direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4⋊D8, C28⋊9D8, C4⋊2(C7×D8), C4⋊C8⋊3C14, D4⋊1(C7×D4), (C4×D4)⋊3C14, (C2×D8)⋊3C14, (C7×D4)⋊12D4, C2.5(C14×D8), (C14×D8)⋊17C2, (D4×C28)⋊32C2, C4⋊1D4⋊2C14, C4.31(D4×C14), C14.77(C2×D8), D4⋊C4⋊6C14, C28.392(C2×D4), (C2×C28).321D4, C42.14(C2×C14), C22.83(D4×C14), C28.341(C4○D4), (C2×C56).256C22, (C4×C28).256C22, (C2×C28).918C23, C14.142(C4⋊D4), C14.134(C8⋊C22), (D4×C14).185C22, (C7×C4⋊C8)⋊22C2, (C2×C8).3(C2×C14), C4.40(C7×C4○D4), C2.9(C7×C8⋊C22), (C7×C4⋊1D4)⋊12C2, C4⋊C4.51(C2×C14), (C2×C4).127(C7×D4), C2.11(C7×C4⋊D4), (C7×D4⋊C4)⋊29C2, (C2×D4).55(C2×C14), (C2×C14).639(C2×D4), (C7×C4⋊C4).372C22, (C2×C4).93(C22×C14), SmallGroup(448,867)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4⋊D8
G = < a,b,c,d | a7=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 314 in 140 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×D4, C28, C28, C28, C2×C14, C2×C14, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, C56, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C4⋊D8, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×D8, C22×C28, D4×C14, D4×C14, D4×C14, C7×D4⋊C4, C7×C4⋊C8, D4×C28, C7×C4⋊1D4, C14×D8, C7×C4⋊D8
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C4○D4, C2×C14, C4⋊D4, C2×D8, C8⋊C22, C7×D4, C22×C14, C4⋊D8, C7×D8, D4×C14, C7×C4○D4, C7×C4⋊D4, C14×D8, C7×C8⋊C22, C7×C4⋊D8
►On 224 points
Generators in S224
(1 199 82 191 74 183 66)(2 200 83 192 75 184 67)(3 193 84 185 76 177 68)(4 194 85 186 77 178 69)(5 195 86 187 78 179 70)(6 196 87 188 79 180 71)(7 197 88 189 80 181 72)(8 198 81 190 73 182 65)(9 125 149 137 134 32 20)(10 126 150 138 135 25 21)(11 127 151 139 136 26 22)(12 128 152 140 129 27 23)(13 121 145 141 130 28 24)(14 122 146 142 131 29 17)(15 123 147 143 132 30 18)(16 124 148 144 133 31 19)(33 57 173 161 158 56 44)(34 58 174 162 159 49 45)(35 59 175 163 160 50 46)(36 60 176 164 153 51 47)(37 61 169 165 154 52 48)(38 62 170 166 155 53 41)(39 63 171 167 156 54 42)(40 64 172 168 157 55 43)(89 118 222 105 214 97 206)(90 119 223 106 215 98 207)(91 120 224 107 216 99 208)(92 113 217 108 209 100 201)(93 114 218 109 210 101 202)(94 115 219 110 211 102 203)(95 116 220 111 212 103 204)(96 117 221 112 213 104 205)
(1 37 121 90)(2 91 122 38)(3 39 123 92)(4 93 124 40)(5 33 125 94)(6 95 126 34)(7 35 127 96)(8 89 128 36)(9 203 70 44)(10 45 71 204)(11 205 72 46)(12 47 65 206)(13 207 66 48)(14 41 67 208)(15 201 68 42)(16 43 69 202)(17 53 184 99)(18 100 177 54)(19 55 178 101)(20 102 179 56)(21 49 180 103)(22 104 181 50)(23 51 182 97)(24 98 183 52)(25 159 79 212)(26 213 80 160)(27 153 73 214)(28 215 74 154)(29 155 75 216)(30 209 76 156)(31 157 77 210)(32 211 78 158)(57 149 115 195)(58 196 116 150)(59 151 117 197)(60 198 118 152)(61 145 119 199)(62 200 120 146)(63 147 113 193)(64 194 114 148)(81 222 140 176)(82 169 141 223)(83 224 142 170)(84 171 143 217)(85 218 144 172)(86 173 137 219)(87 220 138 174)(88 175 139 221)(105 129 164 190)(106 191 165 130)(107 131 166 192)(108 185 167 132)(109 133 168 186)(110 187 161 134)(111 135 162 188)(112 189 163 136)
(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)
