direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4.D8, C28.60D8, C28.51SD16, C4⋊C8⋊2C14, C4.9(C7×D8), (D4×C14).5C4, (C2×D4).2C28, (C2×C28).503D4, C4⋊1D4.1C14, C42.3(C2×C14), C4.11(C7×SD16), (C4×C28).243C22, C14.34(D4⋊C4), C14.13(C4.D4), (C7×C4⋊C8)⋊4C2, (C2×C4).11(C2×C28), (C7×C4⋊1D4).8C2, (C2×C4).109(C7×D4), C2.4(C7×D4⋊C4), C2.4(C7×C4.D4), (C2×C28).178(C2×C4), C22.39(C7×C22⋊C4), (C2×C14).126(C22⋊C4), SmallGroup(448,135)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.D8
G = < a,b,c,d | a7=b4=c8=1, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=bc-1 >
Subgroups: 210 in 84 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C14, C14, C14, C42, C2×C8, C2×D4, C2×D4, C28, C28, C2×C14, C2×C14, C4⋊C8, C4⋊1D4, C56, C2×C28, C2×C28, C7×D4, C22×C14, C4.D8, C4×C28, C2×C56, D4×C14, D4×C14, C7×C4⋊C8, C7×C4⋊1D4, C7×C4.D8
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, D8, SD16, C28, C2×C14, C4.D4, D4⋊C4, C2×C28, C7×D4, C4.D8, C7×C22⋊C4, C7×D8, C7×SD16, C7×C4.D4, C7×D4⋊C4, C7×C4.D8
►On 224 points
Generators in S224
(1 69 57 53 41 37 25)(2 70 58 54 42 38 26)(3 71 59 55 43 39 27)(4 72 60 56 44 40 28)(5 65 61 49 45 33 29)(6 66 62 50 46 34 30)(7 67 63 51 47 35 31)(8 68 64 52 48 36 32)(9 219 214 203 198 187 182)(10 220 215 204 199 188 183)(11 221 216 205 200 189 184)(12 222 209 206 193 190 177)(13 223 210 207 194 191 178)(14 224 211 208 195 192 179)(15 217 212 201 196 185 180)(16 218 213 202 197 186 181)(17 175 162 159 146 143 130)(18 176 163 160 147 144 131)(19 169 164 153 148 137 132)(20 170 165 154 149 138 133)(21 171 166 155 150 139 134)(22 172 167 156 151 140 135)(23 173 168 157 152 141 136)(24 174 161 158 145 142 129)(73 121 115 105 101 89 85)(74 122 116 106 102 90 86)(75 123 117 107 103 91 87)(76 124 118 108 104 92 88)(77 125 119 109 97 93 81)(78 126 120 110 98 94 82)(79 127 113 111 99 95 83)(80 128 114 112 100 96 84)
(1 18 180 78)(2 79 181 19)(3 20 182 80)(4 73 183 21)(5 22 184 74)(6 75 177 23)(7 24 178 76)(8 77 179 17)(9 128 71 170)(10 171 72 121)(11 122 65 172)(12 173 66 123)(13 124 67 174)(14 175 68 125)(15 126 69 176)(16 169 70 127)(25 131 185 82)(26 83 186 132)(27 133 187 84)(28 85 188 134)(29 135 189 86)(30 87 190 136)(31 129 191 88)(32 81 192 130)(33 140 200 90)(34 91 193 141)(35 142 194 92)(36 93 195 143)(37 144 196 94)(38 95 197 137)(39 138 198 96)(40 89 199 139)(41 147 201 98)(42 99 202 148)(43 149 203 100)(44 101 204 150)(45 151 205 102)(46 103 206 152)(47 145 207 104)(48 97 208 146)(49 156 216 106)(50 107 209 157)(51 158 210 108)(52 109 211 159)(53 160 212 110)(54 111 213 153)(55 154 214 112)(56 105 215 155)(57 163 217 120)(58 113 218 164)(59 165 219 114)(60 115 220 166)(61 167 221 116)(62 117 222 168)(63 161 223 118)(64 119 224 162)
(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)
(1 179 18 17 180 8 78 77)(2 76 79 7 181 24 19 178)(3 177 20 23 182 6 80 75)(4 74 73 5 183 22 21 184)(9 66 128 123 71 12 170 173)(10 172 171 11 72 122 121 65)(13 70 124 127 67 16 174 169)(14 176 175 15 68 126 125 69)(25 192 131 130 185 32 82 81)(26 88 83 31 186 129 132 191)(27 190 133 136 187 30 84 87)(28 86 85 29 188 135 134 189)(33 199 140 139 200 40 90 89)(34 96 91 39 193 138 141 198)(35 197 142 137 194 38 92 95)(36 94 93 37 195 144 143 196)(41 208 147 146 201 48 98 97)(42 104 99 47 202 145 148 207)(43 206 149 152 203 46 100 103)(44 102 101 45 204 151 150 205)(49 215 156 155 216 56 106 105)(50 112 107 55 209 154 157 214)(51 213 158 153 210 54 108 111)(52 110 109 53 211 160 159 212)(57 224 163 162 217 64 120 119)(58 118 113 63 218 161 164 223)(59 222 165 168 219 62 114 117)(60 116 115 61 220 167 166 221)
G:=sub<Sym(224)| (1,69,57,53,41,37,25)(2,70,58,54,42,38,26)(3,71,59,55,43,39,27)(4,72,60,56,44,40,28)(5,65,61,49,45,33,29)(6,66,62,50,46,34,30)(7,67,63,51,47,35,31)(8,68,64,52,48,36,32)(9,219,214,203,198,187,182)(10,220,215,204,199,188,183)(11,221,216,205,200,189,184)(12,222,209,206,193,190,177)(13,223,210,207,194,191,178)(14,224,211,208,195,192,179)(15,217,212,201,196,185,180)(16,218,213,202,197,186,181)(17,175,162,159,146,143,130)(18,176,163,160,147,144,131)(19,169,164,153,148,137,132)(20,170,165,154,149,138,133)(21,171,166,155,150,139,134)(22,172,167,156,151,140,135)(23,173,168,157,152,141,136)(24,174,161,158,145,142,129)(73,121,115,105,101,89,85)(74,122,116,106,102,90,86)(75,123,117,107,103,91,87)(76,124,118,108,104,92,88)(77,125,119,109,97,93,81)(78,126,120,110,98,94,82)(79,127,113,111,99,95,83)(80,128,114,112,100,96,84), (1,18,180,78)(2,79,181,19)(3,20,182,80)(4,73,183,21)(5,22,184,74)(6,75,177,23)(7,24,178,76)(8,77,179,17)(9,128,71,170)(10,171,72,121)(11,122,65,172)(12,173,66,123)(13,124,67,174)(14,175,68,125)(15,126,69,176)(16,169,70,127)(25,131,185,82)(26,83,186,132)(27,133,187,84)(28,85,188,134)(29,135,189,86)(30,87,190,136)(31,129,191,88)(32,81,192,130)(33,140,200,90)(34,91,193,141)(35,142,194,92)(36,93,195,143)(37,144,196,94)(38,95,197,137)(39,138,198,96)(40,89,199,139)(41,147,201,98)(42,99,202,148)(43,149,203,100)(44,101,204,150)(45,151,205,102)(46,103,206,152)(47,145,207,104)(48,97,208,146)(49,156,216,106)(50,107,209,157)(51,158,210,108)(52,109,211,159)(53,160,212,110)(54,111,213,153)(55,154,214,112)(56,105,215,155)(57,163,217,120)(58,113,218,164)(59,165,219,114)(60,115,220,166)(61,167,221,116)(62,117,222,168)(63,161,223,118)(64,119,224,162), (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), (1,179,18,17,180,8,78,77)(2,76,79,7,181,24,19,178)(3,177,20,23,182,6,80,75)(4,74,73,5,183,22,21,184)(9,66,128,123,71,12,170,173)(10,172,171,11,72,122,121,65)(13,70,124,127,67,16,174,169)(14,176,175,15,68,126,125,69)(25,192,131,130,185,32,82,81)(26,88,83,31,186,129,132,191)(27,190,133,136,187,30,84,87)(28,86,85,29,188,135,134,189)(33,199,140,139,200,40,90,89)(34,96,91,39,193,138,141,198)(35,197,142,137,194,38,92,95)(36,94,93,37,195,144,143,196)(41,208,147,146,201,48,98,97)(42,104,99,47,202,145,148,207)(43,206,149,152,203,46,100,103)(44,102,101,45,204,151,150,205)(49,215,156,155,216,56,106,105)(50,112,107,55,209,154,157,214)(51,213,158,153,210,54,108,111)(52,110,109,53,211,160,159,212)(57,224,163,162,217,64,120,119)(58,118,113,63,218,161,164,223)(59,222,165,168,219,62,114,117)(60,116,115,61,220,167,166,221)>;
