Copied to
clipboard

G = C7×C4○D16order 448 = 26·7

Direct product of C7 and C4○D16

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C7×C4○D16, D163C14, Q323C14, C56.76D4, C28.71D8, SD323C14, C56.75C23, C112.23C22, (C2×C16)⋊6C14, C4○D81C14, (C7×D16)⋊7C2, (C7×Q32)⋊7C2, C8.13(C7×D4), C4.20(C7×D8), (C2×C112)⋊13C2, C16.6(C2×C14), (C7×SD32)⋊7C2, D8.2(C2×C14), (C2×C14).12D8, C4.10(D4×C14), C2.15(C14×D8), C14.87(C2×D8), C22.1(C7×D8), C28.317(C2×D4), (C2×C28).429D4, C8.6(C22×C14), Q16.2(C2×C14), (C7×D8).12C22, (C2×C56).434C22, (C7×Q16).14C22, (C7×C4○D8)⋊8C2, (C2×C4).85(C7×D4), (C2×C8).91(C2×C14), SmallGroup(448,916)

Series: Derived Chief Lower central Upper central

C1C8 — C7×C4○D16
C1C2C4C8C56C7×D8C7×D16 — C7×C4○D16
C1C2C4C8 — C7×C4○D16
C1C28C2×C28C2×C56 — C7×C4○D16

Generators and relations for C7×C4○D16
 G = < a,b,c,d | a7=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >

Subgroups: 178 in 84 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C7, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C14, C14 [×3], C16 [×2], C2×C8, D8 [×2], SD16 [×2], Q16 [×2], C4○D4 [×2], C28 [×2], C28 [×2], C2×C14, C2×C14 [×2], C2×C16, D16, SD32 [×2], Q32, C4○D8 [×2], C56 [×2], C2×C28, C2×C28 [×2], C7×D4 [×4], C7×Q8 [×2], C4○D16, C112 [×2], C2×C56, C7×D8 [×2], C7×SD16 [×2], C7×Q16 [×2], C7×C4○D4 [×2], C2×C112, C7×D16, C7×SD32 [×2], C7×Q32, C7×C4○D8 [×2], C7×C4○D16
Quotients: C1, C2 [×7], C22 [×7], C7, D4 [×2], C23, C14 [×7], D8 [×2], C2×D4, C2×C14 [×7], C2×D8, C7×D4 [×2], C22×C14, C4○D16, C7×D8 [×2], D4×C14, C14×D8, C7×C4○D16

Smallest permutation representation of C7×C4○D16
On 224 points
Generators in S224
(1 170 138 150 115 208 36)(2 171 139 151 116 193 37)(3 172 140 152 117 194 38)(4 173 141 153 118 195 39)(5 174 142 154 119 196 40)(6 175 143 155 120 197 41)(7 176 144 156 121 198 42)(8 161 129 157 122 199 43)(9 162 130 158 123 200 44)(10 163 131 159 124 201 45)(11 164 132 160 125 202 46)(12 165 133 145 126 203 47)(13 166 134 146 127 204 48)(14 167 135 147 128 205 33)(15 168 136 148 113 206 34)(16 169 137 149 114 207 35)(17 177 211 92 62 66 104)(18 178 212 93 63 67 105)(19 179 213 94 64 68 106)(20 180 214 95 49 69 107)(21 181 215 96 50 70 108)(22 182 216 81 51 71 109)(23 183 217 82 52 72 110)(24 184 218 83 53 73 111)(25 185 219 84 54 74 112)(26 186 220 85 55 75 97)(27 187 221 86 56 76 98)(28 188 222 87 57 77 99)(29 189 223 88 58 78 100)(30 190 224 89 59 79 101)(31 191 209 90 60 80 102)(32 192 210 91 61 65 103)
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)(33 45 41 37)(34 46 42 38)(35 47 43 39)(36 48 44 40)(49 53 57 61)(50 54 58 62)(51 55 59 63)(52 56 60 64)(65 69 73 77)(66 70 74 78)(67 71 75 79)(68 72 76 80)(81 85 89 93)(82 86 90 94)(83 87 91 95)(84 88 92 96)(97 101 105 109)(98 102 106 110)(99 103 107 111)(100 104 108 112)(113 125 121 117)(114 126 122 118)(115 127 123 119)(116 128 124 120)(129 141 137 133)(130 142 138 134)(131 143 139 135)(132 144 140 136)(145 157 153 149)(146 158 154 150)(147 159 155 151)(148 160 156 152)(161 173 169 165)(162 174 170 166)(163 175 171 167)(164 176 172 168)(177 181 185 189)(178 182 186 190)(179 183 187 191)(180 184 188 192)(193 205 201 197)(194 206 202 198)(195 207 203 199)(196 208 204 200)(209 213 217 221)(210 214 218 222)(211 215 219 223)(212 216 220 224)
(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)
(1 112)(2 111)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 102)(12 101)(13 100)(14 99)(15 98)(16 97)(17 162)(18 161)(19 176)(20 175)(21 174)(22 173)(23 172)(24 171)(25 170)(26 169)(27 168)(28 167)(29 166)(30 165)(31 164)(32 163)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 80)(47 79)(48 78)(49 197)(50 196)(51 195)(52 194)(53 193)(54 208)(55 207)(56 206)(57 205)(58 204)(59 203)(60 202)(61 201)(62 200)(63 199)(64 198)(81 118)(82 117)(83 116)(84 115)(85 114)(86 113)(87 128)(88 127)(89 126)(90 125)(91 124)(92 123)(93 122)(94 121)(95 120)(96 119)(129 178)(130 177)(131 192)(132 191)(133 190)(134 189)(135 188)(136 187)(137 186)(138 185)(139 184)(140 183)(141 182)(142 181)(143 180)(144 179)(145 224)(146 223)(147 222)(148 221)(149 220)(150 219)(151 218)(152 217)(153 216)(154 215)(155 214)(156 213)(157 212)(158 211)(159 210)(160 209)

