direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D4.D7, C28⋊11SD16, C42.208D14, C7⋊5(C4×SD16), (C4×D4).7D7, D4.5(C4×D7), (D4×C28).8C2, C14.71(C4×D4), C4⋊C4.245D14, (C2×C28).255D4, (C4×Dic14)⋊19C2, Dic14⋊10(C2×C4), (C2×D4).192D14, C28.52(C4○D4), C14.90(C4○D8), C4.38(C4○D28), C14.Q16⋊44C2, (C4×C28).88C22, C28.23(C22×C4), C4.Dic14⋊44C2, C14.52(C2×SD16), (C2×C28).339C23, D4⋊Dic7.17C2, C2.4(D4.8D14), (D4×C14).234C22, C4⋊Dic7.328C22, (C2×Dic14).264C22, (C4×C7⋊C8)⋊9C2, C7⋊C8⋊15(C2×C4), C4.23(C2×C4×D7), C2.17(C4×C7⋊D4), C2.3(C2×D4.D7), (C7×D4).12(C2×C4), (C2×C14).470(C2×D4), (C2×C7⋊C8).245C22, (C2×D4.D7).10C2, C22.77(C2×C7⋊D4), (C2×C4).102(C7⋊D4), (C7×C4⋊C4).276C22, (C2×C4).439(C22×D7), SmallGroup(448,551)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D4.D7
G = < a,b,c,d,e | a4=b4=c2=d7=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 436 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C28, C2×C14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C7⋊C8, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C4×SD16, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D4.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C22×C28, D4×C14, C4×C7⋊C8, C4.Dic14, C14.Q16, D4⋊Dic7, C4×Dic14, C2×D4.D7, D4×C28, C4×D4.D7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, SD16, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×SD16, C4○D8, C4×D7, C7⋊D4, C22×D7, C4×SD16, D4.D7, C2×C4×D7, C4○D28, C2×C7⋊D4, C4×C7⋊D4, C2×D4.D7, D4.8D14, C4×D4.D7
►On 224 points
Generators in S224
(1 85 29 57)(2 86 30 58)(3 87 31 59)(4 88 32 60)(5 89 33 61)(6 90 34 62)(7 91 35 63)(8 92 36 64)(9 93 37 65)(10 94 38 66)(11 95 39 67)(12 96 40 68)(13 97 41 69)(14 98 42 70)(15 99 43 71)(16 100 44 72)(17 101 45 73)(18 102 46 74)(19 103 47 75)(20 104 48 76)(21 105 49 77)(22 106 50 78)(23 107 51 79)(24 108 52 80)(25 109 53 81)(26 110 54 82)(27 111 55 83)(28 112 56 84)(113 197 141 169)(114 198 142 170)(115 199 143 171)(116 200 144 172)(117 201 145 173)(118 202 146 174)(119 203 147 175)(120 204 148 176)(121 205 149 177)(122 206 150 178)(123 207 151 179)(124 208 152 180)(125 209 153 181)(126 210 154 182)(127 211 155 183)(128 212 156 184)(129 213 157 185)(130 214 158 186)(131 215 159 187)(132 216 160 188)(133 217 161 189)(134 218 162 190)(135 219 163 191)(136 220 164 192)(137 221 165 193)(138 222 166 194)(139 223 167 195)(140 224 168 196)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)(113 127 120 134)(114 128 121 135)(115 129 122 136)(116 130 123 137)(117 131 124 138)(118 132 125 139)(119 133 126 140)(141 155 148 162)(142 156 149 163)(143 157 150 164)(144 158 151 165)(145 159 152 166)(146 160 153 167)(147 161 154 168)(169 183 176 190)(170 184 177 191)(171 185 178 192)(172 186 179 193)(173 187 180 194)(174 188 181 195)(175 189 182 196)(197 211 204 218)(198 212 205 219)(199 213 206 220)(200 214 207 221)(201 215 208 222)(202 216 209 223)(203 217 210 224)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)
(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)
(1 124 8 117)(2 123 9 116)(3 122 10 115)(4 121 11 114)(5 120 12 113)(6 126 13 119)(7 125 14 118)(15 138 22 131)(16 137 23 130)(17 136 24 129)(18 135 25 128)(19 134 26 127)(20 140 27 133)(21 139 28 132)(29 152 36 145)(30 151 37 144)(31 150 38 143)(32 149 39 142)(33 148 40 141)(34 154 41 147)(35 153 42 146)(43 166 50 159)(44 165 51 158)(45 164 52 157)(46 163 53 156)(47 162 54 155)(48 168 55 161)(49 167 56 160)(57 180 64 173)(58 179 65 172)(59 178 66 171)(60 177 67 170)(61 176 68 169)(62 182 69 175)(63 181 70 174)(71 194 78 187)(72 193 79 186)(73 192 80 185)(74 191 81 184)(75 190 82 183)(76 196 83 189)(77 195 84 188)(85 208 92 201)(86 207 93 200)(87 206 94 199)(88 205 95 198)(89 204 96 197)(90 210 97 203)(91 209 98 202)(99 222 106 215)(100 221 107 214)(101 220 108 213)(102 219 109 212)(103 218 110 211)(104 224 111 217)(105 223 112 216)
G:=sub<Sym(224)| (1,85,29,57)(2,86,30,58)(3,87,31,59)(4,88,32,60)(5,89,33,61)(6,90,34,62)(7,91,35,63)(8,92,36,64)(9,93,37,65)(10,94,38,66)(11,95,39,67)(12,96,40,68)(13,97,41,69)(14,98,42,70)(15,99,43,71)(16,100,44,72)(17,101,45,73)(18,102,46,74)(19,103,47,75)(20,104,48,76)(21,105,49,77)(22,106,50,78)(23,107,51,79)(24,108,52,80)(25,109,53,81)(26,110,54,82)(27,111,55,83)(28,112,56,84)(113,197,141,169)(114,198,142,170)(115,199,143,171)(116,200,144,172)(117,201,145,173)(118,202,146,174)(119,203,147,175)(120,204,148,176)(121,205,149,177)(122,206,150,178)(123,207,151,179)(124,208,152,180)(125,209,153,181)(126,210,154,182)(127,211,155,183)(128,212,156,184)(129,213,157,185)(130,214,158,186)(131,215,159,187)(132,216,160,188)(133,217,161,189)(134,218,162,190)(135,219,163,191)(136,220,164,192)(137,221,165,193)(138,222,166,194)(139,223,167,195)(140,224,168,196), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(113,127,120,134)(114,128,121,135)(115,129,122,136)(116,130,123,137)(117,131,124,138)(118,132,125,139)(119,133,126,140)(141,155,148,162)(142,156,149,163)(143,157,150,164)(144,158,151,165)(145,159,152,166)(146,160,153,167)(147,161,154,168)(169,183,176,190)(170,184,177,191)(171,185,178,192)(172,186,179,193)(173,187,180,194)(174,188,181,195)(175,189,182,196)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (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), (1,124,8,117)(2,123,9,116)(3,122,10,115)(4,121,11,114)(5,120,12,113)(6,126,13,119)(7,125,14,118)(15,138,22,131)(16,137,23,130)(17,136,24,129)(18,135,25,128)(19,134,26,127)(20,140,27,133)(21,139,28,132)(29,152,36,145)(30,151,37,144)(31,150,38,143)(32,149,39,142)(33,148,40,141)(34,154,41,147)(35,153,42,146)(43,166,50,159)(44,165,51,158)(45,164,52,157)(46,163,53,156)(47,162,54,155)(48,168,55,161)(49,167,56,160)(57,180,64,173)(58,179,65,172)(59,178,66,171)(60,177,67,170)(61,176,68,169)(62,182,69,175)(63,181,70,174)(71,194,78,187)(72,193,79,186)(73,192,80,185)(74,191,81,184)(75,190,82,183)(76,196,83,189)(77,195,84,188)(85,208,92,201)(86,207,93,200)(87,206,94,199)(88,205,95,198)(89,204,96,197)(90,210,97,203)(91,209,98,202)(99,222,106,215)(100,221,107,214)(101,220,108,213)(102,219,109,212)(103,218,110,211)(104,224,111,217)(105,223,112,216)>;
G:=Group( (1,85,29,57)(2,86,30,58)(3,87,31,59)(4,88,32,60)(5,89,33,61)(6,90,34,62)(7,91,35,63)(8,92,36,64)(9,93,37,65)(10,94,38,66)(11,95,39,67)(12,96,40,68)(13,97,41,69)(14,98,42,70)(15,99,43,71)(16,100,44,72)(17,101,45,73)(18,102,46,74)(19,103,47,75)(20,104,48,76)(21,105,49,77)(22,106,50,78)(23,107,51,79)(24,108,52,80)(25,109,53,81)(26,110,54,82)(27,111,55,83)(28,112,56,84)(113,197,141,169)(114,198,142,170)(115,199,143,171)(116,200,144,172)(117,201,145,173)(118,202,146,174)(119,203,147,175)(120,204,148,176)(121,205,149,177)(122,206,150,178)(123,207,151,179)(124,208,152,180)(125,209,153,181)(126,210,154,182)(127,211,155,183)(128,212,156,184)(129,213,157,185)(130,214,158,186)(131,215,159,187)(132,216,160,188)(133,217,161,189)(134,218,162,190)(135,219,163,191)(136,220,164,192)(137,221,165,193)(138,222,166,194)(139,223,167,195)(140,224,168,196), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(113,127,120,134)(114,128,121,135)(115,129,122,136)(116,130,123,137)(117,131,124,138)(118,132,125,139)(119,133,126,140)(141,155,148,162)(142,156,149,163)(143,157,150,164)(144,158,151,165)(145,159,152,166)(146,160,153,167)(147,161,154,168)(169,183,176,190)(170,184,177,191)(171,185,178,192)(172,186,179,193)(173,187,180,194)(174,188,181,195)(175,189,182,196)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (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), (1,124,8,117)(2,123,9,116)(3,122,10,115)(4,121,11,114)(5,120,12,113)(6,126,13,119)(7,125,14,118)(15,138,22,131)(16,137,23,130)(17,136,24,129)(18,135,25,128)(19,134,26,127)(20,140,27,133)(21,139,28,132)(29,152,36,145)(30,151,37,144)(31,150,38,143)(32,149,39,142)(33,148,40,141)(34,154,41,147)(35,153,42,146)(43,166,50,159)(44,165,51,158)(45,164,52,157)(46,163,53,156)(47,162,54,155)(48,168,55,161)(49,167,56,160)(57,180,64,173)(58,179,65,172)(59,178,66,171)(60,177,67,170)(61,176,68,169)(62,182,69,175)(63,181,70,174)(71,194,78,187)(72,193,79,186)(73,192,80,185)(74,191,81,184)(75,190,82,183)(76,196,83,189)(77,195,84,188)(85,208,92,201)(86,207,93,200)(87,206,94,199)(88,205,95,198)(89,204,96,197)(90,210,97,203)(91,209,98,202)(99,222,106,215)(100,221,107,214)(101,220,108,213)(102,219,109,212)(103,218,110,211)(104,224,111,217)(105,223,112,216) );
G=PermutationGroup([[(1,85,29,57),(2,86,30,58),(3,87,31,59),(4,88,32,60),(5,89,33,61),(6,90,34,62),(7,91,35,63),(8,92,36,64),(9,93,37,65),(10,94,38,66),(11,95,39,67),(12,96,40,68),(13,97,41,69),(14,98,42,70),(15,99,43,71),(16,100,44,72),(17,101,45,73),(18,102,46,74),(19,103,47,75),(20,104,48,76),(21,105,49,77),(22,106,50,78),(23,107,51,79),(24,108,52,80),(25,109,53,81),(26,110,54,82),(27,111,55,83),(28,112,56,84),(113,197,141,169),(114,198,142,170),(115,199,143,171),(116,200,144,172),(117,201,145,173),(118,202,146,174),(119,203,147,175),(120,204,148,176),(121,205,149,177),(122,206,150,178),(123,207,151,179),(124,208,152,180),(125,209,153,181),(126,210,154,182),(127,211,155,183),(128,212,156,184),(129,213,157,185),(130,214,158,186),(131,215,159,187),(132,216,160,188),(133,217,161,189),(134,218,162,190),(135,219,163,191),(136,220,164,192),(137,221,165,193),(138,222,166,194),(139,223,167,195),(140,224,168,196)], [(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21),(29,50,36,43),(30,51,37,44),(31,52,38,45),(32,53,39,46),(33,54,40,47),(34,55,41,48),(35,56,42,49),(57,78,64,71),(58,79,65,72),(59,80,66,73),(60,81,67,74),(61,82,68,75),(62,83,69,76),(63,84,70,77),(85,106,92,99),(86,107,93,100),(87,108,94,101),(88,109,95,102),(89,110,96,103),(90,111,97,104),(91,112,98,105),(113,127,120,134),(114,128,121,135),(115,129,122,136),(116,130,123,137),(117,131,124,138),(118,132,125,139),(119,133,126,140),(141,155,148,162),(142,156,149,163),(143,157,150,164),(144,158,151,165),(145,159,152,166),(146,160,153,167),(147,161,154,168),(169,183,176,190),(170,184,177,191),(171,185,178,192),(172,186,179,193),(173,187,180,194),(174,188,181,195),(175,189,182,196),(197,211,204,218),(198,212,205,219),(199,213,206,220),(200,214,207,221),(201,215,208,222),(202,216,209,223),(203,217,210,224)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210)], [(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)], [(1,124,8,117),(2,123,9,116),(3,122,10,115),(4,121,11,114),(5,120,12,113),(6,126,13,119),(7,125,14,118),(15,138,22,131),(16,137,23,130),(17,136,24,129),(18,135,25,128),(19,134,26,127),(20,140,27,133),(21,139,28,132),(29,152,36,145),(30,151,37,144),(31,150,38,143),(32,149,39,142),(33,148,40,141),(34,154,41,147),(35,153,42,146),(43,166,50,159),(44,165,51,158),(45,164,52,157),(46,163,53,156),(47,162,54,155),(48,168,55,161),(49,167,56,160),(57,180,64,173),(58,179,65,172),(59,178,66,171),(60,177,67,170),(61,176,68,169),(62,182,69,175),(63,181,70,174),(71,194,78,187),(72,193,79,186),(73,192,80,185),(74,191,81,184),(75,190,82,183),(76,196,83,189),(77,195,84,188),(85,208,92,201),(86,207,93,200),(87,206,94,199),(88,205,95,198),(89,204,96,197),(90,210,97,203),(91,209,98,202),(99,222,106,215),(100,221,107,214),(101,220,108,213),(102,219,109,212),(103,218,110,211),(104,224,111,217),(105,223,112,216)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | SD16 | C4○D4 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C4×D7 | C4○D28 | D4.D7 | D4.8D14 |
| kernel | C4×D4.D7 | C4×C7⋊C8 | C4.Dic14 | C14.Q16 | D4⋊Dic7 | C4×Dic14 | C2×D4.D7 | D4×C28 | D4.D7 | C2×C28 | C4×D4 | C28 | C28 | C42 | C4⋊C4 | C2×D4 | C14 | C2×C4 | D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 4 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of C4×D4.D7 ►in GL4(𝔽113) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 111 |
| 0 | 0 | 1 | 112 |
| 112 | 69 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 111 |
| 0 | 0 | 0 | 112 |
| 28 | 26 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 67 | 89 | 0 | 0 |
| 74 | 46 | 0 | 0 |
| 0 | 0 | 0 | 87 |
| 0 | 0 | 100 | 0 |
G:=sub<GL(4,GF(113))| [15,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,1,1,0,0,111,112],[112,0,0,0,69,1,0,0,0,0,1,0,0,0,111,112],[28,0,0,0,26,109,0,0,0,0,1,0,0,0,0,1],[67,74,0,0,89,46,0,0,0,0,0,100,0,0,87,0] >;
C4×D4.D7 in GAP, Magma, Sage, TeX
C_4\times D_4.D_7
% in TeX
G:=Group("C4xD4.D7"); // GroupNames label
G:=SmallGroup(448,551);
// by ID
G=gap.SmallGroup(448,551);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,58,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^7=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations