direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4.4D8, C28.42D8, C28.32SD16, (C4×C8)⋊5C14, C4⋊Q8⋊5C14, C4.4(C7×D8), (C4×C56)⋊10C2, C2.9(C14×D8), C14.81(C2×D8), D4⋊C4⋊3C14, C4.3(C7×SD16), (C2×C28).419D4, C4⋊1D4.4C14, C42.76(C2×C14), C14.94(C2×SD16), C2.14(C14×SD16), C28.268(C4○D4), (C2×C56).366C22, (C2×C28).942C23, (C4×C28).360C22, C22.107(D4×C14), C14.71(C4.4D4), (D4×C14).198C22, (C7×C4⋊Q8)⋊26C2, C4.13(C7×C4○D4), (C2×C4).75(C7×D4), (C7×D4⋊C4)⋊3C2, C4⋊C4.17(C2×C14), (C2×C8).68(C2×C14), C2.9(C7×C4.4D4), (C2×D4).21(C2×C14), (C7×C4⋊1D4).11C2, (C2×C14).663(C2×D4), (C7×C4⋊C4).237C22, (C2×C4).117(C22×C14), SmallGroup(448,894)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.4D8
G = < a,b,c,d | a7=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 258 in 118 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C56, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C4.4D8, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, D4×C14, D4×C14, Q8×C14, C4×C56, C7×D4⋊C4, C7×C4⋊1D4, C7×C4⋊Q8, C7×C4.4D8
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, SD16, C2×D4, C4○D4, C2×C14, C4.4D4, C2×D8, C2×SD16, C7×D4, C22×C14, C4.4D8, C7×D8, C7×SD16, D4×C14, C7×C4○D4, C7×C4.4D4, C14×D8, C14×SD16, C7×C4.4D8
►On 224 points
Generators in S224
(1 36 143 28 135 20 127)(2 37 144 29 136 21 128)(3 38 137 30 129 22 121)(4 39 138 31 130 23 122)(5 40 139 32 131 24 123)(6 33 140 25 132 17 124)(7 34 141 26 133 18 125)(8 35 142 27 134 19 126)(9 57 164 49 156 41 148)(10 58 165 50 157 42 149)(11 59 166 51 158 43 150)(12 60 167 52 159 44 151)(13 61 168 53 160 45 152)(14 62 161 54 153 46 145)(15 63 162 55 154 47 146)(16 64 163 56 155 48 147)(65 172 196 81 188 73 180)(66 173 197 82 189 74 181)(67 174 198 83 190 75 182)(68 175 199 84 191 76 183)(69 176 200 85 192 77 184)(70 169 193 86 185 78 177)(71 170 194 87 186 79 178)(72 171 195 88 187 80 179)(89 117 220 105 212 97 204)(90 118 221 106 213 98 205)(91 119 222 107 214 99 206)(92 120 223 108 215 100 207)(93 113 224 109 216 101 208)(94 114 217 110 209 102 201)(95 115 218 111 210 103 202)(96 116 219 112 211 104 203)
(1 89 171 152)(2 90 172 145)(3 91 173 146)(4 92 174 147)(5 93 175 148)(6 94 176 149)(7 95 169 150)(8 96 170 151)(9 40 113 199)(10 33 114 200)(11 34 115 193)(12 35 116 194)(13 36 117 195)(14 37 118 196)(15 38 119 197)(16 39 120 198)(17 102 184 157)(18 103 177 158)(19 104 178 159)(20 97 179 160)(21 98 180 153)(22 99 181 154)(23 100 182 155)(24 101 183 156)(25 110 192 165)(26 111 185 166)(27 112 186 167)(28 105 187 168)(29 106 188 161)(30 107 189 162)(31 108 190 163)(32 109 191 164)(41 123 208 68)(42 124 201 69)(43 125 202 70)(44 126 203 71)(45 127 204 72)(46 128 205 65)(47 121 206 66)(48 122 207 67)(49 131 216 76)(50 132 209 77)(51 133 210 78)(52 134 211 79)(53 135 212 80)(54 136 213 73)(55 129 214 74)(56 130 215 75)(57 139 224 84)(58 140 217 85)(59 141 218 86)(60 142 219 87)(61 143 220 88)(62 144 221 81)(63 137 222 82)(64 138 223 83)