G:=sub<Sym(224)| (1,170,138,150,115,208,36)(2,171,139,151,116,193,37)(3,172,140,152,117,194,38)(4,173,141,153,118,195,39)(5,174,142,154,119,196,40)(6,175,143,155,120,197,41)(7,176,144,156,121,198,42)(8,161,129,157,122,199,43)(9,162,130,158,123,200,44)(10,163,131,159,124,201,45)(11,164,132,160,125,202,46)(12,165,133,145,126,203,47)(13,166,134,146,127,204,48)(14,167,135,147,128,205,33)(15,168,136,148,113,206,34)(16,169,137,149,114,207,35)(17,177,211,92,62,66,104)(18,178,212,93,63,67,105)(19,179,213,94,64,68,106)(20,180,214,95,49,69,107)(21,181,215,96,50,70,108)(22,182,216,81,51,71,109)(23,183,217,82,52,72,110)(24,184,218,83,53,73,111)(25,185,219,84,54,74,112)(26,186,220,85,55,75,97)(27,187,221,86,56,76,98)(28,188,222,87,57,77,99)(29,189,223,88,58,78,100)(30,190,224,89,59,79,101)(31,191,209,90,60,80,102)(32,192,210,91,61,65,103), (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32)(33,45,41,37)(34,46,42,38)(35,47,43,39)(36,48,44,40)(49,53,57,61)(50,54,58,62)(51,55,59,63)(52,56,60,64)(65,69,73,77)(66,70,74,78)(67,71,75,79)(68,72,76,80)(81,85,89,93)(82,86,90,94)(83,87,91,95)(84,88,92,96)(97,101,105,109)(98,102,106,110)(99,103,107,111)(100,104,108,112)(113,125,121,117)(114,126,122,118)(115,127,123,119)(116,128,124,120)(129,141,137,133)(130,142,138,134)(131,143,139,135)(132,144,140,136)(145,157,153,149)(146,158,154,150)(147,159,155,151)(148,160,156,152)(161,173,169,165)(162,174,170,166)(163,175,171,167)(164,176,172,168)(177,181,185,189)(178,182,186,190)(179,183,187,191)(180,184,188,192)(193,205,201,197)(194,206,202,198)(195,207,203,199)(196,208,204,200)(209,213,217,221)(210,214,218,222)(211,215,219,223)(212,216,220,224), (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), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,162)(18,161)(19,176)(20,175)(21,174)(22,173)(23,172)(24,171)(25,170)(26,169)(27,168)(28,167)(29,166)(30,165)(31,164)(32,163)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,80)(47,79)(48,78)(49,197)(50,196)(51,195)(52,194)(53,193)(54,208)(55,207)(56,206)(57,205)(58,204)(59,203)(60,202)(61,201)(62,200)(63,199)(64,198)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,128)(88,127)(89,126)(90,125)(91,124)(92,123)(93,122)(94,121)(95,120)(96,119)(129,178)(130,177)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,182)(142,181)(143,180)(144,179)(145,224)(146,223)(147,222)(148,221)(149,220)(150,219)(151,218)(152,217)(153,216)(154,215)(155,214)(156,213)(157,212)(158,211)(159,210)(160,209)>;

G:=Group( (1,170,138,150,115,208,36)(2,171,139,151,116,193,37)(3,172,140,152,117,194,38)(4,173,141,153,118,195,39)(5,174,142,154,119,196,40)(6,175,143,155,120,197,41)(7,176,144,156,121,198,42)(8,161,129,157,122,199,43)(9,162,130,158,123,200,44)(10,163,131,159,124,201,45)(11,164,132,160,125,202,46)(12,165,133,145,126,203,47)(13,166,134,146,127,204,48)(14,167,135,147,128,205,33)(15,168,136,148,113,206,34)(16,169,137,149,114,207,35)(17,177,211,92,62,66,104)(18,178,212,93,63,67,105)(19,179,213,94,64,68,106)(20,180,214,95,49,69,107)(21,181,215,96,50,70,108)(22,182,216,81,51,71,109)(23,183,217,82,52,72,110)(24,184,218,83,53,73,111)(25,185,219,84,54,74,112)(26,186,220,85,55,75,97)(27,187,221,86,56,76,98)(28,188,222,87,57,77,99)(29,189,223,88,58,78,100)(30,190,224,89,59,79,101)(31,191,209,90,60,80,102)(32,192,210,91,61,65,103), (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32)(33,45,41,37)(34,46,42,38)(35,47,43,39)(36,48,44,40)(49,53,57,61)(50,54,58,62)(51,55,59,63)(52,56,60,64)(65,69,73,77)(66,70,74,78)(67,71,75,79)(68,72,76,80)(81,85,89,93)(82,86,90,94)(83,87,91,95)(84,88,92,96)(97,101,105,109)(98,102,106,110)(99,103,107,111)(100,104,108,112)(113,125,121,117)(114,126,122,118)(115,127,123,119)(116,128,124,120)(129,141,137,133)(130,142,138,134)(131,143,139,135)(132,144,140,136)(145,157,153,149)(146,158,154,150)(147,159,155,151)(148,160,156,152)(161,173,169,165)(162,174,170,166)(163,175,171,167)(164,176,172,168)(177,181,185,189)(178,182,186,190)(179,183,187,191)(180,184,188,192)(193,205,201,197)(194,206,202,198)(195,207,203,199)(196,208,204,200)(209,213,217,221)(210,214,218,222)(211,215,219,223)(212,216,220,224), (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), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,162)(18,161)(19,176)(20,175)(21,174)(22,173)(23,172)(24,171)(25,170)(26,169)(27,168)(28,167)(29,166)(30,165)(31,164)(32,163)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,80)(47,79)(48,78)(49,197)(50,196)(51,195)(52,194)(53,193)(54,208)(55,207)(56,206)(57,205)(58,204)(59,203)(60,202)(61,201)(62,200)(63,199)(64,198)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,128)(88,127)(89,126)(90,125)(91,124)(92,123)(93,122)(94,121)(95,120)(96,119)(129,178)(130,177)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,182)(142,181)(143,180)(144,179)(145,224)(146,223)(147,222)(148,221)(149,220)(150,219)(151,218)(152,217)(153,216)(154,215)(155,214)(156,213)(157,212)(158,211)(159,210)(160,209) );

G=PermutationGroup([(1,170,138,150,115,208,36),(2,171,139,151,116,193,37),(3,172,140,152,117,194,38),(4,173,141,153,118,195,39),(5,174,142,154,119,196,40),(6,175,143,155,120,197,41),(7,176,144,156,121,198,42),(8,161,129,157,122,199,43),(9,162,130,158,123,200,44),(10,163,131,159,124,201,45),(11,164,132,160,125,202,46),(12,165,133,145,126,203,47),(13,166,134,146,127,204,48),(14,167,135,147,128,205,33),(15,168,136,148,113,206,34),(16,169,137,149,114,207,35),(17,177,211,92,62,66,104),(18,178,212,93,63,67,105),(19,179,213,94,64,68,106),(20,180,214,95,49,69,107),(21,181,215,96,50,70,108),(22,182,216,81,51,71,109),(23,183,217,82,52,72,110),(24,184,218,83,53,73,111),(25,185,219,84,54,74,112),(26,186,220,85,55,75,97),(27,187,221,86,56,76,98),(28,188,222,87,57,77,99),(29,189,223,88,58,78,100),(30,190,224,89,59,79,101),(31,191,209,90,60,80,102),(32,192,210,91,61,65,103)], [(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32),(33,45,41,37),(34,46,42,38),(35,47,43,39),(36,48,44,40),(49,53,57,61),(50,54,58,62),(51,55,59,63),(52,56,60,64),(65,69,73,77),(66,70,74,78),(67,71,75,79),(68,72,76,80),(81,85,89,93),(82,86,90,94),(83,87,91,95),(84,88,92,96),(97,101,105,109),(98,102,106,110),(99,103,107,111),(100,104,108,112),(113,125,121,117),(114,126,122,118),(115,127,123,119),(116,128,124,120),(129,141,137,133),(130,142,138,134),(131,143,139,135),(132,144,140,136),(145,157,153,149),(146,158,154,150),(147,159,155,151),(148,160,156,152),(161,173,169,165),(162,174,170,166),(163,175,171,167),(164,176,172,168),(177,181,185,189),(178,182,186,190),(179,183,187,191),(180,184,188,192),(193,205,201,197),(194,206,202,198),(195,207,203,199),(196,208,204,200),(209,213,217,221),(210,214,218,222),(211,215,219,223),(212,216,220,224)], [(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)], [(1,112),(2,111),(3,110),(4,109),(5,108),(6,107),(7,106),(8,105),(9,104),(10,103),(11,102),(12,101),(13,100),(14,99),(15,98),(16,97),(17,162),(18,161),(19,176),(20,175),(21,174),(22,173),(23,172),(24,171),(25,170),(26,169),(27,168),(28,167),(29,166),(30,165),(31,164),(32,163),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70),(41,69),(42,68),(43,67),(44,66),(45,65),(46,80),(47,79),(48,78),(49,197),(50,196),(51,195),(52,194),(53,193),(54,208),(55,207),(56,206),(57,205),(58,204),(59,203),(60,202),(61,201),(62,200),(63,199),(64,198),(81,118),(82,117),(83,116),(84,115),(85,114),(86,113),(87,128),(88,127),(89,126),(90,125),(91,124),(92,123),(93,122),(94,121),(95,120),(96,119),(129,178),(130,177),(131,192),(132,191),(133,190),(134,189),(135,188),(136,187),(137,186),(138,185),(139,184),(140,183),(141,182),(142,181),(143,180),(144,179),(145,224),(146,223),(147,222),(148,221),(149,220),(150,219),(151,218),(152,217),(153,216),(154,215),(155,214),(156,213),(157,212),(158,211),(159,210),(160,209)])

154 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A···7F8A8B8C8D14A···14F14G···14L14M···14X16A···16H28A···28L28M···28R28S···28AD56A···56X112A···112AV
order12222444447···7888814···1414···1414···1416···1628···2828···2828···2856···56112···112
size11288112881···122221···12···28···82···21···12···28···82···22···2

154 irreducible representations

dim1111111111112222222222
type++++++++++
imageC1C2C2C2C2C2C7C14C14C14C14C14D4D4D8D8C7×D4C7×D4C4○D16C7×D8C7×D8C7×C4○D16
kernelC7×C4○D16C2×C112C7×D16C7×SD32C7×Q32C7×C4○D8C4○D16C2×C16D16SD32Q32C4○D8C56C2×C28C28C2×C14C8C2×C4C7C4C22C1
# reps111212666126121122668121248

Matrix representation of C7×C4○D16 in GL2(𝔽113) generated by

1090
0109
,
980
098
,
650
040
,
040
650
G:=sub<GL(2,GF(113))| [109,0,0,109],[98,0,0,98],[65,0,0,40],[0,65,40,0] >;

C7×C4○D16 in GAP, Magma, Sage, TeX

C_7\times C_4\circ D_{16}
% in TeX

G:=Group("C7xC4oD16");
// GroupNames label

G:=SmallGroup(448,916);
// by ID

G=gap.SmallGroup(448,916);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1192,5884,2951,242,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;
// generators/relations

׿
×
𝔽