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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, C28⋊4C24, C25⋊4C14, C14.21C25, C4⋊(C23×C14), (C23×C4)⋊9C14, (C24×C14)⋊2C2, C24⋊8(C2×C14), (C2×C14)⋊2C24, C22⋊(C23×C14), (C2×C28)⋊17C23, (C23×C28)⋊16C2, C2.1(C24×C14), C23⋊4(C22×C14), (C22×C14)⋊8C23, (C22×C28)⋊66C22, (C23×C14)⋊17C22, (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

Subgroups: 1874 in 1362 conjugacy classes, 850 normal (10 characteristic)
C1, C2, C2 [×14], C2 [×16], C4 [×8], C22 [×51], C22 [×112], C7, C2×C4 [×28], D4 [×64], C23 [×71], C23 [×112], C14, C14 [×14], C14 [×16], C22×C4 [×14], C2×D4 [×112], C24, C24 [×28], C24 [×16], C28 [×8], C2×C14 [×51], C2×C14 [×112], C23×C4, C22×D4 [×28], C25 [×2], C2×C28 [×28], C7×D4 [×64], C22×C14 [×71], C22×C14 [×112], D4×C23, C22×C28 [×14], D4×C14 [×112], C23×C14, C23×C14 [×28], C23×C14 [×16], C23×C28, D4×C2×C14 [×28], C24×C14 [×2], D4×C22×C14

Quotients:
C1, C2 [×31], C22 [×155], C7, D4 [×8], C23 [×155], C14 [×31], C2×D4 [×28], C24 [×31], C2×C14 [×155], C22×D4 [×14], C25, C7×D4 [×8], C22×C14 [×155], D4×C23, D4×C14 [×28], C23×C14 [×31], D4×C2×C14 [×14], C24×C14, D4×C22×C14

Generators and relations
 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 >

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

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

(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)

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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
 

׿
×
⋊
ℤ
𝔽
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