(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 96 171 151)(2 150 172 95)(3 94 173 149)(4 148 174 93)(5 92 175 147)(6 146 176 91)(7 90 169 145)(8 152 170 89)(9 198 113 39)(10 38 114 197)(11 196 115 37)(12 36 116 195)(13 194 117 35)(14 34 118 193)(15 200 119 33)(16 40 120 199)(17 154 184 99)(18 98 177 153)(19 160 178 97)(20 104 179 159)(21 158 180 103)(22 102 181 157)(23 156 182 101)(24 100 183 155)(25 162 192 107)(26 106 185 161)(27 168 186 105)(28 112 187 167)(29 166 188 111)(30 110 189 165)(31 164 190 109)(32 108 191 163)(41 67 208 122)(42 121 201 66)(43 65 202 128)(44 127 203 72)(45 71 204 126)(46 125 205 70)(47 69 206 124)(48 123 207 68)(49 75 216 130)(50 129 209 74)(51 73 210 136)(52 135 211 80)(53 79 212 134)(54 133 213 78)(55 77 214 132)(56 131 215 76)(57 83 224 138)(58 137 217 82)(59 81 218 144)(60 143 219 88)(61 87 220 142)(62 141 221 86)(63 85 222 140)(64 139 223 84)
G:=sub<Sym(224)| (1,36,143,28,135,20,127)(2,37,144,29,136,21,128)(3,38,137,30,129,22,121)(4,39,138,31,130,23,122)(5,40,139,32,131,24,123)(6,33,140,25,132,17,124)(7,34,141,26,133,18,125)(8,35,142,27,134,19,126)(9,57,164,49,156,41,148)(10,58,165,50,157,42,149)(11,59,166,51,158,43,150)(12,60,167,52,159,44,151)(13,61,168,53,160,45,152)(14,62,161,54,153,46,145)(15,63,162,55,154,47,146)(16,64,163,56,155,48,147)(65,172,196,81,188,73,180)(66,173,197,82,189,74,181)(67,174,198,83,190,75,182)(68,175,199,84,191,76,183)(69,176,200,85,192,77,184)(70,169,193,86,185,78,177)(71,170,194,87,186,79,178)(72,171,195,88,187,80,179)(89,117,220,105,212,97,204)(90,118,221,106,213,98,205)(91,119,222,107,214,99,206)(92,120,223,108,215,100,207)(93,113,224,109,216,101,208)(94,114,217,110,209,102,201)(95,115,218,111,210,103,202)(96,116,219,112,211,104,203), (1,89,171,152)(2,90,172,145)(3,91,173,146)(4,92,174,147)(5,93,175,148)(6,94,176,149)(7,95,169,150)(8,96,170,151)(9,40,113,199)(10,33,114,200)(11,34,115,193)(12,35,116,194)(13,36,117,195)(14,37,118,196)(15,38,119,197)(16,39,120,198)(17,102,184,157)(18,103,177,158)(19,104,178,159)(20,97,179,160)(21,98,180,153)(22,99,181,154)(23,100,182,155)(24,101,183,156)(25,110,192,165)(26,111,185,166)(27,112,186,167)(28,105,187,168)(29,106,188,161)(30,107,189,162)(31,108,190,163)(32,109,191,164)(41,123,208,68)(42,124,201,69)(43,125,202,70)(44,126,203,71)(45,127,204,72)(46,128,205,65)(47,121,206,66)(48,122,207,67)(49,131,216,76)(50,132,209,77)(51,133,210,78)(52,134,211,79)(53,135,212,80)(54,136,213,73)(55,129,214,74)(56,130,215,75)(57,139,224,84)(58,140,217,85)(59,141,218,86)(60,142,219,87)(61,143,220,88)(62,144,221,81)(63,137,222,82)(64,138,223,83), (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), (1,96,171,151)(2,150,172,95)(3,94,173,149)(4,148,174,93)(5,92,175,147)(6,146,176,91)(7,90,169,145)(8,152,170,89)(9,198,113,39)(10,38,114,197)(11,196,115,37)(12,36,116,195)(13,194,117,35)(14,34,118,193)(15,200,119,33)(16,40,120,199)(17,154,184,99)(18,98,177,153)(19,160,178,97)(20,104,179,159)(21,158,180,103)(22,102,181,157)(23,156,182,101)(24,100,183,155)(25,162,192,107)(26,106,185,161)(27,168,186,105)(28,112,187,167)(29,166,188,111)(30,110,189,165)(31,164,190,109)(32,108,191,163)(41,67,208,122)(42,121,201,66)(43,65,202,128)(44,127,203,72)(45,71,204,126)(46,125,205,70)(47,69,206,124)(48,123,207,68)(49,75,216,130)(50,129,209,74)(51,73,210,136)(52,135,211,80)(53,79,212,134)(54,133,213,78)(55,77,214,132)(56,131,215,76)(57,83,224,138)(58,137,217,82)(59,81,218,144)(60,143,219,88)(61,87,220,142)(62,141,221,86)(63,85,222,140)(64,139,223,84)>;
G:=Group( (1,36,143,28,135,20,127)(2,37,144,29,136,21,128)(3,38,137,30,129,22,121)(4,39,138,31,130,23,122)(5,40,139,32,131,24,123)(6,33,140,25,132,17,124)(7,34,141,26,133,18,125)(8,35,142,27,134,19,126)(9,57,164,49,156,41,148)(10,58,165,50,157,42,149)(11,59,166,51,158,43,150)(12,60,167,52,159,44,151)(13,61,168,53,160,45,152)(14,62,161,54,153,46,145)(15,63,162,55,154,47,146)(16,64,163,56,155,48,147)(65,172,196,81,188,73,180)(66,173,197,82,189,74,181)(67,174,198,83,190,75,182)(68,175,199,84,191,76,183)(69,176,200,85,192,77,184)(70,169,193,86,185,78,177)(71,170,194,87,186,79,178)(72,171,195,88,187,80,179)(89,117,220,105,212,97,204)(90,118,221,106,213,98,205)(91,119,222,107,214,99,206)(92,120,223,108,215,100,207)(93,113,224,109,216,101,208)(94,114,217,110,209,102,201)(95,115,218,111,210,103,202)(96,116,219,112,211,104,203), (1,89,171,152)(2,90,172,145)(3,91,173,146)(4,92,174,147)(5,93,175,148)(6,94,176,149)(7,95,169,150)(8,96,170,151)(9,40,113,199)(10,33,114,200)(11,34,115,193)(12,35,116,194)(13,36,117,195)(14,37,118,196)(15,38,119,197)(16,39,120,198)(17,102,184,157)(18,103,177,158)(19,104,178,159)(20,97,179,160)(21,98,180,153)(22,99,181,154)(23,100,182,155)(24,101,183,156)(25,110,192,165)(26,111,185,166)(27,112,186,167)(28,105,187,168)(29,106,188,161)(30,107,189,162)(31,108,190,163)(32,109,191,164)(41,123,208,68)(42,124,201,69)(43,125,202,70)(44,126,203,71)(45,127,204,72)(46,128,205,65)(47,121,206,66)(48,122,207,67)(49,131,216,76)(50,132,209,77)(51,133,210,78)(52,134,211,79)(53,135,212,80)(54,136,213,73)(55,129,214,74)(56,130,215,75)(57,139,224,84)(58,140,217,85)(59,141,218,86)(60,142,219,87)(61,143,220,88)(62,144,221,81)(63,137,222,82)(64,138,223,83), (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), (1,96,171,151)(2,150,172,95)(3,94,173,149)(4,148,174,93)(5,92,175,147)(6,146,176,91)(7,90,169,145)(8,152,170,89)(9,198,113,39)(10,38,114,197)(11,196,115,37)(12,36,116,195)(13,194,117,35)(14,34,118,193)(15,200,119,33)(16,40,120,199)(17,154,184,99)(18,98,177,153)(19,160,178,97)(20,104,179,159)(21,158,180,103)(22,102,181,157)(23,156,182,101)(24,100,183,155)(25,162,192,107)(26,106,185,161)(27,168,186,105)(28,112,187,167)(29,166,188,111)(30,110,189,165)(31,164,190,109)(32,108,191,163)(41,67,208,122)(42,121,201,66)(43,65,202,128)(44,127,203,72)(45,71,204,126)(46,125,205,70)(47,69,206,124)(48,123,207,68)(49,75,216,130)(50,129,209,74)(51,73,210,136)(52,135,211,80)(53,79,212,134)(54,133,213,78)(55,77,214,132)(56,131,215,76)(57,83,224,138)(58,137,217,82)(59,81,218,144)(60,143,219,88)(61,87,220,142)(62,141,221,86)(63,85,222,140)(64,139,223,84) );
G=PermutationGroup([[(1,36,143,28,135,20,127),(2,37,144,29,136,21,128),(3,38,137,30,129,22,121),(4,39,138,31,130,23,122),(5,40,139,32,131,24,123),(6,33,140,25,132,17,124),(7,34,141,26,133,18,125),(8,35,142,27,134,19,126),(9,57,164,49,156,41,148),(10,58,165,50,157,42,149),(11,59,166,51,158,43,150),(12,60,167,52,159,44,151),(13,61,168,53,160,45,152),(14,62,161,54,153,46,145),(15,63,162,55,154,47,146),(16,64,163,56,155,48,147),(65,172,196,81,188,73,180),(66,173,197,82,189,74,181),(67,174,198,83,190,75,182),(68,175,199,84,191,76,183),(69,176,200,85,192,77,184),(70,169,193,86,185,78,177),(71,170,194,87,186,79,178),(72,171,195,88,187,80,179),(89,117,220,105,212,97,204),(90,118,221,106,213,98,205),(91,119,222,107,214,99,206),(92,120,223,108,215,100,207),(93,113,224,109,216,101,208),(94,114,217,110,209,102,201),(95,115,218,111,210,103,202),(96,116,219,112,211,104,203)], [(1,89,171,152),(2,90,172,145),(3,91,173,146),(4,92,174,147),(5,93,175,148),(6,94,176,149),(7,95,169,150),(8,96,170,151),(9,40,113,199),(10,33,114,200),(11,34,115,193),(12,35,116,194),(13,36,117,195),(14,37,118,196),(15,38,119,197),(16,39,120,198),(17,102,184,157),(18,103,177,158),(19,104,178,159),(20,97,179,160),(21,98,180,153),(22,99,181,154),(23,100,182,155),(24,101,183,156),(25,110,192,165),(26,111,185,166),(27,112,186,167),(28,105,187,168),(29,106,188,161),(30,107,189,162),(31,108,190,163),(32,109,191,164),(41,123,208,68),(42,124,201,69),(43,125,202,70),(44,126,203,71),(45,127,204,72),(46,128,205,65),(47,121,206,66),(48,122,207,67),(49,131,216,76),(50,132,209,77),(51,133,210,78),(52,134,211,79),(53,135,212,80),(54,136,213,73),(55,129,214,74),(56,130,215,75),(57,139,224,84),(58,140,217,85),(59,141,218,86),(60,142,219,87),(61,143,220,88),(62,144,221,81),(63,137,222,82),(64,138,223,83)], [(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)], [(1,96,171,151),(2,150,172,95),(3,94,173,149),(4,148,174,93),(5,92,175,147),(6,146,176,91),(7,90,169,145),(8,152,170,89),(9,198,113,39),(10,38,114,197),(11,196,115,37),(12,36,116,195),(13,194,117,35),(14,34,118,193),(15,200,119,33),(16,40,120,199),(17,154,184,99),(18,98,177,153),(19,160,178,97),(20,104,179,159),(21,158,180,103),(22,102,181,157),(23,156,182,101),(24,100,183,155),(25,162,192,107),(26,106,185,161),(27,168,186,105),(28,112,187,167),(29,166,188,111),(30,110,189,165),(31,164,190,109),(32,108,191,163),(41,67,208,122),(42,121,201,66),(43,65,202,128),(44,127,203,72),(45,71,204,126),(46,125,205,70),(47,69,206,124),(48,123,207,68),(49,75,216,130),(50,129,209,74),(51,73,210,136),(52,135,211,80),(53,79,212,134),(54,133,213,78),(55,77,214,132),(56,131,215,76),(57,83,224,138),(58,137,217,82),(59,81,218,144),(60,143,219,88),(61,87,220,142),(62,141,221,86),(63,85,222,140),(64,139,223,84)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28AJ | 28AK | ··· | 28AV | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | D4 | D8 | SD16 | C4○D4 | C7×D4 | C7×D8 | C7×SD16 | C7×C4○D4 |
| kernel | C7×C4.4D8 | C4×C56 | C7×D4⋊C4 | C7×C4⋊1D4 | C7×C4⋊Q8 | C4.4D8 | C4×C8 | D4⋊C4 | C4⋊1D4 | C4⋊Q8 | C2×C28 | C28 | C28 | C28 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 6 | 6 | 24 | 6 | 6 | 2 | 4 | 4 | 4 | 12 | 24 | 24 | 24 |
Matrix representation of C7×C4.4D8 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 26 | 83 | 0 | 0 |
| 64 | 87 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 100 | 100 |
| 0 | 0 | 13 | 100 |
| 51 | 2 | 0 | 0 |
| 55 | 62 | 0 | 0 |
| 0 | 0 | 13 | 100 |
| 0 | 0 | 100 | 100 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,30,0,0,0,0,30],[26,64,0,0,83,87,0,0,0,0,0,112,0,0,1,0],[15,0,0,0,0,15,0,0,0,0,100,13,0,0,100,100],[51,55,0,0,2,62,0,0,0,0,13,100,0,0,100,100] >;
C7×C4.4D8 in GAP, Magma, Sage, TeX
C_7\times C_4._4D_8
% in TeX
G:=Group("C7xC4.4D8"); // GroupNames label
G:=SmallGroup(448,894);
// by ID
G=gap.SmallGroup(448,894);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,792,2438,310,9804,172,14117,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^8=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations