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G = D4×C22×C14  order 448 = 26·7

Direct product of C22×C14 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C22×C14, C25⋊4C14, C28⋊4C24, C14.21C25, C4⋊(C23×C14), (C24×C14)⋊2C2, (C23×C4)⋊9C14, C24⋊8(C2×C14), (C2×C14)⋊2C24, C22⋊(C23×C14), (C23×C28)⋊16C2, (C2×C28)⋊17C23, C2.1(C24×C14), C23⋊4(C22×C14), (C22×C14)⋊8C23, (C23×C14)⋊17C22, (C22×C28)⋊66C22, (C2×C4)⋊4(C22×C14), (C22×C4)⋊19(C2×C14), SmallGroup(448,1386)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — D4×C22×C14
Chief series
C1 — C2 — C14 — C2×C14 — C7×D4 — D4×C14 — D4×C2×C14 — D4×C22×C14
Lower central
C1 — C2 — D4×C22×C14
Upper central
C1 — C23×C14 — D4×C22×C14

Generators and relations for D4×C22×C14
 G = < a,b,c,d,e | a2=b2=c14=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1874 in 1362 conjugacy classes, 850 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, C23, C14, C14, C14, C22×C4, C2×D4, C24, C24, C24, C28, C2×C14, C2×C14, C23×C4, C22×D4, C25, C2×C28, C7×D4, C22×C14, C22×C14, D4×C23, C22×C28, D4×C14, C23×C14, C23×C14, C23×C14, C23×C28, D4×C2×C14, C24×C14, D4×C22×C14
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, C25, C7×D4, C22×C14, D4×C23, D4×C14, C23×C14, D4×C2×C14, C24×C14, D4×C22×C14

Smallest permutation representation of D4×C22×C14
►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)(15 145)(16 146)(17 147)(18 148)(19 149)(20 150)(21 151)(22 152)(23 153)(24 154)(25 141)(26 142)(27 143)(28 144)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 85)(41 86)(42 87)(43 157)(44 158)(45 159)(46 160)(47 161)(48 162)(49 163)(50 164)(51 165)(52 166)(53 167)(54 168)(55 155)(56 156)(71 204)(72 205)(73 206)(74 207)(75 208)(76 209)(77 210)(78 197)(79 198)(80 199)(81 200)(82 201)(83 202)(84 203)(99 123)(100 124)(101 125)(102 126)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(111 121)(112 122)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 190)(134 191)(135 192)(136 193)(137 194)(138 195)(139 196)(140 183)(169 216)(170 217)(171 218)(172 219)(173 220)(174 221)(175 222)(176 223)(177 224)(178 211)(179 212)(180 213)(181 214)(182 215)

(1 155)(2 156)(3 157)(4 158)(5 159)(6 160)(7 161)(8 162)(9 163)(10 164)(11 165)(12 166)(13 167)(14 168)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 181)(26 182)(27 169)(28 170)(29 207)(30 208)(31 209)(32 210)(33 197)(34 198)(35 199)(36 200)(37 201)(38 202)(39 203)(40 204)(41 205)(42 206)(43 66)(44 67)(45 68)(46 69)(47 70)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 65)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(99 185)(100 186)(101 187)(102 188)(103 189)(104 190)(105 191)(106 192)(107 193)(108 194)(109 195)(110 196)(111 183)(112 184)(113 132)(114 133)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 140)(122 127)(123 128)(124 129)(125 130)(126 131)(141 214)(142 215)(143 216)(144 217)(145 218)(146 219)(147 220)(148 221)(149 222)(150 223)(151 224)(152 211)(153 212)(154 213)

(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 33 218 190)(2 34 219 191)(3 35 220 192)(4 36 221 193)(5 37 222 194)(6 38 223 195)(7 39 224 196)(8 40 211 183)(9 41 212 184)(10 42 213 185)(11 29 214 186)(12 30 215 187)(13 31 216 188)(14 32 217 189)(15 114 55 78)(16 115 56 79)(17 116 43 80)(18 117 44 81)(19 118 45 82)(20 119 46 83)(21 120 47 84)(22 121 48 71)(23 122 49 72)(24 123 50 73)(25 124 51 74)(26 125 52 75)(27 126 53 76)(28 113 54 77)(57 85 178 140)(58 86 179 127)(59 87 180 128)(60 88 181 129)(61 89 182 130)(62 90 169 131)(63 91 170 132)(64 92 171 133)(65 93 172 134)(66 94 173 135)(67 95 174 136)(68 96 175 137)(69 97 176 138)(70 98 177 139)(99 164 206 154)(100 165 207 141)(101 166 208 142)(102 167 209 143)(103 168 210 144)(104 155 197 145)(105 156 198 146)(106 157 199 147)(107 158 200 148)(108 159 201 149)(109 160 202 150)(110 161 203 151)(111 162 204 152)(112 163 205 153)

(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 81)(30 82)(31 83)(32 84)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 213)(44 214)(45 215)(46 216)(47 217)(48 218)(49 219)(50 220)(51 221)(52 222)(53 223)(54 224)(55 211)(56 212)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 141)(68 142)(69 143)(70 144)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(113 196)(114 183)(115 184)(116 185)(117 186)(118 187)(119 188)(120 189)(121 190)(122 191)(123 192)(124 193)(125 194)(126 195)(155 178)(156 179)(157 180)(158 181)(159 182)(160 169)(161 170)(162 171)(163 172)(164 173)(165 174)(166 175)(167 176)(168 177)
 
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)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,141)(26,142)(27,143)(28,144)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,85)(41,86)(42,87)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,155)(56,156)(71,204)(72,205)(73,206)(74,207)(75,208)(76,209)(77,210)(78,197)(79,198)(80,199)(81,200)(82,201)(83,202)(84,203)(99,123)(100,124)(101,125)(102,126)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,121)(112,122)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)(138,195)(139,196)(140,183)(169,216)(170,217)(171,218)(172,219)(173,220)(174,221)(175,222)(176,223)(177,224)(178,211)(179,212)(180,213)(181,214)(182,215), (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,161)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,181)(26,182)(27,169)(28,170)(29,207)(30,208)(31,209)(32,210)(33,197)(34,198)(35,199)(36,200)(37,201)(38,202)(39,203)(40,204)(41,205)(42,206)(43,66)(44,67)(45,68)(46,69)(47,70)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,183)(112,184)(113,132)(114,133)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,127)(123,128)(124,129)(125,130)(126,131)(141,214)(142,215)(143,216)(144,217)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,211)(153,212)(154,213), (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), (1,33,218,190)(2,34,219,191)(3,35,220,192)(4,36,221,193)(5,37,222,194)(6,38,223,195)(7,39,224,196)(8,40,211,183)(9,41,212,184)(10,42,213,185)(11,29,214,186)(12,30,215,187)(13,31,216,188)(14,32,217,189)(15,114,55,78)(16,115,56,79)(17,116,43,80)(18,117,44,81)(19,118,45,82)(20,119,46,83)(21,120,47,84)(22,121,48,71)(23,122,49,72)(24,123,50,73)(25,124,51,74)(26,125,52,75)(27,126,53,76)(28,113,54,77)(57,85,178,140)(58,86,179,127)(59,87,180,128)(60,88,181,129)(61,89,182,130)(62,90,169,131)(63,91,170,132)(64,92,171,133)(65,93,172,134)(66,94,173,135)(67,95,174,136)(68,96,175,137)(69,97,176,138)(70,98,177,139)(99,164,206,154)(100,165,207,141)(101,166,208,142)(102,167,209,143)(103,168,210,144)(104,155,197,145)(105,156,198,146)(106,157,199,147)(107,158,200,148)(108,159,201,149)(109,160,202,150)(110,161,203,151)(111,162,204,152)(112,163,205,153), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,81)(30,82)(31,83)(32,84)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,213)(44,214)(45,215)(46,216)(47,217)(48,218)(49,219)(50,220)(51,221)(52,222)(53,223)(54,224)(55,211)(56,212)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,141)(68,142)(69,143)(70,144)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,196)(114,183)(115,184)(116,185)(117,186)(118,187)(119,188)(120,189)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)(155,178)(156,179)(157,180)(158,181)(159,182)(160,169)(161,170)(162,171)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)>;
 
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)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,141)(26,142)(27,143)(28,144)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,85)(41,86)(42,87)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,155)(56,156)(71,204)(72,205)(73,206)(74,207)(75,208)(76,209)(77,210)(78,197)(79,198)(80,199)(81,200)(82,201)(83,202)(84,203)(99,123)(100,124)(101,125)(102,126)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,121)(112,122)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)(138,195)(139,196)(140,183)(169,216)(170,217)(171,218)(172,219)(173,220)(174,221)(175,222)(176,223)(177,224)(178,211)(179,212)(180,213)(181,214)(182,215), (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,161)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,181)(26,182)(27,169)(28,170)(29,207)(30,208)(31,209)(32,210)(33,197)(34,198)(35,199)(36,200)(37,201)(38,202)(39,203)(40,204)(41,205)(42,206)(43,66)(44,67)(45,68)(46,69)(47,70)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,183)(112,184)(113,132)(114,133)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,127)(123,128)(124,129)(125,130)(126,131)(141,214)(142,215)(143,216)(144,217)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,211)(153,212)(154,213), (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), (1,33,218,190)(2,34,219,191)(3,35,220,192)(4,36,221,193)(5,37,222,194)(6,38,223,195)(7,39,224,196)(8,40,211,183)(9,41,212,184)(10,42,213,185)(11,29,214,186)(12,30,215,187)(13,31,216,188)(14,32,217,189)(15,114,55,78)(16,115,56,79)(17,116,43,80)(18,117,44,81)(19,118,45,82)(20,119,46,83)(21,120,47,84)(22,121,48,71)(23,122,49,72)(24,123,50,73)(25,124,51,74)(26,125,52,75)(27,126,53,76)(28,113,54,77)(57,85,178,140)(58,86,179,127)(59,87,180,128)(60,88,181,129)(61,89,182,130)(62,90,169,131)(63,91,170,132)(64,92,171,133)(65,93,172,134)(66,94,173,135)(67,95,174,136)(68,96,175,137)(69,97,176,138)(70,98,177,139)(99,164,206,154)(100,165,207,141)(101,166,208,142)(102,167,209,143)(103,168,210,144)(104,155,197,145)(105,156,198,146)(106,157,199,147)(107,158,200,148)(108,159,201,149)(109,160,202,150)(110,161,203,151)(111,162,204,152)(112,163,205,153), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,81)(30,82)(31,83)(32,84)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,213)(44,214)(45,215)(46,216)(47,217)(48,218)(49,219)(50,220)(51,221)(52,222)(53,223)(54,224)(55,211)(56,212)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,141)(68,142)(69,143)(70,144)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,196)(114,183)(115,184)(116,185)(117,186)(118,187)(119,188)(120,189)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)(155,178)(156,179)(157,180)(158,181)(159,182)(160,169)(161,170)(162,171)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177) );
 
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),(15,145),(16,146),(17,147),(18,148),(19,149),(20,150),(21,151),(22,152),(23,153),(24,154),(25,141),(26,142),(27,143),(28,144),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(37,96),(38,97),(39,98),(40,85),(41,86),(42,87),(43,157),(44,158),(45,159),(46,160),(47,161),(48,162),(49,163),(50,164),(51,165),(52,166),(53,167),(54,168),(55,155),(56,156),(71,204),(72,205),(73,206),(74,207),(75,208),(76,209),(77,210),(78,197),(79,198),(80,199),(81,200),(82,201),(83,202),(84,203),(99,123),(100,124),(101,125),(102,126),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(111,121),(112,122),(127,184),(128,185),(129,186),(130,187),(131,188),(132,189),(133,190),(134,191),(135,192),(136,193),(137,194),(138,195),(139,196),(140,183),(169,216),(170,217),(171,218),(172,219),(173,220),(174,221),(175,222),(176,223),(177,224),(178,211),(179,212),(180,213),(181,214),(182,215)], [(1,155),(2,156),(3,157),(4,158),(5,159),(6,160),(7,161),(8,162),(9,163),(10,164),(11,165),(12,166),(13,167),(14,168),(15,171),(16,172),(17,173),(18,174),(19,175),(20,176),(21,177),(22,178),(23,179),(24,180),(25,181),(26,182),(27,169),(28,170),(29,207),(30,208),(31,209),(32,210),(33,197),(34,198),(35,199),(36,200),(37,201),(38,202),(39,203),(40,204),(41,205),(42,206),(43,66),(44,67),(45,68),(46,69),(47,70),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,65),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(99,185),(100,186),(101,187),(102,188),(103,189),(104,190),(105,191),(106,192),(107,193),(108,194),(109,195),(110,196),(111,183),(112,184),(113,132),(114,133),(115,134),(116,135),(117,136),(118,137),(119,138),(120,139),(121,140),(122,127),(123,128),(124,129),(125,130),(126,131),(141,214),(142,215),(143,216),(144,217),(145,218),(146,219),(147,220),(148,221),(149,222),(150,223),(151,224),(152,211),(153,212),(154,213)], [(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)], [(1,33,218,190),(2,34,219,191),(3,35,220,192),(4,36,221,193),(5,37,222,194),(6,38,223,195),(7,39,224,196),(8,40,211,183),(9,41,212,184),(10,42,213,185),(11,29,214,186),(12,30,215,187),(13,31,216,188),(14,32,217,189),(15,114,55,78),(16,115,56,79),(17,116,43,80),(18,117,44,81),(19,118,45,82),(20,119,46,83),(21,120,47,84),(22,121,48,71),(23,122,49,72),(24,123,50,73),(25,124,51,74),(26,125,52,75),(27,126,53,76),(28,113,54,77),(57,85,178,140),(58,86,179,127),(59,87,180,128),(60,88,181,129),(61,89,182,130),(62,90,169,131),(63,91,170,132),(64,92,171,133),(65,93,172,134),(66,94,173,135),(67,95,174,136),(68,96,175,137),(69,97,176,138),(70,98,177,139),(99,164,206,154),(100,165,207,141),(101,166,208,142),(102,167,209,143),(103,168,210,144),(104,155,197,145),(105,156,198,146),(106,157,199,147),(107,158,200,148),(108,159,201,149),(109,160,202,150),(110,161,203,151),(111,162,204,152),(112,163,205,153)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,81),(30,82),(31,83),(32,84),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,213),(44,214),(45,215),(46,216),(47,217),(48,218),(49,219),(50,220),(51,221),(52,222),(53,223),(54,224),(55,211),(56,212),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(64,152),(65,153),(66,154),(67,141),(68,142),(69,143),(70,144),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,135),(100,136),(101,137),(102,138),(103,139),(104,140),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134),(113,196),(114,183),(115,184),(116,185),(117,186),(118,187),(119,188),(120,189),(121,190),(122,191),(123,192),(124,193),(125,194),(126,195),(155,178),(156,179),(157,180),(158,181),(159,182),(160,169),(161,170),(162,171),(163,172),(164,173),(165,174),(166,175),(167,176),(168,177)]])
 

280 conjugacy classes

class 1 2A···2O2P···2AE4A···4H7A···7F14A···14CL14CM···14GD28A···28AV
order12···22···24···47···714···1414···1428···28
size11···12···22···21···11···12···22···2

280 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C7C14C14C14D4C7×D4
kernelD4×C22×C14C23×C28D4×C2×C14C24×C14D4×C23C23×C4C22×D4C25C22×C14C23
# reps112826616812848

Matrix representation of D4×C22×C14 ►in GL5(𝔽29)

10000
028000
00100
00010
00001
,
10000
028000
002800
00010
00001
,
280000
01000
002800
00040
00004
,
280000
028000
00100
000281
000271
,
10000
028000
002800
000128
000028

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,4,0,0,0,0,0,4],[28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28,27,0,0,0,1,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,28,28] >;
 

D4×C22×C14 in GAP, Magma, Sage, TeX 

D_4\times C_2^2\times C_{14}
 
% in TeX
 
G:=Group("D4xC2^2xC14");
 
// GroupNames label
 
G:=SmallGroup(448,1386);
 
// by ID
 
G=gap.SmallGroup(448,1386);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-7,-2,3165]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^14=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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×
⋊
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𝔽
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