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## G = D8.10D14order 448 = 26·7

### The non-split extension by D8 of D14 acting through Inn(D8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D8.10D14
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C4○D28 — D4.10D14 — D8.10D14
 Lower central C7 — C14 — C28 — D8.10D14
 Upper central C1 — C2 — C2×C4 — C4○D8

Generators and relations for D8.10D14
G = < a,b,c,d | a28=b2=1, c4=d2=a14, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a14c3 >

Subgroups: 1108 in 248 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, SD16, SD16, Q16, Q16, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C14, C8○D4, C2×Q16, C4○D8, C4○D8, C8.C22, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, Q8○D8, C8×D7, C8⋊D7, Dic28, C4.Dic7, D4.D7, C7⋊Q16, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×Dic14, C2×Dic14, C4○D28, C4○D28, D42D7, D42D7, Q8×D7, C7×C4○D4, D28.2C4, C2×Dic28, D83D7, SD16⋊D7, D7×Q16, D4.9D14, C7×C4○D8, D4.10D14, D8.10D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, Q8○D8, D4×D7, C23×D7, C2×D4×D7, D8.10D14

Smallest permutation representation of D8.10D14
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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 31)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(74 84)(75 83)(76 82)(77 81)(78 80)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(113 137)(114 136)(115 135)(116 134)(117 133)(118 132)(119 131)(120 130)(121 129)(122 128)(123 127)(124 126)(138 140)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 168)(159 167)(160 166)(161 165)(162 164)(169 173)(170 172)(174 196)(175 195)(176 194)(177 193)(178 192)(179 191)(180 190)(181 189)(182 188)(183 187)(184 186)(197 215)(198 214)(199 213)(200 212)(201 211)(202 210)(203 209)(204 208)(205 207)(216 224)(217 223)(218 222)(219 221)
(1 95 55 146 15 109 41 160)(2 96 56 147 16 110 42 161)(3 97 29 148 17 111 43 162)(4 98 30 149 18 112 44 163)(5 99 31 150 19 85 45 164)(6 100 32 151 20 86 46 165)(7 101 33 152 21 87 47 166)(8 102 34 153 22 88 48 167)(9 103 35 154 23 89 49 168)(10 104 36 155 24 90 50 141)(11 105 37 156 25 91 51 142)(12 106 38 157 26 92 52 143)(13 107 39 158 27 93 53 144)(14 108 40 159 28 94 54 145)(57 131 191 198 71 117 177 212)(58 132 192 199 72 118 178 213)(59 133 193 200 73 119 179 214)(60 134 194 201 74 120 180 215)(61 135 195 202 75 121 181 216)(62 136 196 203 76 122 182 217)(63 137 169 204 77 123 183 218)(64 138 170 205 78 124 184 219)(65 139 171 206 79 125 185 220)(66 140 172 207 80 126 186 221)(67 113 173 208 81 127 187 222)(68 114 174 209 82 128 188 223)(69 115 175 210 83 129 189 224)(70 116 176 211 84 130 190 197)
(1 62 15 76)(2 63 16 77)(3 64 17 78)(4 65 18 79)(5 66 19 80)(6 67 20 81)(7 68 21 82)(8 69 22 83)(9 70 23 84)(10 71 24 57)(11 72 25 58)(12 73 26 59)(13 74 27 60)(14 75 28 61)(29 184 43 170)(30 185 44 171)(31 186 45 172)(32 187 46 173)(33 188 47 174)(34 189 48 175)(35 190 49 176)(36 191 50 177)(37 192 51 178)(38 193 52 179)(39 194 53 180)(40 195 54 181)(41 196 55 182)(42 169 56 183)(85 207 99 221)(86 208 100 222)(87 209 101 223)(88 210 102 224)(89 211 103 197)(90 212 104 198)(91 213 105 199)(92 214 106 200)(93 215 107 201)(94 216 108 202)(95 217 109 203)(96 218 110 204)(97 219 111 205)(98 220 112 206)(113 151 127 165)(114 152 128 166)(115 153 129 167)(116 154 130 168)(117 155 131 141)(118 156 132 142)(119 157 133 143)(120 158 134 144)(121 159 135 145)(122 160 136 146)(123 161 137 147)(124 162 138 148)(125 163 139 149)(126 164 140 150)```

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 8A 8B 8C 8D 8E 14A 14B 14C 14D 14E 14F 14G ··· 14L 28A ··· 28F 28G 28H 28I 28J ··· 28O 56A ··· 56L order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 8 14 14 14 14 14 14 14 ··· 14 28 ··· 28 28 28 28 28 ··· 28 56 ··· 56 size 1 1 2 4 4 14 14 2 2 4 4 14 14 28 28 28 28 2 2 2 2 2 4 28 28 2 2 2 4 4 4 8 ··· 8 2 ··· 2 4 4 4 8 ··· 8 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 D14 D14 Q8○D8 D4×D7 D4×D7 D8.10D14 kernel D8.10D14 D28.2C4 C2×Dic28 D8⋊3D7 SD16⋊D7 D7×Q16 D4.9D14 C7×C4○D8 D4.10D14 Dic14 D28 C7⋊D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C7 C4 C22 C1 # reps 1 1 1 2 4 2 2 1 2 1 1 2 3 3 3 6 3 6 2 3 3 12

Matrix representation of D8.10D14 in GL4(𝔽113) generated by

 4 36 0 0 41 2 0 0 0 0 96 36 0 0 77 23
,
 23 103 0 0 98 90 0 0 0 0 1 0 0 0 89 112
,
 62 0 32 58 0 62 74 94 96 73 0 0 23 100 0 0
,
 97 0 62 70 0 97 101 55 67 38 16 0 29 18 0 16
`G:=sub<GL(4,GF(113))| [4,41,0,0,36,2,0,0,0,0,96,77,0,0,36,23],[23,98,0,0,103,90,0,0,0,0,1,89,0,0,0,112],[62,0,96,23,0,62,73,100,32,74,0,0,58,94,0,0],[97,0,67,29,0,97,38,18,62,101,16,0,70,55,0,16] >;`

D8.10D14 in GAP, Magma, Sage, TeX

`D_8._{10}D_{14}`
`% in TeX`

`G:=Group("D8.10D14");`
`// GroupNames label`

`G:=SmallGroup(448,1224);`
`// by ID`

`G=gap.SmallGroup(448,1224);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,184,570,185,136,438,235,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=1,c^4=d^2=a^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^14*c^3>;`
`// generators/relations`

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