direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4×Dic7, C24.57D14, C14⋊5(C4×D4), (D4×C14)⋊11C4, C28⋊4(C22×C4), C4⋊1(C22×Dic7), C23⋊4(C2×Dic7), (C2×D4).250D14, C14.44(C23×C4), C4⋊Dic7⋊75C22, (C23×Dic7)⋊7C2, (C22×D4).13D7, C22.145(D4×D7), C2.6(C23×Dic7), (C2×C28).539C23, (C2×C14).290C24, (C4×Dic7)⋊66C22, (C22×C4).377D14, C14.128(C22×D4), C23.D7⋊56C22, C22⋊1(C22×Dic7), C22.44(C23×D7), (D4×C14).268C22, (C23×C14).72C22, C23.203(C22×D7), C22.76(D4⋊2D7), (C22×C14).226C23, (C22×C28).272C22, (C2×Dic7).280C23, (C22×Dic7)⋊47C22, C7⋊6(C2×C4×D4), C2.6(C2×D4×D7), (D4×C2×C14).7C2, (C2×C28)⋊14(C2×C4), (C7×D4)⋊19(C2×C4), (C2×C4×Dic7)⋊10C2, (C2×C4)⋊7(C2×Dic7), (C2×C4⋊Dic7)⋊44C2, (C2×C14)⋊4(C22×C4), C2.6(C2×D4⋊2D7), (C22×C14)⋊11(C2×C4), C14.102(C2×C4○D4), (C2×C14).404(C2×D4), (C2×C23.D7)⋊23C2, (C2×C4).622(C22×D7), (C2×C14).174(C4○D4), SmallGroup(448,1248)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4×Dic7
G = < a,b,c,d,e | a2=b4=c2=d14=1, e2=d7, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 1364 in 426 conjugacy classes, 215 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C4×D4, C4×Dic7, C4⋊Dic7, C23.D7, C22×Dic7, C22×Dic7, C22×Dic7, C22×C28, D4×C14, C23×C14, C2×C4×Dic7, C2×C4⋊Dic7, D4×Dic7, C2×C23.D7, C23×Dic7, D4×C2×C14, C2×D4×Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, C24, Dic7, D14, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×Dic7, C22×D7, C2×C4×D4, D4×D7, D4⋊2D7, C22×Dic7, C23×D7, D4×Dic7, C2×D4×D7, C2×D4⋊2D7, C23×Dic7, C2×D4×Dic7
►On 224 points
Generators in S224
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 126)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 121)(25 122)(26 123)(27 124)(28 125)(29 189)(30 190)(31 191)(32 192)(33 193)(34 194)(35 195)(36 196)(37 183)(38 184)(39 185)(40 186)(41 187)(42 188)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 71)(56 72)(85 178)(86 179)(87 180)(88 181)(89 182)(90 169)(91 170)(92 171)(93 172)(94 173)(95 174)(96 175)(97 176)(98 177)(99 222)(100 223)(101 224)(102 211)(103 212)(104 213)(105 214)(106 215)(107 216)(108 217)(109 218)(110 219)(111 220)(112 221)(127 151)(128 152)(129 153)(130 154)(131 141)(132 142)(133 143)(134 144)(135 145)(136 146)(137 147)(138 148)(139 149)(140 150)(155 200)(156 201)(157 202)(158 203)(159 204)(160 205)(161 206)(162 207)(163 208)(164 209)(165 210)(166 197)(167 198)(168 199)
(1 51 32 198)(2 52 33 199)(3 53 34 200)(4 54 35 201)(5 55 36 202)(6 56 37 203)(7 43 38 204)(8 44 39 205)(9 45 40 206)(10 46 41 207)(11 47 42 208)(12 48 29 209)(13 49 30 210)(14 50 31 197)(15 153 177 219)(16 154 178 220)(17 141 179 221)(18 142 180 222)(19 143 181 223)(20 144 182 224)(21 145 169 211)(22 146 170 212)(23 147 171 213)(24 148 172 214)(25 149 173 215)(26 150 174 216)(27 151 175 217)(28 152 176 218)(57 74 185 160)(58 75 186 161)(59 76 187 162)(60 77 188 163)(61 78 189 164)(62 79 190 165)(63 80 191 166)(64 81 192 167)(65 82 193 168)(66 83 194 155)(67 84 195 156)(68 71 196 157)(69 72 183 158)(70 73 184 159)(85 111 113 130)(86 112 114 131)(87 99 115 132)(88 100 116 133)(89 101 117 134)(90 102 118 135)(91 103 119 136)(92 104 120 137)(93 105 121 138)(94 106 122 139)(95 107 123 140)(96 108 124 127)(97 109 125 128)(98 110 126 129)
(43 204)(44 205)(45 206)(46 207)(47 208)(48 209)(49 210)(50 197)(51 198)(52 199)(53 200)(54 201)(55 202)(56 203)(71 157)(72 158)(73 159)(74 160)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(81 167)(82 168)(83 155)(84 156)(99 132)(100 133)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 127)(109 128)(110 129)(111 130)(112 131)(141 221)(142 222)(143 223)(144 224)(145 211)(146 212)(147 213)(148 214)(149 215)(150 216)(151 217)(152 218)(153 219)(154 220)
(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)
(1 178 8 171)(2 177 9 170)(3 176 10 169)(4 175 11 182)(5 174 12 181)(6 173 13 180)(7 172 14 179)(15 40 22 33)(16 39 23 32)(17 38 24 31)(18 37 25 30)(19 36 26 29)(20 35 27 42)(21 34 28 41)(43 214 50 221)(44 213 51 220)(45 212 52 219)(46 211 53 218)(47 224 54 217)(48 223 55 216)(49 222 56 215)(57 92 64 85)(58 91 65 98)(59 90 66 97)(60 89 67 96)(61 88 68 95)(62 87 69 94)(63 86 70 93)(71 107 78 100)(72 106 79 99)(73 105 80 112)(74 104 81 111)(75 103 82 110)(76 102 83 109)(77 101 84 108)(113 185 120 192)(114 184 121 191)(115 183 122 190)(116 196 123 189)(117 195 124 188)(118 194 125 187)(119 193 126 186)(127 163 134 156)(128 162 135 155)(129 161 136 168)(130 160 137 167)(131 159 138 166)(132 158 139 165)(133 157 140 164)(141 204 148 197)(142 203 149 210)(143 202 150 209)(144 201 151 208)(145 200 152 207)(146 199 153 206)(147 198 154 205)
G:=sub<Sym(224)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,126)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,124)(28,125)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,183)(38,184)(39,185)(40,186)(41,187)(42,188)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,71)(56,72)(85,178)(86,179)(87,180)(88,181)(89,182)(90,169)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,222)(100,223)(101,224)(102,211)(103,212)(104,213)(105,214)(106,215)(107,216)(108,217)(109,218)(110,219)(111,220)(112,221)(127,151)(128,152)(129,153)(130,154)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,197)(167,198)(168,199), (1,51,32,198)(2,52,33,199)(3,53,34,200)(4,54,35,201)(5,55,36,202)(6,56,37,203)(7,43,38,204)(8,44,39,205)(9,45,40,206)(10,46,41,207)(11,47,42,208)(12,48,29,209)(13,49,30,210)(14,50,31,197)(15,153,177,219)(16,154,178,220)(17,141,179,221)(18,142,180,222)(19,143,181,223)(20,144,182,224)(21,145,169,211)(22,146,170,212)(23,147,171,213)(24,148,172,214)(25,149,173,215)(26,150,174,216)(27,151,175,217)(28,152,176,218)(57,74,185,160)(58,75,186,161)(59,76,187,162)(60,77,188,163)(61,78,189,164)(62,79,190,165)(63,80,191,166)(64,81,192,167)(65,82,193,168)(66,83,194,155)(67,84,195,156)(68,71,196,157)(69,72,183,158)(70,73,184,159)(85,111,113,130)(86,112,114,131)(87,99,115,132)(88,100,116,133)(89,101,117,134)(90,102,118,135)(91,103,119,136)(92,104,120,137)(93,105,121,138)(94,106,122,139)(95,107,123,140)(96,108,124,127)(97,109,125,128)(98,110,126,129), (43,204)(44,205)(45,206)(46,207)(47,208)(48,209)(49,210)(50,197)(51,198)(52,199)(53,200)(54,201)(55,202)(56,203)(71,157)(72,158)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,155)(84,156)(99,132)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,127)(109,128)(110,129)(111,130)(112,131)(141,221)(142,222)(143,223)(144,224)(145,211)(146,212)(147,213)(148,214)(149,215)(150,216)(151,217)(152,218)(153,219)(154,220), (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), (1,178,8,171)(2,177,9,170)(3,176,10,169)(4,175,11,182)(5,174,12,181)(6,173,13,180)(7,172,14,179)(15,40,22,33)(16,39,23,32)(17,38,24,31)(18,37,25,30)(19,36,26,29)(20,35,27,42)(21,34,28,41)(43,214,50,221)(44,213,51,220)(45,212,52,219)(46,211,53,218)(47,224,54,217)(48,223,55,216)(49,222,56,215)(57,92,64,85)(58,91,65,98)(59,90,66,97)(60,89,67,96)(61,88,68,95)(62,87,69,94)(63,86,70,93)(71,107,78,100)(72,106,79,99)(73,105,80,112)(74,104,81,111)(75,103,82,110)(76,102,83,109)(77,101,84,108)(113,185,120,192)(114,184,121,191)(115,183,122,190)(116,196,123,189)(117,195,124,188)(118,194,125,187)(119,193,126,186)(127,163,134,156)(128,162,135,155)(129,161,136,168)(130,160,137,167)(131,159,138,166)(132,158,139,165)(133,157,140,164)(141,204,148,197)(142,203,149,210)(143,202,150,209)(144,201,151,208)(145,200,152,207)(146,199,153,206)(147,198,154,205)>;
G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,126)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,124)(28,125)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,183)(38,184)(39,185)(40,186)(41,187)(42,188)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,71)(56,72)(85,178)(86,179)(87,180)(88,181)(89,182)(90,169)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,222)(100,223)(101,224)(102,211)(103,212)(104,213)(105,214)(106,215)(107,216)(108,217)(109,218)(110,219)(111,220)(112,221)(127,151)(128,152)(129,153)(130,154)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,197)(167,198)(168,199), (1,51,32,198)(2,52,33,199)(3,53,34,200)(4,54,35,201)(5,55,36,202)(6,56,37,203)(7,43,38,204)(8,44,39,205)(9,45,40,206)(10,46,41,207)(11,47,42,208)(12,48,29,209)(13,49,30,210)(14,50,31,197)(15,153,177,219)(16,154,178,220)(17,141,179,221)(18,142,180,222)(19,143,181,223)(20,144,182,224)(21,145,169,211)(22,146,170,212)(23,147,171,213)(24,148,172,214)(25,149,173,215)(26,150,174,216)(27,151,175,217)(28,152,176,218)(57,74,185,160)(58,75,186,161)(59,76,187,162)(60,77,188,163)(61,78,189,164)(62,79,190,165)(63,80,191,166)(64,81,192,167)(65,82,193,168)(66,83,194,155)(67,84,195,156)(68,71,196,157)(69,72,183,158)(70,73,184,159)(85,111,113,130)(86,112,114,131)(87,99,115,132)(88,100,116,133)(89,101,117,134)(90,102,118,135)(91,103,119,136)(92,104,120,137)(93,105,121,138)(94,106,122,139)(95,107,123,140)(96,108,124,127)(97,109,125,128)(98,110,126,129), (43,204)(44,205)(45,206)(46,207)(47,208)(48,209)(49,210)(50,197)(51,198)(52,199)(53,200)(54,201)(55,202)(56,203)(71,157)(72,158)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,155)(84,156)(99,132)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,127)(109,128)(110,129)(111,130)(112,131)(141,221)(142,222)(143,223)(144,224)(145,211)(146,212)(147,213)(148,214)(149,215)(150,216)(151,217)(152,218)(153,219)(154,220), (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), (1,178,8,171)(2,177,9,170)(3,176,10,169)(4,175,11,182)(5,174,12,181)(6,173,13,180)(7,172,14,179)(15,40,22,33)(16,39,23,32)(17,38,24,31)(18,37,25,30)(19,36,26,29)(20,35,27,42)(21,34,28,41)(43,214,50,221)(44,213,51,220)(45,212,52,219)(46,211,53,218)(47,224,54,217)(48,223,55,216)(49,222,56,215)(57,92,64,85)(58,91,65,98)(59,90,66,97)(60,89,67,96)(61,88,68,95)(62,87,69,94)(63,86,70,93)(71,107,78,100)(72,106,79,99)(73,105,80,112)(74,104,81,111)(75,103,82,110)(76,102,83,109)(77,101,84,108)(113,185,120,192)(114,184,121,191)(115,183,122,190)(116,196,123,189)(117,195,124,188)(118,194,125,187)(119,193,126,186)(127,163,134,156)(128,162,135,155)(129,161,136,168)(130,160,137,167)(131,159,138,166)(132,158,139,165)(133,157,140,164)(141,204,148,197)(142,203,149,210)(143,202,150,209)(144,201,151,208)(145,200,152,207)(146,199,153,206)(147,198,154,205) );
G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,126),(16,113),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,121),(25,122),(26,123),(27,124),(28,125),(29,189),(30,190),(31,191),(32,192),(33,193),(34,194),(35,195),(36,196),(37,183),(38,184),(39,185),(40,186),(41,187),(42,188),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,71),(56,72),(85,178),(86,179),(87,180),(88,181),(89,182),(90,169),(91,170),(92,171),(93,172),(94,173),(95,174),(96,175),(97,176),(98,177),(99,222),(100,223),(101,224),(102,211),(103,212),(104,213),(105,214),(106,215),(107,216),(108,217),(109,218),(110,219),(111,220),(112,221),(127,151),(128,152),(129,153),(130,154),(131,141),(132,142),(133,143),(134,144),(135,145),(136,146),(137,147),(138,148),(139,149),(140,150),(155,200),(156,201),(157,202),(158,203),(159,204),(160,205),(161,206),(162,207),(163,208),(164,209),(165,210),(166,197),(167,198),(168,199)], [(1,51,32,198),(2,52,33,199),(3,53,34,200),(4,54,35,201),(5,55,36,202),(6,56,37,203),(7,43,38,204),(8,44,39,205),(9,45,40,206),(10,46,41,207),(11,47,42,208),(12,48,29,209),(13,49,30,210),(14,50,31,197),(15,153,177,219),(16,154,178,220),(17,141,179,221),(18,142,180,222),(19,143,181,223),(20,144,182,224),(21,145,169,211),(22,146,170,212),(23,147,171,213),(24,148,172,214),(25,149,173,215),(26,150,174,216),(27,151,175,217),(28,152,176,218),(57,74,185,160),(58,75,186,161),(59,76,187,162),(60,77,188,163),(61,78,189,164),(62,79,190,165),(63,80,191,166),(64,81,192,167),(65,82,193,168),(66,83,194,155),(67,84,195,156),(68,71,196,157),(69,72,183,158),(70,73,184,159),(85,111,113,130),(86,112,114,131),(87,99,115,132),(88,100,116,133),(89,101,117,134),(90,102,118,135),(91,103,119,136),(92,104,120,137),(93,105,121,138),(94,106,122,139),(95,107,123,140),(96,108,124,127),(97,109,125,128),(98,110,126,129)], [(43,204),(44,205),(45,206),(46,207),(47,208),(48,209),(49,210),(50,197),(51,198),(52,199),(53,200),(54,201),(55,202),(56,203),(71,157),(72,158),(73,159),(74,160),(75,161),(76,162),(77,163),(78,164),(79,165),(80,166),(81,167),(82,168),(83,155),(84,156),(99,132),(100,133),(101,134),(102,135),(103,136),(104,137),(105,138),(106,139),(107,140),(108,127),(109,128),(110,129),(111,130),(112,131),(141,221),(142,222),(143,223),(144,224),(145,211),(146,212),(147,213),(148,214),(149,215),(150,216),(151,217),(152,218),(153,219),(154,220)], [(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)], [(1,178,8,171),(2,177,9,170),(3,176,10,169),(4,175,11,182),(5,174,12,181),(6,173,13,180),(7,172,14,179),(15,40,22,33),(16,39,23,32),(17,38,24,31),(18,37,25,30),(19,36,26,29),(20,35,27,42),(21,34,28,41),(43,214,50,221),(44,213,51,220),(45,212,52,219),(46,211,53,218),(47,224,54,217),(48,223,55,216),(49,222,56,215),(57,92,64,85),(58,91,65,98),(59,90,66,97),(60,89,67,96),(61,88,68,95),(62,87,69,94),(63,86,70,93),(71,107,78,100),(72,106,79,99),(73,105,80,112),(74,104,81,111),(75,103,82,110),(76,102,83,109),(77,101,84,108),(113,185,120,192),(114,184,121,191),(115,183,122,190),(116,196,123,189),(117,195,124,188),(118,194,125,187),(119,193,126,186),(127,163,134,156),(128,162,135,155),(129,161,136,168),(130,160,137,167),(131,159,138,166),(132,158,139,165),(133,157,140,164),(141,204,148,197),(142,203,149,210),(143,202,150,209),(144,201,151,208),(145,200,152,207),(146,199,153,206),(147,198,154,205)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4X | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | Dic7 | D14 | D14 | D4×D7 | D4⋊2D7 |
| kernel | C2×D4×Dic7 | C2×C4×Dic7 | C2×C4⋊Dic7 | D4×Dic7 | C2×C23.D7 | C23×Dic7 | D4×C2×C14 | D4×C14 | C2×Dic7 | C22×D4 | C2×C14 | C22×C4 | C2×D4 | C2×D4 | C24 | C22 | C22 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 16 | 4 | 3 | 4 | 3 | 24 | 12 | 6 | 6 | 6 |
Matrix representation of C2×D4×Dic7 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 22 |
| 0 | 0 | 0 | 21 | 1 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 8 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 28 | 0 | 0 |
| 0 | 5 | 25 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 10 | 7 | 0 | 0 |
| 0 | 27 | 19 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,21,0,0,0,22,1],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,8,0,0,0,0,28],[1,0,0,0,0,0,1,5,0,0,0,28,25,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,10,27,0,0,0,7,19,0,0,0,0,0,12,0,0,0,0,0,12] >;
C2×D4×Dic7 in GAP, Magma, Sage, TeX
C_2\times D_4\times {\rm Dic}_7 % in TeX
G:=Group("C2xD4xDic7"); // GroupNames label
G:=SmallGroup(448,1248);
// by ID
G=gap.SmallGroup(448,1248);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^14=1,e^2=d^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations