(4 11 66 84 28 76 20)(5 12 67 85 29 77 21)(6 13 68 86 30 78 22)(7 14 69 87 31 79 23)(8 15 70 88 32 80 24)(33 120 105 49 97 41 89)(34 113 106 50 98 42 90)(35 114 107 51 99 43 91)(36 115 108 52 100 44 92)(37 116 109 53 101 45 93)(38 117 110 54 102 46 94)(39 118 111 55 103 47 95)(40 119 112 56 104 48 96)(57 221 164 213 156 205 148)(58 222 165 214 157 206 149)(59 223 166 215 158 207 150)(60 224 167 216 159 208 151)(61 217 168 209 160 201 152)(62 218 161 210 153 202 145)(63 219 162 211 154 203 146)(64 220 163 212 155 204 147)(121 173 194 137 186 129 178)(122 174 195 138 187 130 179)(123 175 196 139 188 131 180)(124 176 197 140 189 132 181)(125 169 198 141 190 133 182)(126 170 199 142 191 134 183)(127 171 200 143 192 135 184)(128 172 193 144 185 136 177)
(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)
(1 149 39 123)(2 146 40 128)(3 151 33 125)(4 148 34 122)(5 145 35 127)(6 150 36 124)(7 147 37 121)(8 152 38 126)(9 63 119 172)(10 60 120 169)(11 57 113 174)(12 62 114 171)(13 59 115 176)(14 64 116 173)(15 61 117 170)(16 58 118 175)(17 206 95 180)(18 203 96 177)(19 208 89 182)(20 205 90 179)(21 202 91 184)(22 207 92 181)(23 204 93 178)(24 201 94 183)(25 214 103 188)(26 211 104 185)(27 216 97 190)(28 213 98 187)(29 210 99 192)(30 215 100 189)(31 212 101 186)(32 209 102 191)(41 133 75 159)(42 130 76 156)(43 135 77 153)(44 132 78 158)(45 129 79 155)(46 134 80 160)(47 131 73 157)(48 136 74 154)(49 141 83 167)(50 138 84 164)(51 143 85 161)(52 140 86 166)(53 137 87 163)(54 142 88 168)(55 139 81 165)(56 144 82 162)(65 224 105 198)(66 221 106 195)(67 218 107 200)(68 223 108 197)(69 220 109 194)(70 217 110 199)(71 222 111 196)(72 219 112 193)
(1 123)(2 128)(3 125)(4 122)(5 127)(6 124)(7 121)(8 126)(9 172)(10 169)(11 174)(12 171)(13 176)(14 173)(15 170)(16 175)(17 180)(18 177)(19 182)(20 179)(21 184)(22 181)(23 178)(24 183)(25 188)(26 185)(27 190)(28 187)(29 192)(30 189)(31 186)(32 191)(33 151)(34 148)(35 145)(36 150)(37 147)(38 152)(39 149)(40 146)(41 159)(42 156)(43 153)(44 158)(45 155)(46 160)(47 157)(48 154)(49 167)(50 164)(51 161)(52 166)(53 163)(54 168)(55 165)(56 162)(57 113)(58 118)(59 115)(60 120)(61 117)(62 114)(63 119)(64 116)(65 198)(66 195)(67 200)(68 197)(69 194)(70 199)(71 196)(72 193)(73 131)(74 136)(75 133)(76 130)(77 135)(78 132)(79 129)(80 134)(81 139)(82 144)(83 141)(84 138)(85 143)(86 140)(87 137)(88 142)(89 208)(90 205)(91 202)(92 207)(93 204)(94 201)(95 206)(96 203)(97 216)(98 213)(99 210)(100 215)(101 212)(102 209)(103 214)(104 211)(105 224)(106 221)(107 218)(108 223)(109 220)(110 217)(111 222)(112 219)
G:=sub<Sym(224)| (1,16,71,81,25,73,17)(2,9,72,82,26,74,18)(3,10,65,83,27,75,19)(4,11,66,84,28,76,20)(5,12,67,85,29,77,21)(6,13,68,86,30,78,22)(7,14,69,87,31,79,23)(8,15,70,88,32,80,24)(33,120,105,49,97,41,89)(34,113,106,50,98,42,90)(35,114,107,51,99,43,91)(36,115,108,52,100,44,92)(37,116,109,53,101,45,93)(38,117,110,54,102,46,94)(39,118,111,55,103,47,95)(40,119,112,56,104,48,96)(57,221,164,213,156,205,148)(58,222,165,214,157,206,149)(59,223,166,215,158,207,150)(60,224,167,216,159,208,151)(61,217,168,209,160,201,152)(62,218,161,210,153,202,145)(63,219,162,211,154,203,146)(64,220,163,212,155,204,147)(121,173,194,137,186,129,178)(122,174,195,138,187,130,179)(123,175,196,139,188,131,180)(124,176,197,140,189,132,181)(125,169,198,141,190,133,182)(126,170,199,142,191,134,183)(127,171,200,143,192,135,184)(128,172,193,144,185,136,177), (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), (1,149,39,123)(2,146,40,128)(3,151,33,125)(4,148,34,122)(5,145,35,127)(6,150,36,124)(7,147,37,121)(8,152,38,126)(9,63,119,172)(10,60,120,169)(11,57,113,174)(12,62,114,171)(13,59,115,176)(14,64,116,173)(15,61,117,170)(16,58,118,175)(17,206,95,180)(18,203,96,177)(19,208,89,182)(20,205,90,179)(21,202,91,184)(22,207,92,181)(23,204,93,178)(24,201,94,183)(25,214,103,188)(26,211,104,185)(27,216,97,190)(28,213,98,187)(29,210,99,192)(30,215,100,189)(31,212,101,186)(32,209,102,191)(41,133,75,159)(42,130,76,156)(43,135,77,153)(44,132,78,158)(45,129,79,155)(46,134,80,160)(47,131,73,157)(48,136,74,154)(49,141,83,167)(50,138,84,164)(51,143,85,161)(52,140,86,166)(53,137,87,163)(54,142,88,168)(55,139,81,165)(56,144,82,162)(65,224,105,198)(66,221,106,195)(67,218,107,200)(68,223,108,197)(69,220,109,194)(70,217,110,199)(71,222,111,196)(72,219,112,193), (1,123)(2,128)(3,125)(4,122)(5,127)(6,124)(7,121)(8,126)(9,172)(10,169)(11,174)(12,171)(13,176)(14,173)(15,170)(16,175)(17,180)(18,177)(19,182)(20,179)(21,184)(22,181)(23,178)(24,183)(25,188)(26,185)(27,190)(28,187)(29,192)(30,189)(31,186)(32,191)(33,151)(34,148)(35,145)(36,150)(37,147)(38,152)(39,149)(40,146)(41,159)(42,156)(43,153)(44,158)(45,155)(46,160)(47,157)(48,154)(49,167)(50,164)(51,161)(52,166)(53,163)(54,168)(55,165)(56,162)(57,113)(58,118)(59,115)(60,120)(61,117)(62,114)(63,119)(64,116)(65,198)(66,195)(67,200)(68,197)(69,194)(70,199)(71,196)(72,193)(73,131)(74,136)(75,133)(76,130)(77,135)(78,132)(79,129)(80,134)(81,139)(82,144)(83,141)(84,138)(85,143)(86,140)(87,137)(88,142)(89,208)(90,205)(91,202)(92,207)(93,204)(94,201)(95,206)(96,203)(97,216)(98,213)(99,210)(100,215)(101,212)(102,209)(103,214)(104,211)(105,224)(106,221)(107,218)(108,223)(109,220)(110,217)(111,222)(112,219)>;
G:=Group( (1,16,71,81,25,73,17)(2,9,72,82,26,74,18)(3,10,65,83,27,75,19)(4,11,66,84,28,76,20)(5,12,67,85,29,77,21)(6,13,68,86,30,78,22)(7,14,69,87,31,79,23)(8,15,70,88,32,80,24)(33,120,105,49,97,41,89)(34,113,106,50,98,42,90)(35,114,107,51,99,43,91)(36,115,108,52,100,44,92)(37,116,109,53,101,45,93)(38,117,110,54,102,46,94)(39,118,111,55,103,47,95)(40,119,112,56,104,48,96)(57,221,164,213,156,205,148)(58,222,165,214,157,206,149)(59,223,166,215,158,207,150)(60,224,167,216,159,208,151)(61,217,168,209,160,201,152)(62,218,161,210,153,202,145)(63,219,162,211,154,203,146)(64,220,163,212,155,204,147)(121,173,194,137,186,129,178)(122,174,195,138,187,130,179)(123,175,196,139,188,131,180)(124,176,197,140,189,132,181)(125,169,198,141,190,133,182)(126,170,199,142,191,134,183)(127,171,200,143,192,135,184)(128,172,193,144,185,136,177), (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), (1,149,39,123)(2,146,40,128)(3,151,33,125)(4,148,34,122)(5,145,35,127)(6,150,36,124)(7,147,37,121)(8,152,38,126)(9,63,119,172)(10,60,120,169)(11,57,113,174)(12,62,114,171)(13,59,115,176)(14,64,116,173)(15,61,117,170)(16,58,118,175)(17,206,95,180)(18,203,96,177)(19,208,89,182)(20,205,90,179)(21,202,91,184)(22,207,92,181)(23,204,93,178)(24,201,94,183)(25,214,103,188)(26,211,104,185)(27,216,97,190)(28,213,98,187)(29,210,99,192)(30,215,100,189)(31,212,101,186)(32,209,102,191)(41,133,75,159)(42,130,76,156)(43,135,77,153)(44,132,78,158)(45,129,79,155)(46,134,80,160)(47,131,73,157)(48,136,74,154)(49,141,83,167)(50,138,84,164)(51,143,85,161)(52,140,86,166)(53,137,87,163)(54,142,88,168)(55,139,81,165)(56,144,82,162)(65,224,105,198)(66,221,106,195)(67,218,107,200)(68,223,108,197)(69,220,109,194)(70,217,110,199)(71,222,111,196)(72,219,112,193), (1,123)(2,128)(3,125)(4,122)(5,127)(6,124)(7,121)(8,126)(9,172)(10,169)(11,174)(12,171)(13,176)(14,173)(15,170)(16,175)(17,180)(18,177)(19,182)(20,179)(21,184)(22,181)(23,178)(24,183)(25,188)(26,185)(27,190)(28,187)(29,192)(30,189)(31,186)(32,191)(33,151)(34,148)(35,145)(36,150)(37,147)(38,152)(39,149)(40,146)(41,159)(42,156)(43,153)(44,158)(45,155)(46,160)(47,157)(48,154)(49,167)(50,164)(51,161)(52,166)(53,163)(54,168)(55,165)(56,162)(57,113)(58,118)(59,115)(60,120)(61,117)(62,114)(63,119)(64,116)(65,198)(66,195)(67,200)(68,197)(69,194)(70,199)(71,196)(72,193)(73,131)(74,136)(75,133)(76,130)(77,135)(78,132)(79,129)(80,134)(81,139)(82,144)(83,141)(84,138)(85,143)(86,140)(87,137)(88,142)(89,208)(90,205)(91,202)(92,207)(93,204)(94,201)(95,206)(96,203)(97,216)(98,213)(99,210)(100,215)(101,212)(102,209)(103,214)(104,211)(105,224)(106,221)(107,218)(108,223)(109,220)(110,217)(111,222)(112,219) );
G=PermutationGroup([[(1,16,71,81,25,73,17),(2,9,72,82,26,74,18),(3,10,65,83,27,75,19),(4,11,66,84,28,76,20),(5,12,67,85,29,77,21),(6,13,68,86,30,78,22),(7,14,69,87,31,79,23),(8,15,70,88,32,80,24),(33,120,105,49,97,41,89),(34,113,106,50,98,42,90),(35,114,107,51,99,43,91),(36,115,108,52,100,44,92),(37,116,109,53,101,45,93),(38,117,110,54,102,46,94),(39,118,111,55,103,47,95),(40,119,112,56,104,48,96),(57,221,164,213,156,205,148),(58,222,165,214,157,206,149),(59,223,166,215,158,207,150),(60,224,167,216,159,208,151),(61,217,168,209,160,201,152),(62,218,161,210,153,202,145),(63,219,162,211,154,203,146),(64,220,163,212,155,204,147),(121,173,194,137,186,129,178),(122,174,195,138,187,130,179),(123,175,196,139,188,131,180),(124,176,197,140,189,132,181),(125,169,198,141,190,133,182),(126,170,199,142,191,134,183),(127,171,200,143,192,135,184),(128,172,193,144,185,136,177)], [(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)], [(1,149,39,123),(2,146,40,128),(3,151,33,125),(4,148,34,122),(5,145,35,127),(6,150,36,124),(7,147,37,121),(8,152,38,126),(9,63,119,172),(10,60,120,169),(11,57,113,174),(12,62,114,171),(13,59,115,176),(14,64,116,173),(15,61,117,170),(16,58,118,175),(17,206,95,180),(18,203,96,177),(19,208,89,182),(20,205,90,179),(21,202,91,184),(22,207,92,181),(23,204,93,178),(24,201,94,183),(25,214,103,188),(26,211,104,185),(27,216,97,190),(28,213,98,187),(29,210,99,192),(30,215,100,189),(31,212,101,186),(32,209,102,191),(41,133,75,159),(42,130,76,156),(43,135,77,153),(44,132,78,158),(45,129,79,155),(46,134,80,160),(47,131,73,157),(48,136,74,154),(49,141,83,167),(50,138,84,164),(51,143,85,161),(52,140,86,166),(53,137,87,163),(54,142,88,168),(55,139,81,165),(56,144,82,162),(65,224,105,198),(66,221,106,195),(67,218,107,200),(68,223,108,197),(69,220,109,194),(70,217,110,199),(71,222,111,196),(72,219,112,193)], [(1,123),(2,128),(3,125),(4,122),(5,127),(6,124),(7,121),(8,126),(9,172),(10,169),(11,174),(12,171),(13,176),(14,173),(15,170),(16,175),(17,180),(18,177),(19,182),(20,179),(21,184),(22,181),(23,178),(24,183),(25,188),(26,185),(27,190),(28,187),(29,192),(30,189),(31,186),(32,191),(33,151),(34,148),(35,145),(36,150),(37,147),(38,152),(39,149),(40,146),(41,159),(42,156),(43,153),(44,158),(45,155),(46,160),(47,157),(48,154),(49,167),(50,164),(51,161),(52,166),(53,163),(54,168),(55,165),(56,162),(57,113),(58,118),(59,115),(60,120),(61,117),(62,114),(63,119),(64,116),(65,198),(66,195),(67,200),(68,197),(69,194),(70,199),(71,196),(72,193),(73,131),(74,136),(75,133),(76,130),(77,135),(78,132),(79,129),(80,134),(81,139),(82,144),(83,141),(84,138),(85,143),(86,140),(87,137),(88,142),(89,208),(90,205),(91,202),(92,207),(93,204),(94,201),(95,206),(96,203),(97,216),(98,213),(99,210),(100,215),(101,212),(102,209),(103,214),(104,211),(105,224),(106,221),(107,218),(108,223),(109,220),(110,217),(111,222),(112,219)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | ··· | 7F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AJ | 28A | ··· | 28X | 28Y | ··· | 28AJ | 28AK | ··· | 28BB | 56A | ··· | 56AV | 56AW | ··· | 56BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C28 | C28 | C28 | D4 | C4○D4 | M4(2) | C8○D4 | C7×D4 | C7×C4○D4 | C7×M4(2) | C7×C8○D4 |
| kernel | C7×C8⋊9D4 | C7×C8⋊C4 | C7×C22⋊C8 | C7×C4⋊C8 | D4×C28 | C22×C56 | C14×M4(2) | C7×C22⋊C4 | C7×C4⋊C4 | D4×C14 | C8⋊9D4 | C8⋊C4 | C22⋊C8 | C4⋊C8 | C4×D4 | C22×C8 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×D4 | C56 | C28 | C2×C14 | C14 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 6 | 6 | 12 | 6 | 6 | 6 | 6 | 24 | 12 | 12 | 2 | 2 | 4 | 4 | 12 | 12 | 24 | 24 |
Matrix representation of C7×C8⋊9D4 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 18 | 108 | 0 | 0 |
| 0 | 95 | 0 | 0 |
| 0 | 0 | 18 | 2 |
| 0 | 0 | 0 | 95 |
| 5 | 106 | 0 | 0 |
| 36 | 108 | 0 | 0 |
| 0 | 0 | 17 | 94 |
| 0 | 0 | 33 | 96 |
| 108 | 76 | 0 | 0 |
| 77 | 5 | 0 | 0 |
| 0 | 0 | 17 | 94 |
| 0 | 0 | 33 | 96 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,109,0,0,0,0,109],[18,0,0,0,108,95,0,0,0,0,18,0,0,0,2,95],[5,36,0,0,106,108,0,0,0,0,17,33,0,0,94,96],[108,77,0,0,76,5,0,0,0,0,17,33,0,0,94,96] >;
C7×C8⋊9D4 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes_9D_4
% in TeX
G:=Group("C7xC8:9D4"); // GroupNames label
G:=SmallGroup(448,843);
// by ID
G=gap.SmallGroup(448,843);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,4790,604,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^5,d*c*d=c^-1>;
// generators/relations