(1 128)(2 127)(3 126)(4 125)(5 124)(6 123)(7 122)(8 121)(9 69)(10 68)(11 67)(12 66)(13 65)(14 72)(15 71)(16 70)(17 181)(18 180)(19 179)(20 178)(21 177)(22 184)(23 183)(24 182)(25 76)(26 75)(27 74)(28 73)(29 80)(30 79)(31 78)(32 77)(33 93)(34 92)(35 91)(36 90)(37 89)(38 96)(39 95)(40 94)(41 205)(42 204)(43 203)(44 202)(45 201)(46 208)(47 207)(48 206)(49 100)(50 99)(51 98)(52 97)(53 104)(54 103)(55 102)(56 101)(57 114)(58 113)(59 120)(60 119)(61 118)(62 117)(63 116)(64 115)(81 141)(82 140)(83 139)(84 138)(85 137)(86 144)(87 143)(88 142)(105 165)(106 164)(107 163)(108 162)(109 161)(110 168)(111 167)(112 166)(129 191)(130 190)(131 189)(132 188)(133 187)(134 186)(135 185)(136 192)(145 198)(146 197)(147 196)(148 195)(149 194)(150 193)(151 200)(152 199)(153 215)(154 214)(155 213)(156 212)(157 211)(158 210)(159 209)(160 216)(169 222)(170 221)(171 220)(172 219)(173 218)(174 217)(175 224)(176 223)
G:=sub<Sym(224)| (1,199,82,191,74,183,66)(2,200,83,192,75,184,67)(3,193,84,185,76,177,68)(4,194,85,186,77,178,69)(5,195,86,187,78,179,70)(6,196,87,188,79,180,71)(7,197,88,189,80,181,72)(8,198,81,190,73,182,65)(9,125,149,137,134,32,20)(10,126,150,138,135,25,21)(11,127,151,139,136,26,22)(12,128,152,140,129,27,23)(13,121,145,141,130,28,24)(14,122,146,142,131,29,17)(15,123,147,143,132,30,18)(16,124,148,144,133,31,19)(33,57,173,161,158,56,44)(34,58,174,162,159,49,45)(35,59,175,163,160,50,46)(36,60,176,164,153,51,47)(37,61,169,165,154,52,48)(38,62,170,166,155,53,41)(39,63,171,167,156,54,42)(40,64,172,168,157,55,43)(89,118,222,105,214,97,206)(90,119,223,106,215,98,207)(91,120,224,107,216,99,208)(92,113,217,108,209,100,201)(93,114,218,109,210,101,202)(94,115,219,110,211,102,203)(95,116,220,111,212,103,204)(96,117,221,112,213,104,205), (1,37,121,90)(2,91,122,38)(3,39,123,92)(4,93,124,40)(5,33,125,94)(6,95,126,34)(7,35,127,96)(8,89,128,36)(9,203,70,44)(10,45,71,204)(11,205,72,46)(12,47,65,206)(13,207,66,48)(14,41,67,208)(15,201,68,42)(16,43,69,202)(17,53,184,99)(18,100,177,54)(19,55,178,101)(20,102,179,56)(21,49,180,103)(22,104,181,50)(23,51,182,97)(24,98,183,52)(25,159,79,212)(26,213,80,160)(27,153,73,214)(28,215,74,154)(29,155,75,216)(30,209,76,156)(31,157,77,210)(32,211,78,158)(57,149,115,195)(58,196,116,150)(59,151,117,197)(60,198,118,152)(61,145,119,199)(62,200,120,146)(63,147,113,193)(64,194,114,148)(81,222,140,176)(82,169,141,223)(83,224,142,170)(84,171,143,217)(85,218,144,172)(86,173,137,219)(87,220,138,174)(88,175,139,221)(105,129,164,190)(106,191,165,130)(107,131,166,192)(108,185,167,132)(109,133,168,186)(110,187,161,134)(111,135,162,188)(112,189,163,136), (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), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,69)(10,68)(11,67)(12,66)(13,65)(14,72)(15,71)(16,70)(17,181)(18,180)(19,179)(20,178)(21,177)(22,184)(23,183)(24,182)(25,76)(26,75)(27,74)(28,73)(29,80)(30,79)(31,78)(32,77)(33,93)(34,92)(35,91)(36,90)(37,89)(38,96)(39,95)(40,94)(41,205)(42,204)(43,203)(44,202)(45,201)(46,208)(47,207)(48,206)(49,100)(50,99)(51,98)(52,97)(53,104)(54,103)(55,102)(56,101)(57,114)(58,113)(59,120)(60,119)(61,118)(62,117)(63,116)(64,115)(81,141)(82,140)(83,139)(84,138)(85,137)(86,144)(87,143)(88,142)(105,165)(106,164)(107,163)(108,162)(109,161)(110,168)(111,167)(112,166)(129,191)(130,190)(131,189)(132,188)(133,187)(134,186)(135,185)(136,192)(145,198)(146,197)(147,196)(148,195)(149,194)(150,193)(151,200)(152,199)(153,215)(154,214)(155,213)(156,212)(157,211)(158,210)(159,209)(160,216)(169,222)(170,221)(171,220)(172,219)(173,218)(174,217)(175,224)(176,223)>;
