direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8.D14, C56.8C23, C28.59C24, C23.53D28, M4(2)⋊18D14, Dic28⋊8C22, D28.22C23, Dic14.22C23, (C2×C4).58D28, C4.49(C2×D28), C8.8(C22×D7), (C2×C8).101D14, C28.293(C2×D4), (C2×C28).204D4, C56⋊C2⋊9C22, (C2×M4(2))⋊4D7, C4.56(C23×D7), (C2×Dic28)⋊14C2, C14⋊1(C8.C22), (C14×M4(2))⋊4C2, (C2×C56).69C22, C22.74(C2×D28), C2.28(C22×D28), C14.26(C22×D4), (C2×C28).512C23, C4○D28.50C22, (C22×C4).266D14, (C22×C14).119D4, (C2×Dic14)⋊63C22, (C22×Dic14)⋊18C2, (C2×D28).230C22, (C7×M4(2))⋊20C22, (C22×C28).267C22, C7⋊1(C2×C8.C22), (C2×C56⋊C2)⋊5C2, (C2×C14).63(C2×D4), (C2×C4○D28).23C2, (C2×C4).224(C22×D7), SmallGroup(448,1200)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1252 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×13], C23, C23, D7 [×2], C14, C14 [×2], C14 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×10], C4○D4 [×6], Dic7 [×6], C28 [×2], C28 [×2], D14 [×4], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C56 [×4], Dic14 [×6], Dic14 [×7], C4×D7 [×4], D28 [×2], D28, C2×Dic7 [×7], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C22×D7, C22×C14, C2×C8.C22, C56⋊C2 [×8], Dic28 [×8], C2×C56 [×2], C7×M4(2) [×4], C2×Dic14, C2×Dic14 [×6], C2×Dic14 [×3], C2×C4×D7, C2×D28, C4○D28 [×4], C4○D28 [×2], C22×Dic7, C2×C7⋊D4, C22×C28, C2×C56⋊C2 [×2], C2×Dic28 [×2], C8.D14 [×8], C14×M4(2), C22×Dic14, C2×C4○D28, C2×C8.D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8.C22 [×2], C22×D4, D28 [×4], C22×D7 [×7], C2×C8.C22, C2×D28 [×6], C23×D7, C8.D14 [×2], C22×D28, C2×C8.D14
Generators and relations
G = < a,b,c,d | a2=b8=1, c14=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c13 >
►On 224 points
Generators in S224
(1 84)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 112)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(113 174)(114 175)(115 176)(116 177)(117 178)(118 179)(119 180)(120 181)(121 182)(122 183)(123 184)(124 185)(125 186)(126 187)(127 188)(128 189)(129 190)(130 191)(131 192)(132 193)(133 194)(134 195)(135 196)(136 169)(137 170)(138 171)(139 172)(140 173)(141 202)(142 203)(143 204)(144 205)(145 206)(146 207)(147 208)(148 209)(149 210)(150 211)(151 212)(152 213)(153 214)(154 215)(155 216)(156 217)(157 218)(158 219)(159 220)(160 221)(161 222)(162 223)(163 224)(164 197)(165 198)(166 199)(167 200)(168 201)
(1 42 8 35 15 56 22 49)(2 29 9 50 16 43 23 36)(3 44 10 37 17 30 24 51)(4 31 11 52 18 45 25 38)(5 46 12 39 19 32 26 53)(6 33 13 54 20 47 27 40)(7 48 14 41 21 34 28 55)(57 112 64 105 71 98 78 91)(58 99 65 92 72 85 79 106)(59 86 66 107 73 100 80 93)(60 101 67 94 74 87 81 108)(61 88 68 109 75 102 82 95)(62 103 69 96 76 89 83 110)(63 90 70 111 77 104 84 97)(113 158 134 165 127 144 120 151)(114 145 135 152 128 159 121 166)(115 160 136 167 129 146 122 153)(116 147 137 154 130 161 123 168)(117 162 138 141 131 148 124 155)(118 149 139 156 132 163 125 142)(119 164 140 143 133 150 126 157)(169 200 190 207 183 214 176 221)(170 215 191 222 184 201 177 208)(171 202 192 209 185 216 178 223)(172 217 193 224 186 203 179 210)(173 204 194 211 187 218 180 197)(174 219 195 198 188 205 181 212)(175 206 196 213 189 220 182 199)
(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)
(1 125 15 139)(2 138 16 124)(3 123 17 137)(4 136 18 122)(5 121 19 135)(6 134 20 120)(7 119 21 133)(8 132 22 118)(9 117 23 131)(10 130 24 116)(11 115 25 129)(12 128 26 114)(13 113 27 127)(14 126 28 140)(29 162 43 148)(30 147 44 161)(31 160 45 146)(32 145 46 159)(33 158 47 144)(34 143 48 157)(35 156 49 142)(36 141 50 155)(37 154 51 168)(38 167 52 153)(39 152 53 166)(40 165 54 151)(41 150 55 164)(42 163 56 149)(57 171 71 185)(58 184 72 170)(59 169 73 183)(60 182 74 196)(61 195 75 181)(62 180 76 194)(63 193 77 179)(64 178 78 192)(65 191 79 177)(66 176 80 190)(67 189 81 175)(68 174 82 188)(69 187 83 173)(70 172 84 186)(85 208 99 222)(86 221 100 207)(87 206 101 220)(88 219 102 205)(89 204 103 218)(90 217 104 203)(91 202 105 216)(92 215 106 201)(93 200 107 214)(94 213 108 199)(95 198 109 212)(96 211 110 197)(97 224 111 210)(98 209 112 223)
G:=sub<Sym(224)| (1,84)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,112)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(113,174)(114,175)(115,176)(116,177)(117,178)(118,179)(119,180)(120,181)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(127,188)(128,189)(129,190)(130,191)(131,192)(132,193)(133,194)(134,195)(135,196)(136,169)(137,170)(138,171)(139,172)(140,173)(141,202)(142,203)(143,204)(144,205)(145,206)(146,207)(147,208)(148,209)(149,210)(150,211)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217)(157,218)(158,219)(159,220)(160,221)(161,222)(162,223)(163,224)(164,197)(165,198)(166,199)(167,200)(168,201), (1,42,8,35,15,56,22,49)(2,29,9,50,16,43,23,36)(3,44,10,37,17,30,24,51)(4,31,11,52,18,45,25,38)(5,46,12,39,19,32,26,53)(6,33,13,54,20,47,27,40)(7,48,14,41,21,34,28,55)(57,112,64,105,71,98,78,91)(58,99,65,92,72,85,79,106)(59,86,66,107,73,100,80,93)(60,101,67,94,74,87,81,108)(61,88,68,109,75,102,82,95)(62,103,69,96,76,89,83,110)(63,90,70,111,77,104,84,97)(113,158,134,165,127,144,120,151)(114,145,135,152,128,159,121,166)(115,160,136,167,129,146,122,153)(116,147,137,154,130,161,123,168)(117,162,138,141,131,148,124,155)(118,149,139,156,132,163,125,142)(119,164,140,143,133,150,126,157)(169,200,190,207,183,214,176,221)(170,215,191,222,184,201,177,208)(171,202,192,209,185,216,178,223)(172,217,193,224,186,203,179,210)(173,204,194,211,187,218,180,197)(174,219,195,198,188,205,181,212)(175,206,196,213,189,220,182,199), (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), (1,125,15,139)(2,138,16,124)(3,123,17,137)(4,136,18,122)(5,121,19,135)(6,134,20,120)(7,119,21,133)(8,132,22,118)(9,117,23,131)(10,130,24,116)(11,115,25,129)(12,128,26,114)(13,113,27,127)(14,126,28,140)(29,162,43,148)(30,147,44,161)(31,160,45,146)(32,145,46,159)(33,158,47,144)(34,143,48,157)(35,156,49,142)(36,141,50,155)(37,154,51,168)(38,167,52,153)(39,152,53,166)(40,165,54,151)(41,150,55,164)(42,163,56,149)(57,171,71,185)(58,184,72,170)(59,169,73,183)(60,182,74,196)(61,195,75,181)(62,180,76,194)(63,193,77,179)(64,178,78,192)(65,191,79,177)(66,176,80,190)(67,189,81,175)(68,174,82,188)(69,187,83,173)(70,172,84,186)(85,208,99,222)(86,221,100,207)(87,206,101,220)(88,219,102,205)(89,204,103,218)(90,217,104,203)(91,202,105,216)(92,215,106,201)(93,200,107,214)(94,213,108,199)(95,198,109,212)(96,211,110,197)(97,224,111,210)(98,209,112,223)>;
G:=Group( (1,84)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,112)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(113,174)(114,175)(115,176)(116,177)(117,178)(118,179)(119,180)(120,181)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(127,188)(128,189)(129,190)(130,191)(131,192)(132,193)(133,194)(134,195)(135,196)(136,169)(137,170)(138,171)(139,172)(140,173)(141,202)(142,203)(143,204)(144,205)(145,206)(146,207)(147,208)(148,209)(149,210)(150,211)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217)(157,218)(158,219)(159,220)(160,221)(161,222)(162,223)(163,224)(164,197)(165,198)(166,199)(167,200)(168,201), (1,42,8,35,15,56,22,49)(2,29,9,50,16,43,23,36)(3,44,10,37,17,30,24,51)(4,31,11,52,18,45,25,38)(5,46,12,39,19,32,26,53)(6,33,13,54,20,47,27,40)(7,48,14,41,21,34,28,55)(57,112,64,105,71,98,78,91)(58,99,65,92,72,85,79,106)(59,86,66,107,73,100,80,93)(60,101,67,94,74,87,81,108)(61,88,68,109,75,102,82,95)(62,103,69,96,76,89,83,110)(63,90,70,111,77,104,84,97)(113,158,134,165,127,144,120,151)(114,145,135,152,128,159,121,166)(115,160,136,167,129,146,122,153)(116,147,137,154,130,161,123,168)(117,162,138,141,131,148,124,155)(118,149,139,156,132,163,125,142)(119,164,140,143,133,150,126,157)(169,200,190,207,183,214,176,221)(170,215,191,222,184,201,177,208)(171,202,192,209,185,216,178,223)(172,217,193,224,186,203,179,210)(173,204,194,211,187,218,180,197)(174,219,195,198,188,205,181,212)(175,206,196,213,189,220,182,199), (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), (1,125,15,139)(2,138,16,124)(3,123,17,137)(4,136,18,122)(5,121,19,135)(6,134,20,120)(7,119,21,133)(8,132,22,118)(9,117,23,131)(10,130,24,116)(11,115,25,129)(12,128,26,114)(13,113,27,127)(14,126,28,140)(29,162,43,148)(30,147,44,161)(31,160,45,146)(32,145,46,159)(33,158,47,144)(34,143,48,157)(35,156,49,142)(36,141,50,155)(37,154,51,168)(38,167,52,153)(39,152,53,166)(40,165,54,151)(41,150,55,164)(42,163,56,149)(57,171,71,185)(58,184,72,170)(59,169,73,183)(60,182,74,196)(61,195,75,181)(62,180,76,194)(63,193,77,179)(64,178,78,192)(65,191,79,177)(66,176,80,190)(67,189,81,175)(68,174,82,188)(69,187,83,173)(70,172,84,186)(85,208,99,222)(86,221,100,207)(87,206,101,220)(88,219,102,205)(89,204,103,218)(90,217,104,203)(91,202,105,216)(92,215,106,201)(93,200,107,214)(94,213,108,199)(95,198,109,212)(96,211,110,197)(97,224,111,210)(98,209,112,223) );
G=PermutationGroup([(1,84),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,112),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(113,174),(114,175),(115,176),(116,177),(117,178),(118,179),(119,180),(120,181),(121,182),(122,183),(123,184),(124,185),(125,186),(126,187),(127,188),(128,189),(129,190),(130,191),(131,192),(132,193),(133,194),(134,195),(135,196),(136,169),(137,170),(138,171),(139,172),(140,173),(141,202),(142,203),(143,204),(144,205),(145,206),(146,207),(147,208),(148,209),(149,210),(150,211),(151,212),(152,213),(153,214),(154,215),(155,216),(156,217),(157,218),(158,219),(159,220),(160,221),(161,222),(162,223),(163,224),(164,197),(165,198),(166,199),(167,200),(168,201)], [(1,42,8,35,15,56,22,49),(2,29,9,50,16,43,23,36),(3,44,10,37,17,30,24,51),(4,31,11,52,18,45,25,38),(5,46,12,39,19,32,26,53),(6,33,13,54,20,47,27,40),(7,48,14,41,21,34,28,55),(57,112,64,105,71,98,78,91),(58,99,65,92,72,85,79,106),(59,86,66,107,73,100,80,93),(60,101,67,94,74,87,81,108),(61,88,68,109,75,102,82,95),(62,103,69,96,76,89,83,110),(63,90,70,111,77,104,84,97),(113,158,134,165,127,144,120,151),(114,145,135,152,128,159,121,166),(115,160,136,167,129,146,122,153),(116,147,137,154,130,161,123,168),(117,162,138,141,131,148,124,155),(118,149,139,156,132,163,125,142),(119,164,140,143,133,150,126,157),(169,200,190,207,183,214,176,221),(170,215,191,222,184,201,177,208),(171,202,192,209,185,216,178,223),(172,217,193,224,186,203,179,210),(173,204,194,211,187,218,180,197),(174,219,195,198,188,205,181,212),(175,206,196,213,189,220,182,199)], [(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)], [(1,125,15,139),(2,138,16,124),(3,123,17,137),(4,136,18,122),(5,121,19,135),(6,134,20,120),(7,119,21,133),(8,132,22,118),(9,117,23,131),(10,130,24,116),(11,115,25,129),(12,128,26,114),(13,113,27,127),(14,126,28,140),(29,162,43,148),(30,147,44,161),(31,160,45,146),(32,145,46,159),(33,158,47,144),(34,143,48,157),(35,156,49,142),(36,141,50,155),(37,154,51,168),(38,167,52,153),(39,152,53,166),(40,165,54,151),(41,150,55,164),(42,163,56,149),(57,171,71,185),(58,184,72,170),(59,169,73,183),(60,182,74,196),(61,195,75,181),(62,180,76,194),(63,193,77,179),(64,178,78,192),(65,191,79,177),(66,176,80,190),(67,189,81,175),(68,174,82,188),(69,187,83,173),(70,172,84,186),(85,208,99,222),(86,221,100,207),(87,206,101,220),(88,219,102,205),(89,204,103,218),(90,217,104,203),(91,202,105,216),(92,215,106,201),(93,200,107,214),(94,213,108,199),(95,198,109,212),(96,211,110,197),(97,224,111,210),(98,209,112,223)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 104 | 46 | 0 | 0 | 0 | 0 |
| 67 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 69 | 0 |
| 0 | 0 | 0 | 37 | 0 | 36 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 31 | 0 | 76 |
| 34 | 25 | 0 | 0 | 0 | 0 |
| 88 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 44 | 0 | 0 |
| 0 | 0 | 77 | 1 | 0 | 0 |
| 0 | 0 | 36 | 36 | 1 | 36 |
| 0 | 0 | 36 | 67 | 69 | 112 |
| 18 | 42 | 0 | 0 | 0 | 0 |
| 108 | 95 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 66 | 58 | 52 |
| 0 | 0 | 36 | 95 | 2 | 68 |
| 0 | 0 | 0 | 29 | 86 | 81 |
| 0 | 0 | 75 | 104 | 67 | 18 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[104,67,0,0,0,0,46,9,0,0,0,0,0,0,112,0,0,1,0,0,0,37,1,31,0,0,69,0,1,0,0,0,0,36,0,76],[34,88,0,0,0,0,25,88,0,0,0,0,0,0,112,77,36,36,0,0,44,1,36,67,0,0,0,0,1,69,0,0,0,0,36,112],[18,108,0,0,0,0,42,95,0,0,0,0,0,0,27,36,0,75,0,0,66,95,29,104,0,0,58,2,86,67,0,0,52,68,81,18] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 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 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | C8.C22 | C8.D14 |
| kernel | C2×C8.D14 | C2×C56⋊C2 | C2×Dic28 | C8.D14 | C14×M4(2) | C22×Dic14 | C2×C4○D28 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 3 | 1 | 3 | 6 | 12 | 3 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_2\times C_8.D_{14} % in TeX
G:=Group("C2xC8.D14"); // GroupNames label
G:=SmallGroup(448,1200);
// by ID
G=gap.SmallGroup(448,1200);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,297,80,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^14=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^13>;
// generators/relations
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