direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C2.D8, C56⋊5C4, C8⋊1C28, C14.14D8, C14.7Q16, C28.10Q8, C2.2(C7×D8), C4⋊C4.3C14, C4.2(C7×Q8), (C2×C8).3C14, C4.7(C2×C28), C2.2(C7×Q16), (C2×C56).13C2, C28.44(C2×C4), (C2×C14).49D4, C14.13(C4⋊C4), C22.11(C7×D4), (C2×C28).118C22, C2.4(C7×C4⋊C4), (C7×C4⋊C4).10C2, (C2×C4).21(C2×C14), SmallGroup(224,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C2.D8
G = < a,b,c,d | a7=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of C7×C2.D8
►Regular action on 224 points
Generators in S224
(1 22 67 79 13 87 31)(2 23 68 80 14 88 32)(3 24 69 73 15 81 25)(4 17 70 74 16 82 26)(5 18 71 75 9 83 27)(6 19 72 76 10 84 28)(7 20 65 77 11 85 29)(8 21 66 78 12 86 30)(33 117 105 49 97 41 89)(34 118 106 50 98 42 90)(35 119 107 51 99 43 91)(36 120 108 52 100 44 92)(37 113 109 53 101 45 93)(38 114 110 54 102 46 94)(39 115 111 55 103 47 95)(40 116 112 56 104 48 96)(57 220 168 212 160 204 152)(58 221 161 213 153 205 145)(59 222 162 214 154 206 146)(60 223 163 215 155 207 147)(61 224 164 216 156 208 148)(62 217 165 209 157 201 149)(63 218 166 210 158 202 150)(64 219 167 211 159 203 151)(121 174 197 137 189 129 181)(122 175 198 138 190 130 182)(123 176 199 139 191 131 183)(124 169 200 140 192 132 184)(125 170 193 141 185 133 177)(126 171 194 142 186 134 178)(127 172 195 143 187 135 179)(128 173 196 144 188 136 180)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 103)(10 104)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 113)(25 93)(26 94)(27 95)(28 96)(29 89)(30 90)(31 91)(32 92)(41 85)(42 86)(43 87)(44 88)(45 81)(46 82)(47 83)(48 84)(49 77)(50 78)(51 79)(52 80)(53 73)(54 74)(55 75)(56 76)(57 175)(58 176)(59 169)(60 170)(61 171)(62 172)(63 173)(64 174)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(121 151)(122 152)(123 145)(124 146)(125 147)(126 148)(127 149)(128 150)(129 159)(130 160)(131 153)(132 154)(133 155)(134 156)(135 157)(136 158)(137 167)(138 168)(139 161)(140 162)(141 163)(142 164)(143 165)(144 166)(177 207)(178 208)(179 201)(180 202)(181 203)(182 204)(183 205)(184 206)(185 215)(186 216)(187 209)(188 210)(189 211)(190 212)(191 213)(192 214)(193 223)(194 224)(195 217)(196 218)(197 219)(198 220)(199 221)(200 222)
(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)
(1 152 35 122)(2 151 36 121)(3 150 37 128)(4 149 38 127)(5 148 39 126)(6 147 40 125)(7 146 33 124)(8 145 34 123)(9 216 103 186)(10 215 104 185)(11 214 97 192)(12 213 98 191)(13 212 99 190)(14 211 100 189)(15 210 101 188)(16 209 102 187)(17 62 114 172)(18 61 115 171)(19 60 116 170)(20 59 117 169)(21 58 118 176)(22 57 119 175)(23 64 120 174)(24 63 113 173)(25 202 93 180)(26 201 94 179)(27 208 95 178)(28 207 96 177)(29 206 89 184)(30 205 90 183)(31 204 91 182)(32 203 92 181)(41 132 85 154)(42 131 86 153)(43 130 87 160)(44 129 88 159)(45 136 81 158)(46 135 82 157)(47 134 83 156)(48 133 84 155)(49 140 77 162)(50 139 78 161)(51 138 79 168)(52 137 80 167)(53 144 73 166)(54 143 74 165)(55 142 75 164)(56 141 76 163)(65 222 105 200)(66 221 106 199)(67 220 107 198)(68 219 108 197)(69 218 109 196)(70 217 110 195)(71 224 111 194)(72 223 112 193)
G:=sub<Sym(224)| (1,22,67,79,13,87,31)(2,23,68,80,14,88,32)(3,24,69,73,15,81,25)(4,17,70,74,16,82,26)(5,18,71,75,9,83,27)(6,19,72,76,10,84,28)(7,20,65,77,11,85,29)(8,21,66,78,12,86,30)(33,117,105,49,97,41,89)(34,118,106,50,98,42,90)(35,119,107,51,99,43,91)(36,120,108,52,100,44,92)(37,113,109,53,101,45,93)(38,114,110,54,102,46,94)(39,115,111,55,103,47,95)(40,116,112,56,104,48,96)(57,220,168,212,160,204,152)(58,221,161,213,153,205,145)(59,222,162,214,154,206,146)(60,223,163,215,155,207,147)(61,224,164,216,156,208,148)(62,217,165,209,157,201,149)(63,218,166,210,158,202,150)(64,219,167,211,159,203,151)(121,174,197,137,189,129,181)(122,175,198,138,190,130,182)(123,176,199,139,191,131,183)(124,169,200,140,192,132,184)(125,170,193,141,185,133,177)(126,171,194,142,186,134,178)(127,172,195,143,187,135,179)(128,173,196,144,188,136,180), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,103)(10,104)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,113)(25,93)(26,94)(27,95)(28,96)(29,89)(30,90)(31,91)(32,92)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(49,77)(50,78)(51,79)(52,80)(53,73)(54,74)(55,75)(56,76)(57,175)(58,176)(59,169)(60,170)(61,171)(62,172)(63,173)(64,174)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(121,151)(122,152)(123,145)(124,146)(125,147)(126,148)(127,149)(128,150)(129,159)(130,160)(131,153)(132,154)(133,155)(134,156)(135,157)(136,158)(137,167)(138,168)(139,161)(140,162)(141,163)(142,164)(143,165)(144,166)(177,207)(178,208)(179,201)(180,202)(181,203)(182,204)(183,205)(184,206)(185,215)(186,216)(187,209)(188,210)(189,211)(190,212)(191,213)(192,214)(193,223)(194,224)(195,217)(196,218)(197,219)(198,220)(199,221)(200,222), (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), (1,152,35,122)(2,151,36,121)(3,150,37,128)(4,149,38,127)(5,148,39,126)(6,147,40,125)(7,146,33,124)(8,145,34,123)(9,216,103,186)(10,215,104,185)(11,214,97,192)(12,213,98,191)(13,212,99,190)(14,211,100,189)(15,210,101,188)(16,209,102,187)(17,62,114,172)(18,61,115,171)(19,60,116,170)(20,59,117,169)(21,58,118,176)(22,57,119,175)(23,64,120,174)(24,63,113,173)(25,202,93,180)(26,201,94,179)(27,208,95,178)(28,207,96,177)(29,206,89,184)(30,205,90,183)(31,204,91,182)(32,203,92,181)(41,132,85,154)(42,131,86,153)(43,130,87,160)(44,129,88,159)(45,136,81,158)(46,135,82,157)(47,134,83,156)(48,133,84,155)(49,140,77,162)(50,139,78,161)(51,138,79,168)(52,137,80,167)(53,144,73,166)(54,143,74,165)(55,142,75,164)(56,141,76,163)(65,222,105,200)(66,221,106,199)(67,220,107,198)(68,219,108,197)(69,218,109,196)(70,217,110,195)(71,224,111,194)(72,223,112,193)>;
G:=Group( (1,22,67,79,13,87,31)(2,23,68,80,14,88,32)(3,24,69,73,15,81,25)(4,17,70,74,16,82,26)(5,18,71,75,9,83,27)(6,19,72,76,10,84,28)(7,20,65,77,11,85,29)(8,21,66,78,12,86,30)(33,117,105,49,97,41,89)(34,118,106,50,98,42,90)(35,119,107,51,99,43,91)(36,120,108,52,100,44,92)(37,113,109,53,101,45,93)(38,114,110,54,102,46,94)(39,115,111,55,103,47,95)(40,116,112,56,104,48,96)(57,220,168,212,160,204,152)(58,221,161,213,153,205,145)(59,222,162,214,154,206,146)(60,223,163,215,155,207,147)(61,224,164,216,156,208,148)(62,217,165,209,157,201,149)(63,218,166,210,158,202,150)(64,219,167,211,159,203,151)(121,174,197,137,189,129,181)(122,175,198,138,190,130,182)(123,176,199,139,191,131,183)(124,169,200,140,192,132,184)(125,170,193,141,185,133,177)(126,171,194,142,186,134,178)(127,172,195,143,187,135,179)(128,173,196,144,188,136,180), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,103)(10,104)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,113)(25,93)(26,94)(27,95)(28,96)(29,89)(30,90)(31,91)(32,92)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(49,77)(50,78)(51,79)(52,80)(53,73)(54,74)(55,75)(56,76)(57,175)(58,176)(59,169)(60,170)(61,171)(62,172)(63,173)(64,174)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(121,151)(122,152)(123,145)(124,146)(125,147)(126,148)(127,149)(128,150)(129,159)(130,160)(131,153)(132,154)(133,155)(134,156)(135,157)(136,158)(137,167)(138,168)(139,161)(140,162)(141,163)(142,164)(143,165)(144,166)(177,207)(178,208)(179,201)(180,202)(181,203)(182,204)(183,205)(184,206)(185,215)(186,216)(187,209)(188,210)(189,211)(190,212)(191,213)(192,214)(193,223)(194,224)(195,217)(196,218)(197,219)(198,220)(199,221)(200,222), (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), (1,152,35,122)(2,151,36,121)(3,150,37,128)(4,149,38,127)(5,148,39,126)(6,147,40,125)(7,146,33,124)(8,145,34,123)(9,216,103,186)(10,215,104,185)(11,214,97,192)(12,213,98,191)(13,212,99,190)(14,211,100,189)(15,210,101,188)(16,209,102,187)(17,62,114,172)(18,61,115,171)(19,60,116,170)(20,59,117,169)(21,58,118,176)(22,57,119,175)(23,64,120,174)(24,63,113,173)(25,202,93,180)(26,201,94,179)(27,208,95,178)(28,207,96,177)(29,206,89,184)(30,205,90,183)(31,204,91,182)(32,203,92,181)(41,132,85,154)(42,131,86,153)(43,130,87,160)(44,129,88,159)(45,136,81,158)(46,135,82,157)(47,134,83,156)(48,133,84,155)(49,140,77,162)(50,139,78,161)(51,138,79,168)(52,137,80,167)(53,144,73,166)(54,143,74,165)(55,142,75,164)(56,141,76,163)(65,222,105,200)(66,221,106,199)(67,220,107,198)(68,219,108,197)(69,218,109,196)(70,217,110,195)(71,224,111,194)(72,223,112,193) );
G=PermutationGroup([[(1,22,67,79,13,87,31),(2,23,68,80,14,88,32),(3,24,69,73,15,81,25),(4,17,70,74,16,82,26),(5,18,71,75,9,83,27),(6,19,72,76,10,84,28),(7,20,65,77,11,85,29),(8,21,66,78,12,86,30),(33,117,105,49,97,41,89),(34,118,106,50,98,42,90),(35,119,107,51,99,43,91),(36,120,108,52,100,44,92),(37,113,109,53,101,45,93),(38,114,110,54,102,46,94),(39,115,111,55,103,47,95),(40,116,112,56,104,48,96),(57,220,168,212,160,204,152),(58,221,161,213,153,205,145),(59,222,162,214,154,206,146),(60,223,163,215,155,207,147),(61,224,164,216,156,208,148),(62,217,165,209,157,201,149),(63,218,166,210,158,202,150),(64,219,167,211,159,203,151),(121,174,197,137,189,129,181),(122,175,198,138,190,130,182),(123,176,199,139,191,131,183),(124,169,200,140,192,132,184),(125,170,193,141,185,133,177),(126,171,194,142,186,134,178),(127,172,195,143,187,135,179),(128,173,196,144,188,136,180)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,103),(10,104),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,113),(25,93),(26,94),(27,95),(28,96),(29,89),(30,90),(31,91),(32,92),(41,85),(42,86),(43,87),(44,88),(45,81),(46,82),(47,83),(48,84),(49,77),(50,78),(51,79),(52,80),(53,73),(54,74),(55,75),(56,76),(57,175),(58,176),(59,169),(60,170),(61,171),(62,172),(63,173),(64,174),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(121,151),(122,152),(123,145),(124,146),(125,147),(126,148),(127,149),(128,150),(129,159),(130,160),(131,153),(132,154),(133,155),(134,156),(135,157),(136,158),(137,167),(138,168),(139,161),(140,162),(141,163),(142,164),(143,165),(144,166),(177,207),(178,208),(179,201),(180,202),(181,203),(182,204),(183,205),(184,206),(185,215),(186,216),(187,209),(188,210),(189,211),(190,212),(191,213),(192,214),(193,223),(194,224),(195,217),(196,218),(197,219),(198,220),(199,221),(200,222)], [(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)], [(1,152,35,122),(2,151,36,121),(3,150,37,128),(4,149,38,127),(5,148,39,126),(6,147,40,125),(7,146,33,124),(8,145,34,123),(9,216,103,186),(10,215,104,185),(11,214,97,192),(12,213,98,191),(13,212,99,190),(14,211,100,189),(15,210,101,188),(16,209,102,187),(17,62,114,172),(18,61,115,171),(19,60,116,170),(20,59,117,169),(21,58,118,176),(22,57,119,175),(23,64,120,174),(24,63,113,173),(25,202,93,180),(26,201,94,179),(27,208,95,178),(28,207,96,177),(29,206,89,184),(30,205,90,183),(31,204,91,182),(32,203,92,181),(41,132,85,154),(42,131,86,153),(43,130,87,160),(44,129,88,159),(45,136,81,158),(46,135,82,157),(47,134,83,156),(48,133,84,155),(49,140,77,162),(50,139,78,161),(51,138,79,168),(52,137,80,167),(53,144,73,166),(54,143,74,165),(55,142,75,164),(56,141,76,163),(65,222,105,200),(66,221,106,199),(67,220,107,198),(68,219,108,197),(69,218,109,196),(70,217,110,195),(71,224,111,194),(72,223,112,193)]])
C7×C2.D8 is a maximal subgroup of
C8.4Dic14 C8.5Dic14 C14.D16 C56.6D4 Dic7⋊5D8 Dic28⋊6C4 C56⋊2Q8 Dic14⋊2Q8 C56⋊4Q8 Dic14.2Q8 C56.4Q8 C8.27(C4×D7) C56⋊(C2×C4) D14.5D8 C8⋊7D28 D14.2Q16 C2.D8⋊D7 C8⋊3D28 D14⋊2Q16 C2.D8⋊7D7 C56⋊C2⋊C4 D28⋊2Q8 D28.2Q8 D8×C28 Q16×C28
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 28A | ··· | 28L | 28M | ··· | 28AJ | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | Q8 | D4 | D8 | Q16 | C7×Q8 | C7×D4 | C7×D8 | C7×Q16 |
| kernel | C7×C2.D8 | C7×C4⋊C4 | C2×C56 | C56 | C2.D8 | C4⋊C4 | C2×C8 | C8 | C28 | C2×C14 | C14 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 |
Matrix representation of C7×C2.D8 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 0 | 30 |
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 31 | 82 |
| 0 | 31 | 31 |
| 98 | 0 | 0 |
| 0 | 34 | 38 |
| 0 | 38 | 79 |
G:=sub<GL(3,GF(113))| [1,0,0,0,30,0,0,0,30],[112,0,0,0,1,0,0,0,1],[1,0,0,0,31,31,0,82,31],[98,0,0,0,34,38,0,38,79] >;
C7×C2.D8 in GAP, Magma, Sage, TeX
C_7\times C_2.D_8
% in TeX
G:=Group("C7xC2.D8"); // GroupNames label
G:=SmallGroup(224,56);
// by ID
G=gap.SmallGroup(224,56);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,847,3363,117]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export