G:=Group( (1,199,82,191,74,183,66)(2,200,83,192,75,184,67)(3,193,84,185,76,177,68)(4,194,85,186,77,178,69)(5,195,86,187,78,179,70)(6,196,87,188,79,180,71)(7,197,88,189,80,181,72)(8,198,81,190,73,182,65)(9,125,149,137,134,32,20)(10,126,150,138,135,25,21)(11,127,151,139,136,26,22)(12,128,152,140,129,27,23)(13,121,145,141,130,28,24)(14,122,146,142,131,29,17)(15,123,147,143,132,30,18)(16,124,148,144,133,31,19)(33,57,173,161,158,56,44)(34,58,174,162,159,49,45)(35,59,175,163,160,50,46)(36,60,176,164,153,51,47)(37,61,169,165,154,52,48)(38,62,170,166,155,53,41)(39,63,171,167,156,54,42)(40,64,172,168,157,55,43)(89,118,222,105,214,97,206)(90,119,223,106,215,98,207)(91,120,224,107,216,99,208)(92,113,217,108,209,100,201)(93,114,218,109,210,101,202)(94,115,219,110,211,102,203)(95,116,220,111,212,103,204)(96,117,221,112,213,104,205), (1,37,121,90)(2,91,122,38)(3,39,123,92)(4,93,124,40)(5,33,125,94)(6,95,126,34)(7,35,127,96)(8,89,128,36)(9,203,70,44)(10,45,71,204)(11,205,72,46)(12,47,65,206)(13,207,66,48)(14,41,67,208)(15,201,68,42)(16,43,69,202)(17,53,184,99)(18,100,177,54)(19,55,178,101)(20,102,179,56)(21,49,180,103)(22,104,181,50)(23,51,182,97)(24,98,183,52)(25,159,79,212)(26,213,80,160)(27,153,73,214)(28,215,74,154)(29,155,75,216)(30,209,76,156)(31,157,77,210)(32,211,78,158)(57,149,115,195)(58,196,116,150)(59,151,117,197)(60,198,118,152)(61,145,119,199)(62,200,120,146)(63,147,113,193)(64,194,114,148)(81,222,140,176)(82,169,141,223)(83,224,142,170)(84,171,143,217)(85,218,144,172)(86,173,137,219)(87,220,138,174)(88,175,139,221)(105,129,164,190)(106,191,165,130)(107,131,166,192)(108,185,167,132)(109,133,168,186)(110,187,161,134)(111,135,162,188)(112,189,163,136), (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), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,69)(10,68)(11,67)(12,66)(13,65)(14,72)(15,71)(16,70)(17,181)(18,180)(19,179)(20,178)(21,177)(22,184)(23,183)(24,182)(25,76)(26,75)(27,74)(28,73)(29,80)(30,79)(31,78)(32,77)(33,93)(34,92)(35,91)(36,90)(37,89)(38,96)(39,95)(40,94)(41,205)(42,204)(43,203)(44,202)(45,201)(46,208)(47,207)(48,206)(49,100)(50,99)(51,98)(52,97)(53,104)(54,103)(55,102)(56,101)(57,114)(58,113)(59,120)(60,119)(61,118)(62,117)(63,116)(64,115)(81,141)(82,140)(83,139)(84,138)(85,137)(86,144)(87,143)(88,142)(105,165)(106,164)(107,163)(108,162)(109,161)(110,168)(111,167)(112,166)(129,191)(130,190)(131,189)(132,188)(133,187)(134,186)(135,185)(136,192)(145,198)(146,197)(147,196)(148,195)(149,194)(150,193)(151,200)(152,199)(153,215)(154,214)(155,213)(156,212)(157,211)(158,210)(159,209)(160,216)(169,222)(170,221)(171,220)(172,219)(173,218)(174,217)(175,224)(176,223) );
G=PermutationGroup([[(1,199,82,191,74,183,66),(2,200,83,192,75,184,67),(3,193,84,185,76,177,68),(4,194,85,186,77,178,69),(5,195,86,187,78,179,70),(6,196,87,188,79,180,71),(7,197,88,189,80,181,72),(8,198,81,190,73,182,65),(9,125,149,137,134,32,20),(10,126,150,138,135,25,21),(11,127,151,139,136,26,22),(12,128,152,140,129,27,23),(13,121,145,141,130,28,24),(14,122,146,142,131,29,17),(15,123,147,143,132,30,18),(16,124,148,144,133,31,19),(33,57,173,161,158,56,44),(34,58,174,162,159,49,45),(35,59,175,163,160,50,46),(36,60,176,164,153,51,47),(37,61,169,165,154,52,48),(38,62,170,166,155,53,41),(39,63,171,167,156,54,42),(40,64,172,168,157,55,43),(89,118,222,105,214,97,206),(90,119,223,106,215,98,207),(91,120,224,107,216,99,208),(92,113,217,108,209,100,201),(93,114,218,109,210,101,202),(94,115,219,110,211,102,203),(95,116,220,111,212,103,204),(96,117,221,112,213,104,205)], [(1,37,121,90),(2,91,122,38),(3,39,123,92),(4,93,124,40),(5,33,125,94),(6,95,126,34),(7,35,127,96),(8,89,128,36),(9,203,70,44),(10,45,71,204),(11,205,72,46),(12,47,65,206),(13,207,66,48),(14,41,67,208),(15,201,68,42),(16,43,69,202),(17,53,184,99),(18,100,177,54),(19,55,178,101),(20,102,179,56),(21,49,180,103),(22,104,181,50),(23,51,182,97),(24,98,183,52),(25,159,79,212),(26,213,80,160),(27,153,73,214),(28,215,74,154),(29,155,75,216),(30,209,76,156),(31,157,77,210),(32,211,78,158),(57,149,115,195),(58,196,116,150),(59,151,117,197),(60,198,118,152),(61,145,119,199),(62,200,120,146),(63,147,113,193),(64,194,114,148),(81,222,140,176),(82,169,141,223),(83,224,142,170),(84,171,143,217),(85,218,144,172),(86,173,137,219),(87,220,138,174),(88,175,139,221),(105,129,164,190),(106,191,165,130),(107,131,166,192),(108,185,167,132),(109,133,168,186),(110,187,161,134),(111,135,162,188),(112,189,163,136)], [(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)], [(1,128),(2,127),(3,126),(4,125),(5,124),(6,123),(7,122),(8,121),(9,69),(10,68),(11,67),(12,66),(13,65),(14,72),(15,71),(16,70),(17,181),(18,180),(19,179),(20,178),(21,177),(22,184),(23,183),(24,182),(25,76),(26,75),(27,74),(28,73),(29,80),(30,79),(31,78),(32,77),(33,93),(34,92),(35,91),(36,90),(37,89),(38,96),(39,95),(40,94),(41,205),(42,204),(43,203),(44,202),(45,201),(46,208),(47,207),(48,206),(49,100),(50,99),(51,98),(52,97),(53,104),(54,103),(55,102),(56,101),(57,114),(58,113),(59,120),(60,119),(61,118),(62,117),(63,116),(64,115),(81,141),(82,140),(83,139),(84,138),(85,137),(86,144),(87,143),(88,142),(105,165),(106,164),(107,163),(108,162),(109,161),(110,168),(111,167),(112,166),(129,191),(130,190),(131,189),(132,188),(133,187),(134,186),(135,185),(136,192),(145,198),(146,197),(147,196),(148,195),(149,194),(150,193),(151,200),(152,199),(153,215),(154,214),(155,213),(156,212),(157,211),(158,210),(159,209),(160,216),(169,222),(170,221),(171,220),(172,219),(173,218),(174,217),(175,224),(176,223)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28AP | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D8 | C4○D4 | C7×D4 | C7×D4 | C7×D8 | C7×C4○D4 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×C4⋊D8 | C7×D4⋊C4 | C7×C4⋊C8 | D4×C28 | C7×C4⋊1D4 | C14×D8 | C4⋊D8 | D4⋊C4 | C4⋊C8 | C4×D4 | C4⋊1D4 | C2×D8 | C2×C28 | C7×D4 | C28 | C28 | C2×C4 | D4 | C4 | C4 | C14 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 6 | 12 | 6 | 6 | 6 | 12 | 2 | 2 | 4 | 2 | 12 | 12 | 24 | 12 | 1 | 6 |
Matrix representation of C7×C4⋊D8 ►in GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 106 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 47 | 98 |
| 62 | 31 | 0 | 0 |
| 51 | 0 | 0 | 0 |
| 0 | 0 | 27 | 2 |
| 0 | 0 | 87 | 86 |
| 0 | 31 | 0 | 0 |
| 62 | 0 | 0 | 0 |
| 0 | 0 | 86 | 111 |
| 0 | 0 | 25 | 27 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,106,0,0,0,0,106],[1,0,0,0,0,1,0,0,0,0,15,47,0,0,0,98],[62,51,0,0,31,0,0,0,0,0,27,87,0,0,2,86],[0,62,0,0,31,0,0,0,0,0,86,25,0,0,111,27] >;
C7×C4⋊D8 in GAP, Magma, Sage, TeX
C_7\times C_4\rtimes D_8
% in TeX
G:=Group("C7xC4:D8"); // GroupNames label
G:=SmallGroup(448,867);
// by ID
G=gap.SmallGroup(448,867);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,400,2438,14117,3547,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations