direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8⋊7D4, C56⋊33D4, C8⋊7(C7×D4), (C2×C14)⋊4D8, (C2×D8)⋊4C14, C2.D8⋊2C14, C2.6(C14×D8), C22⋊1(C7×D8), (C14×D8)⋊18C2, C4⋊D4⋊3C14, (C22×C8)⋊6C14, C14.78(C2×D8), C4.58(D4×C14), D4⋊C4⋊2C14, (C22×C56)⋊20C2, (C2×C28).364D4, C28.465(C2×D4), C23.26(C7×D4), C22.90(D4×C14), C28.263(C4○D4), C14.124(C4○D8), (C2×C56).364C22, (C2×C28).925C23, (C22×C14).130D4, C14.149(C4⋊D4), (D4×C14).190C22, (C22×C28).592C22, C4.8(C7×C4○D4), C4⋊C4.6(C2×C14), (C7×C2.D8)⋊17C2, C2.11(C7×C4○D8), (C2×C4).54(C7×D4), (C7×D4⋊C4)⋊2C2, (C7×C4⋊D4)⋊30C2, (C2×C8).77(C2×C14), C2.18(C7×C4⋊D4), (C2×D4).13(C2×C14), (C2×C14).646(C2×D4), (C7×C4⋊C4).228C22, (C2×C4).100(C22×C14), (C22×C4).121(C2×C14), SmallGroup(448,874)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊7D4
G = < a,b,c,d | a7=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 282 in 134 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C28, C28, C2×C14, C2×C14, C2×C14, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C56, C56, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C8⋊7D4, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C7×D8, C22×C28, D4×C14, D4×C14, C7×D4⋊C4, C7×C2.D8, C7×C4⋊D4, C22×C56, C14×D8, C7×C8⋊7D4
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C4○D4, C2×C14, C4⋊D4, C2×D8, C4○D8, C7×D4, C22×C14, C8⋊7D4, C7×D8, D4×C14, C7×C4○D4, C7×C4⋊D4, C14×D8, C7×C4○D8, C7×C8⋊7D4
►On 224 points
Generators in S224
(1 117 107 49 99 41 91)(2 118 108 50 100 42 92)(3 119 109 51 101 43 93)(4 120 110 52 102 44 94)(5 113 111 53 103 45 95)(6 114 112 54 104 46 96)(7 115 105 55 97 47 89)(8 116 106 56 98 48 90)(9 84 26 37 79 68 19)(10 85 27 38 80 69 20)(11 86 28 39 73 70 21)(12 87 29 40 74 71 22)(13 88 30 33 75 72 23)(14 81 31 34 76 65 24)(15 82 32 35 77 66 17)(16 83 25 36 78 67 18)(57 218 164 210 156 202 148)(58 219 165 211 157 203 149)(59 220 166 212 158 204 150)(60 221 167 213 159 205 151)(61 222 168 214 160 206 152)(62 223 161 215 153 207 145)(63 224 162 216 154 208 146)(64 217 163 209 155 201 147)(121 175 199 137 191 129 183)(122 176 200 138 192 130 184)(123 169 193 139 185 131 177)(124 170 194 140 186 132 178)(125 171 195 141 187 133 179)(126 172 196 142 188 134 180)(127 173 197 143 189 135 181)(128 174 198 144 190 136 182)
(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)
(1 174 15 151)(2 173 16 150)(3 172 9 149)(4 171 10 148)(5 170 11 147)(6 169 12 146)(7 176 13 145)(8 175 14 152)(17 205 91 128)(18 204 92 127)(19 203 93 126)(20 202 94 125)(21 201 95 124)(22 208 96 123)(23 207 89 122)(24 206 90 121)(25 220 108 143)(26 219 109 142)(27 218 110 141)(28 217 111 140)(29 224 112 139)(30 223 105 138)(31 222 106 137)(32 221 107 144)(33 161 55 192)(34 168 56 191)(35 167 49 190)(36 166 50 189)(37 165 51 188)(38 164 52 187)(39 163 53 186)(40 162 54 185)(41 182 66 159)(42 181 67 158)(43 180 68 157)(44 179 69 156)(45 178 70 155)(46 177 71 154)(47 184 72 153)(48 183 65 160)(57 120 195 85)(58 119 196 84)(59 118 197 83)(60 117 198 82)(61 116 199 81)(62 115 200 88)(63 114 193 87)(64 113 194 86)(73 209 103 132)(74 216 104 131)(75 215 97 130)(76 214 98 129)(77 213 99 136)(78 212 100 135)(79 211 101 134)(80 210 102 133)
(1 151)(2 150)(3 149)(4 148)(5 147)(6 146)(7 145)(8 152)(9 172)(10 171)(11 170)(12 169)(13 176)(14 175)(15 174)(16 173)(17 128)(18 127)(19 126)(20 125)(21 124)(22 123)(23 122)(24 121)(25 143)(26 142)(27 141)(28 140)(29 139)(30 138)(31 137)(32 144)(33 192)(34 191)(35 190)(36 189)(37 188)(38 187)(39 186)(40 185)(41 159)(42 158)(43 157)(44 156)(45 155)(46 154)(47 153)(48 160)(49 167)(50 166)(51 165)(52 164)(53 163)(54 162)(55 161)(56 168)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 183)(66 182)(67 181)(68 180)(69 179)(70 178)(71 177)(72 184)(73 132)(74 131)(75 130)(76 129)(77 136)(78 135)(79 134)(80 133)(81 199)(82 198)(83 197)(84 196)(85 195)(86 194)(87 193)(88 200)(89 207)(90 206)(91 205)(92 204)(93 203)(94 202)(95 201)(96 208)(97 215)(98 214)(99 213)(100 212)(101 211)(102 210)(103 209)(104 216)(105 223)(106 222)(107 221)(108 220)(109 219)(110 218)(111 217)(112 224)
G:=sub<Sym(224)| (1,117,107,49,99,41,91)(2,118,108,50,100,42,92)(3,119,109,51,101,43,93)(4,120,110,52,102,44,94)(5,113,111,53,103,45,95)(6,114,112,54,104,46,96)(7,115,105,55,97,47,89)(8,116,106,56,98,48,90)(9,84,26,37,79,68,19)(10,85,27,38,80,69,20)(11,86,28,39,73,70,21)(12,87,29,40,74,71,22)(13,88,30,33,75,72,23)(14,81,31,34,76,65,24)(15,82,32,35,77,66,17)(16,83,25,36,78,67,18)(57,218,164,210,156,202,148)(58,219,165,211,157,203,149)(59,220,166,212,158,204,150)(60,221,167,213,159,205,151)(61,222,168,214,160,206,152)(62,223,161,215,153,207,145)(63,224,162,216,154,208,146)(64,217,163,209,155,201,147)(121,175,199,137,191,129,183)(122,176,200,138,192,130,184)(123,169,193,139,185,131,177)(124,170,194,140,186,132,178)(125,171,195,141,187,133,179)(126,172,196,142,188,134,180)(127,173,197,143,189,135,181)(128,174,198,144,190,136,182), (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), (1,174,15,151)(2,173,16,150)(3,172,9,149)(4,171,10,148)(5,170,11,147)(6,169,12,146)(7,176,13,145)(8,175,14,152)(17,205,91,128)(18,204,92,127)(19,203,93,126)(20,202,94,125)(21,201,95,124)(22,208,96,123)(23,207,89,122)(24,206,90,121)(25,220,108,143)(26,219,109,142)(27,218,110,141)(28,217,111,140)(29,224,112,139)(30,223,105,138)(31,222,106,137)(32,221,107,144)(33,161,55,192)(34,168,56,191)(35,167,49,190)(36,166,50,189)(37,165,51,188)(38,164,52,187)(39,163,53,186)(40,162,54,185)(41,182,66,159)(42,181,67,158)(43,180,68,157)(44,179,69,156)(45,178,70,155)(46,177,71,154)(47,184,72,153)(48,183,65,160)(57,120,195,85)(58,119,196,84)(59,118,197,83)(60,117,198,82)(61,116,199,81)(62,115,200,88)(63,114,193,87)(64,113,194,86)(73,209,103,132)(74,216,104,131)(75,215,97,130)(76,214,98,129)(77,213,99,136)(78,212,100,135)(79,211,101,134)(80,210,102,133), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,152)(9,172)(10,171)(11,170)(12,169)(13,176)(14,175)(15,174)(16,173)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,143)(26,142)(27,141)(28,140)(29,139)(30,138)(31,137)(32,144)(33,192)(34,191)(35,190)(36,189)(37,188)(38,187)(39,186)(40,185)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,160)(49,167)(50,166)(51,165)(52,164)(53,163)(54,162)(55,161)(56,168)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,183)(66,182)(67,181)(68,180)(69,179)(70,178)(71,177)(72,184)(73,132)(74,131)(75,130)(76,129)(77,136)(78,135)(79,134)(80,133)(81,199)(82,198)(83,197)(84,196)(85,195)(86,194)(87,193)(88,200)(89,207)(90,206)(91,205)(92,204)(93,203)(94,202)(95,201)(96,208)(97,215)(98,214)(99,213)(100,212)(101,211)(102,210)(103,209)(104,216)(105,223)(106,222)(107,221)(108,220)(109,219)(110,218)(111,217)(112,224)>;
G:=Group( (1,117,107,49,99,41,91)(2,118,108,50,100,42,92)(3,119,109,51,101,43,93)(4,120,110,52,102,44,94)(5,113,111,53,103,45,95)(6,114,112,54,104,46,96)(7,115,105,55,97,47,89)(8,116,106,56,98,48,90)(9,84,26,37,79,68,19)(10,85,27,38,80,69,20)(11,86,28,39,73,70,21)(12,87,29,40,74,71,22)(13,88,30,33,75,72,23)(14,81,31,34,76,65,24)(15,82,32,35,77,66,17)(16,83,25,36,78,67,18)(57,218,164,210,156,202,148)(58,219,165,211,157,203,149)(59,220,166,212,158,204,150)(60,221,167,213,159,205,151)(61,222,168,214,160,206,152)(62,223,161,215,153,207,145)(63,224,162,216,154,208,146)(64,217,163,209,155,201,147)(121,175,199,137,191,129,183)(122,176,200,138,192,130,184)(123,169,193,139,185,131,177)(124,170,194,140,186,132,178)(125,171,195,141,187,133,179)(126,172,196,142,188,134,180)(127,173,197,143,189,135,181)(128,174,198,144,190,136,182), (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), (1,174,15,151)(2,173,16,150)(3,172,9,149)(4,171,10,148)(5,170,11,147)(6,169,12,146)(7,176,13,145)(8,175,14,152)(17,205,91,128)(18,204,92,127)(19,203,93,126)(20,202,94,125)(21,201,95,124)(22,208,96,123)(23,207,89,122)(24,206,90,121)(25,220,108,143)(26,219,109,142)(27,218,110,141)(28,217,111,140)(29,224,112,139)(30,223,105,138)(31,222,106,137)(32,221,107,144)(33,161,55,192)(34,168,56,191)(35,167,49,190)(36,166,50,189)(37,165,51,188)(38,164,52,187)(39,163,53,186)(40,162,54,185)(41,182,66,159)(42,181,67,158)(43,180,68,157)(44,179,69,156)(45,178,70,155)(46,177,71,154)(47,184,72,153)(48,183,65,160)(57,120,195,85)(58,119,196,84)(59,118,197,83)(60,117,198,82)(61,116,199,81)(62,115,200,88)(63,114,193,87)(64,113,194,86)(73,209,103,132)(74,216,104,131)(75,215,97,130)(76,214,98,129)(77,213,99,136)(78,212,100,135)(79,211,101,134)(80,210,102,133), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,152)(9,172)(10,171)(11,170)(12,169)(13,176)(14,175)(15,174)(16,173)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,143)(26,142)(27,141)(28,140)(29,139)(30,138)(31,137)(32,144)(33,192)(34,191)(35,190)(36,189)(37,188)(38,187)(39,186)(40,185)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,160)(49,167)(50,166)(51,165)(52,164)(53,163)(54,162)(55,161)(56,168)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,183)(66,182)(67,181)(68,180)(69,179)(70,178)(71,177)(72,184)(73,132)(74,131)(75,130)(76,129)(77,136)(78,135)(79,134)(80,133)(81,199)(82,198)(83,197)(84,196)(85,195)(86,194)(87,193)(88,200)(89,207)(90,206)(91,205)(92,204)(93,203)(94,202)(95,201)(96,208)(97,215)(98,214)(99,213)(100,212)(101,211)(102,210)(103,209)(104,216)(105,223)(106,222)(107,221)(108,220)(109,219)(110,218)(111,217)(112,224) );
G=PermutationGroup([[(1,117,107,49,99,41,91),(2,118,108,50,100,42,92),(3,119,109,51,101,43,93),(4,120,110,52,102,44,94),(5,113,111,53,103,45,95),(6,114,112,54,104,46,96),(7,115,105,55,97,47,89),(8,116,106,56,98,48,90),(9,84,26,37,79,68,19),(10,85,27,38,80,69,20),(11,86,28,39,73,70,21),(12,87,29,40,74,71,22),(13,88,30,33,75,72,23),(14,81,31,34,76,65,24),(15,82,32,35,77,66,17),(16,83,25,36,78,67,18),(57,218,164,210,156,202,148),(58,219,165,211,157,203,149),(59,220,166,212,158,204,150),(60,221,167,213,159,205,151),(61,222,168,214,160,206,152),(62,223,161,215,153,207,145),(63,224,162,216,154,208,146),(64,217,163,209,155,201,147),(121,175,199,137,191,129,183),(122,176,200,138,192,130,184),(123,169,193,139,185,131,177),(124,170,194,140,186,132,178),(125,171,195,141,187,133,179),(126,172,196,142,188,134,180),(127,173,197,143,189,135,181),(128,174,198,144,190,136,182)], [(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)], [(1,174,15,151),(2,173,16,150),(3,172,9,149),(4,171,10,148),(5,170,11,147),(6,169,12,146),(7,176,13,145),(8,175,14,152),(17,205,91,128),(18,204,92,127),(19,203,93,126),(20,202,94,125),(21,201,95,124),(22,208,96,123),(23,207,89,122),(24,206,90,121),(25,220,108,143),(26,219,109,142),(27,218,110,141),(28,217,111,140),(29,224,112,139),(30,223,105,138),(31,222,106,137),(32,221,107,144),(33,161,55,192),(34,168,56,191),(35,167,49,190),(36,166,50,189),(37,165,51,188),(38,164,52,187),(39,163,53,186),(40,162,54,185),(41,182,66,159),(42,181,67,158),(43,180,68,157),(44,179,69,156),(45,178,70,155),(46,177,71,154),(47,184,72,153),(48,183,65,160),(57,120,195,85),(58,119,196,84),(59,118,197,83),(60,117,198,82),(61,116,199,81),(62,115,200,88),(63,114,193,87),(64,113,194,86),(73,209,103,132),(74,216,104,131),(75,215,97,130),(76,214,98,129),(77,213,99,136),(78,212,100,135),(79,211,101,134),(80,210,102,133)], [(1,151),(2,150),(3,149),(4,148),(5,147),(6,146),(7,145),(8,152),(9,172),(10,171),(11,170),(12,169),(13,176),(14,175),(15,174),(16,173),(17,128),(18,127),(19,126),(20,125),(21,124),(22,123),(23,122),(24,121),(25,143),(26,142),(27,141),(28,140),(29,139),(30,138),(31,137),(32,144),(33,192),(34,191),(35,190),(36,189),(37,188),(38,187),(39,186),(40,185),(41,159),(42,158),(43,157),(44,156),(45,155),(46,154),(47,153),(48,160),(49,167),(50,166),(51,165),(52,164),(53,163),(54,162),(55,161),(56,168),(57,120),(58,119),(59,118),(60,117),(61,116),(62,115),(63,114),(64,113),(65,183),(66,182),(67,181),(68,180),(69,179),(70,178),(71,177),(72,184),(73,132),(74,131),(75,130),(76,129),(77,136),(78,135),(79,134),(80,133),(81,199),(82,198),(83,197),(84,196),(85,195),(86,194),(87,193),(88,200),(89,207),(90,206),(91,205),(92,204),(93,203),(94,202),(95,201),(96,208),(97,215),(98,214),(99,213),(100,212),(101,211),(102,210),(103,209),(104,216),(105,223),(106,222),(107,221),(108,220),(109,219),(110,218),(111,217),(112,224)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D4 | C4○D4 | D8 | C4○D8 | C7×D4 | C7×D4 | C7×D4 | C7×C4○D4 | C7×D8 | C7×C4○D8 |
| kernel | C7×C8⋊7D4 | C7×D4⋊C4 | C7×C2.D8 | C7×C4⋊D4 | C22×C56 | C14×D8 | C8⋊7D4 | D4⋊C4 | C2.D8 | C4⋊D4 | C22×C8 | C2×D8 | C56 | C2×C28 | C22×C14 | C28 | C2×C14 | C14 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 6 | 12 | 6 | 12 | 6 | 6 | 2 | 1 | 1 | 2 | 4 | 4 | 12 | 6 | 6 | 12 | 24 | 24 |
Matrix representation of C7×C8⋊7D4 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 98 | 4 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 26 | 44 |
| 83 | 3 | 0 | 0 |
| 1 | 30 | 0 | 0 |
| 0 | 0 | 111 | 111 |
| 0 | 0 | 59 | 2 |
| 30 | 108 | 0 | 0 |
| 112 | 83 | 0 | 0 |
| 0 | 0 | 111 | 111 |
| 0 | 0 | 58 | 2 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,30,0,0,0,0,30],[98,0,0,0,4,15,0,0,0,0,18,26,0,0,0,44],[83,1,0,0,3,30,0,0,0,0,111,59,0,0,111,2],[30,112,0,0,108,83,0,0,0,0,111,58,0,0,111,2] >;
C7×C8⋊7D4 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes_7D_4
% in TeX
G:=Group("C7xC8:7D4"); // GroupNames label
G:=SmallGroup(448,874);
// by ID
G=gap.SmallGroup(448,874);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1968,2438,9804,172]);
// 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^-1,d*c*d=c^-1>;
// generators/relations