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## G = C2×D14.D4order 448 = 26·7

### Direct product of C2 and D14.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×D14.D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22×C4 — C2×D14.D4
 Lower central C7 — C2×C14 — C2×D14.D4
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×D14.D4
G = < a,b,c,d,e | a2=b14=c2=d4=1, e2=b7, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b7c, ce=ec, ede-1=b7d-1 >

Subgroups: 1556 in 342 conjugacy classes, 119 normal (31 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C22.D4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×C4×D7, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, D14.D4, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×D14⋊C4, C2×C23.D7, C14×C22⋊C4, D7×C22×C4, C22×C7⋊D4, C2×D14.D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22.D4, C22×D4, C2×C4○D4, C22×D7, C2×C22.D4, C4○D28, D4×D7, D42D7, C23×D7, D14.D4, C2×C4○D28, C2×D4×D7, C2×D42D7, C2×D14.D4

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

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

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

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

88 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 7A 7B 7C 14A ··· 14U 14V ··· 14AG 28A ··· 28X order 1 2 ··· 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 ··· 1 4 4 14 14 14 14 2 2 2 2 4 4 14 14 14 14 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 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 C2 C2 D4 D7 C4○D4 D14 D14 D14 C4○D28 D4×D7 D4⋊2D7 kernel C2×D14.D4 D14.D4 C2×Dic7⋊C4 C2×C4⋊Dic7 C2×D14⋊C4 C2×C23.D7 C14×C22⋊C4 D7×C22×C4 C22×C7⋊D4 C22×D7 C2×C22⋊C4 C2×C14 C22⋊C4 C22×C4 C24 C22 C22 C22 # reps 1 8 1 1 1 1 1 1 1 4 3 8 12 6 3 24 6 6

Matrix representation of C2×D14.D4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 10 19 0 0 0 0 10 22 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 7 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 15 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 14 2 0 0 0 0 4 15
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 17 0 0 0 0 0 0 17 0 0 0 0 0 0 12 0 0 0 0 0 6 17

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[10,10,0,0,0,0,19,22,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,7,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,15,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,14,4,0,0,0,0,2,15],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,12,6,0,0,0,0,0,17] >;

C2×D14.D4 in GAP, Magma, Sage, TeX

C_2\times D_{14}.D_4
% in TeX

G:=Group("C2xD14.D4");
// GroupNames label

G:=SmallGroup(448,941);
// by ID

G=gap.SmallGroup(448,941);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,675,297,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^14=c^2=d^4=1,e^2=b^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,d*c*d^-1=b^7*c,c*e=e*c,e*d*e^-1=b^7*d^-1>;
// generators/relations

׿
×
𝔽