G:=Group( (1,69,57,53,41,37,25)(2,70,58,54,42,38,26)(3,71,59,55,43,39,27)(4,72,60,56,44,40,28)(5,65,61,49,45,33,29)(6,66,62,50,46,34,30)(7,67,63,51,47,35,31)(8,68,64,52,48,36,32)(9,219,214,203,198,187,182)(10,220,215,204,199,188,183)(11,221,216,205,200,189,184)(12,222,209,206,193,190,177)(13,223,210,207,194,191,178)(14,224,211,208,195,192,179)(15,217,212,201,196,185,180)(16,218,213,202,197,186,181)(17,175,162,159,146,143,130)(18,176,163,160,147,144,131)(19,169,164,153,148,137,132)(20,170,165,154,149,138,133)(21,171,166,155,150,139,134)(22,172,167,156,151,140,135)(23,173,168,157,152,141,136)(24,174,161,158,145,142,129)(73,121,115,105,101,89,85)(74,122,116,106,102,90,86)(75,123,117,107,103,91,87)(76,124,118,108,104,92,88)(77,125,119,109,97,93,81)(78,126,120,110,98,94,82)(79,127,113,111,99,95,83)(80,128,114,112,100,96,84), (1,18,180,78)(2,79,181,19)(3,20,182,80)(4,73,183,21)(5,22,184,74)(6,75,177,23)(7,24,178,76)(8,77,179,17)(9,128,71,170)(10,171,72,121)(11,122,65,172)(12,173,66,123)(13,124,67,174)(14,175,68,125)(15,126,69,176)(16,169,70,127)(25,131,185,82)(26,83,186,132)(27,133,187,84)(28,85,188,134)(29,135,189,86)(30,87,190,136)(31,129,191,88)(32,81,192,130)(33,140,200,90)(34,91,193,141)(35,142,194,92)(36,93,195,143)(37,144,196,94)(38,95,197,137)(39,138,198,96)(40,89,199,139)(41,147,201,98)(42,99,202,148)(43,149,203,100)(44,101,204,150)(45,151,205,102)(46,103,206,152)(47,145,207,104)(48,97,208,146)(49,156,216,106)(50,107,209,157)(51,158,210,108)(52,109,211,159)(53,160,212,110)(54,111,213,153)(55,154,214,112)(56,105,215,155)(57,163,217,120)(58,113,218,164)(59,165,219,114)(60,115,220,166)(61,167,221,116)(62,117,222,168)(63,161,223,118)(64,119,224,162), (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), (1,179,18,17,180,8,78,77)(2,76,79,7,181,24,19,178)(3,177,20,23,182,6,80,75)(4,74,73,5,183,22,21,184)(9,66,128,123,71,12,170,173)(10,172,171,11,72,122,121,65)(13,70,124,127,67,16,174,169)(14,176,175,15,68,126,125,69)(25,192,131,130,185,32,82,81)(26,88,83,31,186,129,132,191)(27,190,133,136,187,30,84,87)(28,86,85,29,188,135,134,189)(33,199,140,139,200,40,90,89)(34,96,91,39,193,138,141,198)(35,197,142,137,194,38,92,95)(36,94,93,37,195,144,143,196)(41,208,147,146,201,48,98,97)(42,104,99,47,202,145,148,207)(43,206,149,152,203,46,100,103)(44,102,101,45,204,151,150,205)(49,215,156,155,216,56,106,105)(50,112,107,55,209,154,157,214)(51,213,158,153,210,54,108,111)(52,110,109,53,211,160,159,212)(57,224,163,162,217,64,120,119)(58,118,113,63,218,161,164,223)(59,222,165,168,219,62,114,117)(60,116,115,61,220,167,166,221) );
G=PermutationGroup([[(1,69,57,53,41,37,25),(2,70,58,54,42,38,26),(3,71,59,55,43,39,27),(4,72,60,56,44,40,28),(5,65,61,49,45,33,29),(6,66,62,50,46,34,30),(7,67,63,51,47,35,31),(8,68,64,52,48,36,32),(9,219,214,203,198,187,182),(10,220,215,204,199,188,183),(11,221,216,205,200,189,184),(12,222,209,206,193,190,177),(13,223,210,207,194,191,178),(14,224,211,208,195,192,179),(15,217,212,201,196,185,180),(16,218,213,202,197,186,181),(17,175,162,159,146,143,130),(18,176,163,160,147,144,131),(19,169,164,153,148,137,132),(20,170,165,154,149,138,133),(21,171,166,155,150,139,134),(22,172,167,156,151,140,135),(23,173,168,157,152,141,136),(24,174,161,158,145,142,129),(73,121,115,105,101,89,85),(