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## G = C4○D4×C2×C14order 448 = 26·7

### Direct product of C2×C14 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C2×C14
 Chief series C1 — C2 — C14 — C2×C14 — C7×D4 — C7×C4○D4 — C14×C4○D4 — C4○D4×C2×C14
 Lower central C1 — C2 — C4○D4×C2×C14
 Upper central C1 — C22×C28 — C4○D4×C2×C14

Generators and relations for C4○D4×C2×C14
G = < a,b,c,d,e | a2=b14=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 1010 in 890 conjugacy classes, 770 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, D4, Q8, C23, C23, C23, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C28, C2×C14, C2×C14, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C22×C14, C22×C4○D4, C22×C28, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C23×C14, C23×C28, D4×C2×C14, Q8×C2×C14, C14×C4○D4, C4○D4×C2×C14
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, C25, C22×C14, C22×C4○D4, C7×C4○D4, C23×C14, C14×C4○D4, C24×C14, C4○D4×C2×C14

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

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

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

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

280 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 4A ··· 4H 4I ··· 4T 7A ··· 7F 14A ··· 14AP 14AQ ··· 14DJ 28A ··· 28AV 28AW ··· 28DP order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

280 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C7 C14 C14 C14 C14 C4○D4 C7×C4○D4 kernel C4○D4×C2×C14 C23×C28 D4×C2×C14 Q8×C2×C14 C14×C4○D4 C22×C4○D4 C23×C4 C22×D4 C22×Q8 C2×C4○D4 C2×C14 C22 # reps 1 3 3 1 24 6 18 18 6 144 8 48

Matrix representation of C4○D4×C2×C14 in GL5(𝔽29)

 28 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 28
,
 1 0 0 0 0 0 20 0 0 0 0 0 28 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 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 28 2 0 0 0 28 1
,
 1 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 28

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,20,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28,28,0,0,0,2,1],[1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,28] >;

C4○D4×C2×C14 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_2\times C_{14}
% in TeX

G:=Group("C4oD4xC2xC14");
// GroupNames label

G:=SmallGroup(448,1388);
// by ID

G=gap.SmallGroup(448,1388);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-7,-2,3165,1192]);
// Polycyclic

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

׿
×
𝔽