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
Subgroups: 1364 in 426 conjugacy classes, 215 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×10], C22, C22 [×14], C22 [×24], C7, C2×C4 [×6], C2×C4 [×34], D4 [×16], C23, C23 [×12], C23 [×8], C14 [×3], C14 [×4], C14 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×20], C2×D4 [×12], C24 [×2], Dic7 [×4], Dic7 [×6], C28 [×4], C2×C14, C2×C14 [×14], C2×C14 [×24], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C2×Dic7 [×12], C2×Dic7 [×22], C2×C28 [×6], C7×D4 [×16], C22×C14, C22×C14 [×12], C22×C14 [×8], C2×C4×D4, C4×Dic7 [×4], C4⋊Dic7 [×4], C23.D7 [×8], C22×Dic7 [×2], C22×Dic7 [×10], C22×Dic7 [×8], C22×C28, D4×C14 [×12], C23×C14 [×2], C2×C4×Dic7, C2×C4⋊Dic7, D4×Dic7 [×8], C2×C23.D7 [×2], C23×Dic7 [×2], D4×C2×C14, C2×D4×Dic7
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D7, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, Dic7 [×8], D14 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×Dic7 [×28], C22×D7 [×7], C2×C4×D4, D4×D7 [×2], D4⋊2D7 [×2], C22×Dic7 [×14], C23×D7, D4×Dic7 [×4], C2×D4×D7, C2×D4⋊2D7, C23×Dic7, C2×D4×Dic7
Generators and relations
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 >
►On 224 points
Generators in S224
(1 196)(2 183)(3 184)(4 185)(5 186)(6 187)(7 188)(8 189)(9 190)(10 191)(11 192)(12 193)(13 194)(14 195)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 133)(42 134)(43 175)(44 176)(45 177)(46 178)(47 179)(48 180)(49 181)(50 182)(51 169)(52 170)(53 171)(54 172)(55 173)(56 174)(57 199)(58 200)(59 201)(60 202)(61 203)(62 204)(63 205)(64 206)(65 207)(66 208)(67 209)(68 210)(69 197)(70 198)(71 216)(72 217)(73 218)(74 219)(75 220)(76 221)(77 222)(78 223)(79 224)(80 211)(81 212)(82 213)(83 214)(84 215)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 141)(95 142)(96 143)(97 144)(98 145)(99 167)(100 168)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)
(1 91 136 71)(2 92 137 72)(3 93 138 73)(4 94 139 74)(5 95 140 75)(6 96 127 76)(7 97 128 77)(8 98 129 78)(9 85 130 79)(10 86 131 80)(11 87 132 81)(12 88 133 82)(13 89 134 83)(14 90 135 84)(15 106 57 51)(16 107 58 52)(17 108 59 53)(18 109 60 54)(19 110 61 55)(20 111 62 56)(21 112 63 43)(22 99 64 44)(23 100 65 45)(24 101 66 46)(25 102 67 47)(26 103 68 48)(27 104 69 49)(28 105 70 50)(29 215 195 151)(30 216 196 152)(31 217 183 153)(32 218 184 154)(33 219 185 141)(34 220 186 142)(35 221 187 143)(36 222 188 144)(37 223 189 145)(38 224 190 146)(39 211 191 147)(40 212 192 148)(41 213 193 149)(42 214 194 150)(113 166 205 175)(114 167 206 176)(115 168 207 177)(116 155 208 178)(117 156 209 179)(118 157 210 180)(119 158 197 181)(120 159 198 182)(121 160 199 169)(122 161 200 170)(123 162 201 171)(124 163 202 172)(125 164 203 173)(126 165 204 174)
(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(113 205)(114 206)(115 207)(116 208)(117 209)(118 210)(119 197)(120 198)(121 199)(122 200)(123 201)(124 202)(125 203)(126 204)(141 219)(142 220)(143 221)(144 222)(145 223)(146 224)(147 211)(148 212)(149 213)(150 214)(151 215)(152 216)(153 217)(154 218)
(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)
(1 103 8 110)(2 102 9 109)(3 101 10 108)(4 100 11 107)(5 99 12 106)(6 112 13 105)(7 111 14 104)(15 75 22 82)(16 74 23 81)(17 73 24 80)(18 72 25 79)(19 71 26 78)(20 84 27 77)(21 83 28 76)(29 181 36 174)(30 180 37 173)(31 179 38 172)(32 178 39 171)(33 177 40 170)(34 176 41 169)(35 175 42 182)(43 134 50 127)(44 133 51 140)(45 132 52 139)(46 131 53 138)(47 130 54 137)(48 129 55 136)(49 128 56 135)(57 95 64 88)(58 94 65 87)(59 93 66 86)(60 92 67 85)(61 91 68 98)(62 90 69 97)(63 89 70 96)(113 214 120 221)(114 213 121 220)(115 212 122 219)(116 211 123 218)(117 224 124 217)(118 223 125 216)(119 222 126 215)(141 207 148 200)(142 206 149 199)(143 205 150 198)(144 204 151 197)(145 203 152 210)(146 202 153 209)(147 201 154 208)(155 191 162 184)(156 190 163 183)(157 189 164 196)(158 188 165 195)(159 187 166 194)(160 186 167 193)(161 185 168 192)
G:=sub<Sym(224)| (1,196)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,133)(42,134)(43,175)(44,176)(45,177)(46,178)(47,179)(48,180)(49,181)(50,182)(51,169)(52,170)(53,171)(54,172)(55,173)(56,174)(57,199)(58,200)(59,201)(60,202)(61,203)(62,204)(63,205)(64,206)(65,207)(66,208)(67,209)(68,210)(69,197)(70,198)(71,216)(72,217)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,211)(81,212)(82,213)(83,214)(84,215)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,141)(95,142)(96,143)(97,144)(98,145)(99,167)(100,168)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166), (1,91,136,71)(2,92,137,72)(3,93,138,73)(4,94,139,74)(5,95,140,75)(6,96,127,76)(7,97,128,77)(8,98,129,78)(9,85,130,79)(10,86,131,80)(11,87,132,81)(12,88,133,82)(13,89,134,83)(14,90,135,84)(15,106,57,51)(16,107,58,52)(17,108,59,53)(18,109,60,54)(19,110,61,55)(20,111,62,56)(21,112,63,43)(22,99,64,44)(23,100,65,45)(24,101,66,46)(25,102,67,47)(26,103,68,48)(27,104,69,49)(28,105,70,50)(29,215,195,151)(30,216,196,152)(31,217,183,153)(32,218,184,154)(33,219,185,141)(34,220,186,142)(35,221,187,143)(36,222,188,144)(37,223,189,145)(38,224,190,146)(39,211,191,147)(40,212,192,148)(41,213,193,149)(42,214,194,150)(113,166,205,175)(114,167,206,176)(115,168,207,177)(116,155,208,178)(117,156,209,179)(118,157,210,180)(119,158,197,181)(120,159,198,182)(121,160,199,169)(122,161,200,170)(123,162,201,171)(124,163,202,172)(125,164,203,173)(126,165,204,174), (15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(113,205)(114,206)(115,207)(116,208)(117,209)(118,210)(119,197)(120,198)(121,199)(122,200)(123,201)(124,202)(125,203)(126,204)(141,219)(142,220)(143,221)(144,222)(145,223)(146,224)(147,211)(148,212)(149,213)(150,214)(151,215)(152,216)(153,217)(154,218), (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), (1,103,8,110)(2,102,9,109)(3,101,10,108)(4,100,11,107)(5,99,12,106)(6,112,13,105)(7,111,14,104)(15,75,22,82)(16,74,23,81)(17,73,24,80)(18,72,25,79)(19,71,26,78)(20,84,27,77)(21,83,28,76)(29,181,36,174)(30,180,37,173)(31,179,38,172)(32,178,39,171)(33,177,40,170)(34,176,41,169)(35,175,42,182)(43,134,50,127)(44,133,51,140)(45,132,52,139)(46,131,53,138)(47,130,54,137)(48,129,55,136)(49,128,56,135)(57,95,64,88)(58,94,65,87)(59,93,66,86)(60,92,67,85)(61,91,68,98)(62,90,69,97)(63,89,70,96)(113,214,120,221)(114,213,121,220)(115,212,122,219)(116,211,123,218)(117,224,124,217)(118,223,125,216)(119,222,126,215)(141,207,148,200)(142,206,149,199)(143,205,150,198)(144,204,151,197)(145,203,152,210)(146,202,153,209)(147,201,154,208)(155,191,162,184)(156,190,163,183)(157,189,164,196)(158,188,165,195)(159,187,166,194)(160,186,167,193)(161,185,168,192)>;
G:=Group( (1,196)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,133)(42,134)(43,175)(44,176)(45,177)(46,178)(47,179)(48,180)(49,181)(50,182)(51,169)(52,170)(53,171)(54,172)(55,173)(56,174)(57,199)(58,200)(59,201)(60,202)(61,203)(62,204)(63,205)(64,206)(65,207)(66,208)(67,209)(68,210)(69,197)(70,198)(71,216)(72,217)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,211)(81,212)(82,213)(83,214)(84,215)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,141)(95,142)(96,143)(97,144)(98,145)(99,167)(100,168)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166), (1,91,136,71)(2,92,137,72)(3,93,138,73)(4,94,139,74)(5,95,140,75)(6,96,127,76)(7,97,128,77)(8,98,129,78)(9,85,130,79)(10,86,131,80)(11,87,132,81)(12,88,133,82)(13,89,134,83)(14,90,135,84)(15,106,57,51)(16,107,58,52)(17,108,59,53)(18,109,60,54)(19,110,61,55)(20,111,62,56)(21,112,63,43)(22,99,64,44)(23,100,65,45)(24,101,66,46)(25,102,67,47)(26,103,68,48)(27,104,69,49)(28,105,70,50)(29,215,195,151)(30,216,196,152)(31,217,183,153)(32,218,184,154)(33,219,185,141)(34,220,186,142)(35,221,187,143)(36,222,188,144)(37,223,189,145)(38,224,190,146)(39,211,191,147)(40,212,192,148)(41,213,193,149)(42,214,194,150)(113,166,205,175)(114,167,206,176)(115,168,207,177)(116,155,208,178)(117,156,209,179)(118,157,210,180)(119,158,197,181)(120,159,198,182)(121,160,199,169)(122,161,200,170)(123,162,201,171)(124,163,202,172)(125,164,203,173)(126,165,204,174), (15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(113,205)(114,206)(115,207)(116,208)(117,209)(118,210)(119,197)(120,198)(121,199)(122,200)(123,201)(124,202)(125,203)(126,204)(141,219)(142,220)(143,221)(144,222)(145,223)(146,224)(147,211)(148,212)(149,213)(150,214)(151,215)(152,216)(153,217)(154,218), (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), (1,103,8,110)(2,102,9,109)(3,101,10,108)(4,100,11,107)(5,99,12,106)(6,112,13,105)(7,111,14,104)(15,75,22,82)(16,74,23,81)(17,73,24,80)(18,72,25,79)(19,71,26,78)(20,84,27,77)(21,83,28,76)(29,181,36,174)(30,180,37,173)(31,179,38,172)(32,178,39,171)(33,177,40,170)(34,176,41,169)(35,175,42,182)(43,134,50,127)(44,133,51,140)(45,132,52,139)(46,131,53,138)(47,130,54,137)(48,129,55,136)(49,128,56,135)(57,95,64,88)(58,94,65,87)(59,93,66,86)(60,92,67,85)(61,91,68,98)(62,90,69,97)(63,89,70,96)(113,214,120,221)(114,213,121,220)(115,212,122,219)(116,211,123,218)(117,224,124,217)(118,223,125,216)(119,222,126,215)(141,207,148,200)(142,206,149,199)(143,205,150,198)(144,204,151,197)(145,203,152,210)(146,202,153,209)(147,201,154,208)(155,191,162,184)(156,190,163,183)(157,189,164,196)(158,188,165,195)(159,187,166,194)(160,186,167,193)(161,185,168,192) );
G=PermutationGroup([(1,196),(2,183),(3,184),(4,185),(5,186),(6,187),(7,188),(8,189),(9,190),(10,191),(11,192),(12,193),(13,194),(14,195),(15,121),(16,122),(17,123),(18,124),(19,125),(20,126),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,135),(30,136),(31,137),(32,138),(33,139),(34,140),(35,127),(36,128),(37,129),(38,130),(39,131),(40,132),(41,133),(42,134),(43,175),(44,176),(45,177),(46,178),(47,179),(48,180),(49,181),(50,182),(51,169),(52,170),(53,171),(54,172),(55,173),(56,174),(57,199),(58,200),(59,201),(60,202),(61,203),(62,204),(63,205),(64,206),(65,207),(66,208),(67,209),(68,210),(69,197),(70,198),(71,216),(72,217),(73,218),(74,219),(75,220),(76,221),(77,222),(78,223),(79,224),(80,211),(81,212),(82,213),(83,214),(84,215),(85,146),(86,147),(87,148),(88,149),(89,150),(90,151),(91,152),(92,153),(93,154),(94,141),(95,142),(96,143),(97,144),(98,145),(99,167),(100,168),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,161),(108,162),(109,163),(110,164),(111,165),(112,166)], [(1,91,136,71),(2,92,137,72),(3,93,138,73),(4,94,139,74),(5,95,140,75),(6,96,127,76),(7,97,128,77),(8,98,129,78),(9,85,130,79),(10,86,131,80),(11,87,132,81),(12,88,133,82),(13,89,134,83),(14,90,135,84),(15,106,57,51),(16,107,58,52),(17,108,59,53),(18,109,60,54),(19,110,61,55),(20,111,62,56),(21,112,63,43),(22,99,64,44),(23,100,65,45),(24,101,66,46),(25,102,67,47),(26,103,68,48),(27,104,69,49),(28,105,70,50),(29,215,195,151),(30,216,196,152),(31,217,183,153),(32,218,184,154),(33,219,185,141),(34,220,186,142),(35,221,187,143),(36,222,188,144),(37,223,189,145),(38,224,190,146),(39,211,191,147),(40,212,192,148),(41,213,193,149),(42,214,194,150),(113,166,205,175),(114,167,206,176),(115,168,207,177),(116,155,208,178),(117,156,209,179),(118,157,210,180),(119,158,197,181),(120,159,198,182),(121,160,199,169),(122,161,200,170),(123,162,201,171),(124,163,202,172),(125,164,203,173),(126,165,204,174)], [(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(71,91),(72,92),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(113,205),(114,206),(115,207),(116,208),(117,209),(118,210),(119,197),(120,198),(121,199),(122,200),(123,201),(124,202),(125,203),(126,204),(141,219),(142,220),(143,221),(144,222),(145,223),(146,224),(147,211),(148,212),(149,213),(150,214),(151,215),(152,216),(153,217),(154,218)], [(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)], [(1,103,8,110),(2,102,9,109),(3,101,10,108),(4,100,11,107),(5,99,12,106),(6,112,13,105),(7,111,14,104),(15,75,22,82),(16,74,23,81),(17,73,24,80),(18,72,25,79),(19,71,26,78),(20,84,27,77),(21,83,28,76),(29,181,36,174),(30,180,37,173),(31,179,38,172),(32,178,39,171),(33,177,40,170),(34,176,41,169),(35,175,42,182),(43,134,50,127),(44,133,51,140),(45,132,52,139),(46,131,53,138),(47,130,54,137),(48,129,55,136),(49,128,56,135),(57,95,64,88),(58,94,65,87),(59,93,66,86),(60,92,67,85),(61,91,68,98),(62,90,69,97),(63,89,70,96),(113,214,120,221),(114,213,121,220),(115,212,122,219),(116,211,123,218),(117,224,124,217),(118,223,125,216),(119,222,126,215),(141,207,148,200),(142,206,149,199),(143,205,150,198),(144,204,151,197),(145,203,152,210),(146,202,153,209),(147,201,154,208),(155,191,162,184),(156,190,163,183),(157,189,164,196),(158,188,165,195),(159,187,166,194),(160,186,167,193),(161,185,168,192)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
C_2\times D_4\times 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
×
⋊
ℤ
𝔽
○
≀