74,122,116,106,102,90,86),(75,123,117,107,103,91,87),(76,124,118,108,104,92,88),(77,125,119,109,97,93,81),(78,126,120,110,98,94,82),(79,127,113,111,99,95,83),(80,128,114,112,100,96,84)], [(1,18,180,78),(2,79,181,19),(3,20,182,80),(4,73,183,21),(5,22,184,74),(6,75,177,23),(7,24,178,76),(8,77,179,17),(9,128,71,170),(10,171,72,121),(11,122,65,172),(12,173,66,123),(13,124,67,174),(14,175,68,125),(15,126,69,176),(16,169,70,127),(25,131,185,82),(26,83,186,132),(27,133,187,84),(28,85,188,134),(29,135,189,86),(30,87,190,136),(31,129,191,88),(32,81,192,130),(33,140,200,90),(34,91,193,141),(35,142,194,92),(36,93,195,143),(37,144,196,94),(38,95,197,137),(39,138,198,96),(40,89,199,139),(41,147,201,98),(42,99,202,148),(43,149,203,100),(44,101,204,150),(45,151,205,102),(46,103,206,152),(47,145,207,104),(48,97,208,146),(49,156,216,106),(50,107,209,157),(51,158,210,108),(52,109,211,159),(53,160,212,110),(54,111,213,153),(55,154,214,112),(56,105,215,155),(57,163,217,120),(58,113,218,164),(59,165,219,114),(60,115,220,166),(61,167,221,116),(62,117,222,168),(63,161,223,118),(64,119,224,162)], [(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)], [(1,179,18,17,180,8,78,77),(2,76,79,7,181,24,19,178),(3,177,20,23,182,6,80,75),(4,74,73,5,183,22,21,184),(9,66,128,123,71,12,170,173),(10,172,171,11,72,122,121,65),(13,70,124,127,67,16,174,169),(14,176,175,15,68,126,125,69),(25,192,131,130,185,32,82,81),(26,88,83,31,186,129,132,191),(27,190,133,136,187,30,84,87),(28,86,85,29,188,135,134,189),(33,199,140,139,200,40,90,89),(34,96,91,39,193,138,141,198),(35,197,142,137,194,38,92,95),(36,94,93,37,195,144,143,196),(41,208,147,146,201,48,98,97),(42,104,99,47,202,145,148,207),(43,206,149,152,203,46,100,103),(44,102,101,45,204,151,150,205),(49,215,156,155,216,56,106,105),(50,112,107,55,209,154,157,214),(51,213,158,153,210,54,108,111),(52,110,109,53,211,160,159,212),(57,224,163,162,217,64,120,119),(58,118,113,63,218,161,164,223),(59,222,165,168,219,62,114,117),(60,116,115,61,220,167,166,221)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AD | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | D8 | SD16 | C7×D4 | C7×D8 | C7×SD16 | C4.D4 | C7×C4.D4 |
| kernel | C7×C4.D8 | C7×C4⋊C8 | C7×C4⋊1D4 | D4×C14 | C4.D8 | C4⋊C8 | C4⋊1D4 | C2×D4 | C2×C28 | C28 | C28 | C2×C4 | C4 | C4 | C14 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 2 | 4 | 4 | 12 | 24 | 24 | 1 | 6 |
Matrix representation of C7×C4.D8 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 13 | 0 | 0 |
| 13 | 100 | 0 | 0 |
| 0 | 0 | 62 | 51 |
| 0 | 0 | 31 | 0 |
| 100 | 13 | 0 | 0 |
| 100 | 100 | 0 | 0 |
| 0 | 0 | 0 | 62 |
| 0 | 0 | 31 | 0 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,16,0,0,0,0,16],[0,112,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,13,0,0,13,100,0,0,0,0,62,31,0,0,51,0],[100,100,0,0,13,100,0,0,0,0,0,31,0,0,62,0] >;
C7×C4.D8 in GAP, Magma, Sage, TeX
C_7\times C_4.D_8
% in TeX
G:=Group("C7xC4.D8"); // GroupNames label
G:=SmallGroup(448,135);
// by ID
G=gap.SmallGroup(448,135);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,3538,248,6871,242]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations