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## G = C14×C4.10D4order 448 = 26·7

### Direct product of C14 and C4.10D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C14×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C4.10D4 — C14×C4.10D4
 Lower central C1 — C2 — C22 — C14×C4.10D4
 Upper central C1 — C2×C14 — C22×C28 — C14×C4.10D4

Generators and relations for C14×C4.10D4
G = < a,b,c,d | a14=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 210 in 146 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C14, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C28, C28, C2×C14, C2×C14, C4.10D4, C2×M4(2), C22×Q8, C56, C2×C28, C2×C28, C2×C28, C7×Q8, C22×C14, C2×C4.10D4, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C22×C28, Q8×C14, Q8×C14, C7×C4.10D4, C14×M4(2), Q8×C2×C14, C14×C4.10D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C4.10D4, C2×C22⋊C4, C2×C28, C7×D4, C22×C14, C2×C4.10D4, C7×C22⋊C4, C22×C28, D4×C14, C7×C4.10D4, C14×C22⋊C4, C14×C4.10D4

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

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A ··· 7F 8A ··· 8H 14A ··· 14R 14S ··· 14AD 28A ··· 28X 28Y ··· 28AV 56A ··· 56AV order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + - image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 D4 C7×D4 C4.10D4 C7×C4.10D4 kernel C14×C4.10D4 C7×C4.10D4 C14×M4(2) Q8×C2×C14 C22×C28 Q8×C14 C2×C4.10D4 C4.10D4 C2×M4(2) C22×Q8 C22×C4 C2×Q8 C2×C28 C2×C4 C14 C2 # reps 1 4 2 1 4 4 6 24 12 6 24 24 4 24 2 12

Matrix representation of C14×C4.10D4 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 106 0 0 0 0 0 0 106 0 0 0 0 0 0 106 0 0 0 0 0 0 106
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 112 0 0 0 0 1 0 0 0 0 0 75 32 1 111 0 0 110 35 1 112
,
 59 88 0 0 0 0 76 54 0 0 0 0 0 0 0 0 112 0 0 0 38 81 112 2 0 0 0 112 0 0 0 0 99 109 3 32
,
 28 13 0 0 0 0 18 85 0 0 0 0 0 0 20 1 97 7 0 0 90 61 9 88 0 0 53 69 0 0 0 0 35 1 109 32

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,75,110,0,0,112,0,32,35,0,0,0,0,1,1,0,0,0,0,111,112],[59,76,0,0,0,0,88,54,0,0,0,0,0,0,0,38,0,99,0,0,0,81,112,109,0,0,112,112,0,3,0,0,0,2,0,32],[28,18,0,0,0,0,13,85,0,0,0,0,0,0,20,90,53,35,0,0,1,61,69,1,0,0,97,9,0,109,0,0,7,88,0,32] >;

C14×C4.10D4 in GAP, Magma, Sage, TeX

C_{14}\times C_4._{10}D_4
% in TeX

G:=Group("C14xC4.10D4");
// GroupNames label

G:=SmallGroup(448,820);
// by ID

G=gap.SmallGroup(448,820);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1576,9804,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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