(C2xC8).D14 - GroupNames

G = (C₂×C₈).D₁₄ order 448 = 2⁶·7

125^th non-split extension by C₂×C₈ of D₁₄ acting via D₁₄/C₇=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₄⋊C₈.₆C₂, C₈⋊Dic₇⋊₁₅C₂, C₄⋊C₄.₁₅₄D₁₄, Q₈⋊C₄⋊₁₃D₇, (C₂×C₈).₁₂₅D₁₄, (C₂×Q₈).₂₁D₁₄, C₄.₅₈(C₄○D₂₈), C₁₄.₅₀(C₄○D₈), C₂₈.Q₈⋊₁₂C₂, Q₈⋊Dic₇⋊₁₀C₂, D₁₄⋊₃Q₈.₅C₂, (C₂²×D₇).₂₁D₄, C₂².₂₀₅(D₄×D₇), C₂₈.₁₆₄(C₄○D₄), C₄.₈₉(D₄⋊₂D₇), (C₂×C₂₈).₂₅₅C₂³, (C₂×C₅₆).₁₃₆C₂², (C₂×Dic₇).₁₅₇D₄, C₇⋊₃(C₂³.₂₀D₄), C₄⋊Dic₇.₉₉C₂², (Q₈×C₁₄).₃₈C₂², C₂.₁₇(Q₁₆⋊D₇), C₁₄.₆₃(C₈.C₂²), C₂.₁₉(SD₁₆⋊₃D₇), C₂.₁₉(D₁₄.D₄), C₁₄.₂₇(C₂².D₄), C₄⋊C₄⋊₇D₇.₃C₂, (C₂×C₇⋊C₈).₄₅C₂², (C₂×C₄×D₇).₂₇C₂², (C₇×Q₈⋊C₄)⋊₁₃C₂, (C₂×C₁₄).₂₆₈(C₂×D₄), (C₇×C₄⋊C₄).₅₆C₂², (C₂×C₄).₃₆₂(C₂²×D₇), SmallGroup(448,349)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — (C₂×C₈).D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — C₂×C₄×D₇ — C₄⋊C₄⋊₇D₇ — (C₂×C₈).D₁₄

Lower central

C₇ — C₁₄ — C₂×C₂₈ — (C₂×C₈).D₁₄

Upper central

C₁ — C₂² — C₂×C₄ — Q₈⋊C₄

Generators and relations for (C₂×C₈).D₁₄
G = < a,b,c,d | a²=b⁸=1, c¹⁴=a, d²=ab⁴, ab=ba, ac=ca, ad=da, cbc^-1=ab³, dbd^-1=b³, dcd^-1=b⁴c¹³ >

Subgroups: 468 in 96 conjugacy classes, 37 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, D₇, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂×Q₈, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂²⋊C₈, Q₈⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₂²⋊Q₈, C₇⋊C₈, C₅₆, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂³.₂₀D₄, C₂×C₇⋊C₈, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₇×C₄⋊C₄, C₂×C₅₆, C₂×C₄×D₇, Q₈×C₁₄, C₂₈.Q₈, C₈⋊Dic₇, D₁₄⋊C₈, Q₈⋊Dic₇, C₇×Q₈⋊C₄, C₄⋊C₄⋊₇D₇, D₁₄⋊₃Q₈, (C₂×C₈).D₁₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₄○D₄, D₁₄, C₂².D₄, C₄○D₈, C₈.C₂², C₂²×D₇, C₂³.₂₀D₄, C₄○D₂₈, D₄×D₇, D₄⋊₂D₇, D₁₄.D₄, SD₁₆⋊₃D₇, Q₁₆⋊D₇, (C₂×C₈).D₁₄

Smallest permutation representation of (C₂×C₈).D₁₄
►On 224 points

Generators in S₂₂₄

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

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

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

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

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

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

61 conjugacy classes

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

61 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+		+	+	+			-	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₇	C₄○D₄	D₁₄	D₁₄	D₁₄	C₄○D₈	C₄○D₂₈	C₈.C₂²	D₄⋊₂D₇	D₄×D₇	SD₁₆⋊₃D₇	Q₁₆⋊D₇
kernel	(C₂×C₈).D₁₄	C₂₈.Q₈	C₈⋊Dic₇	D₁₄⋊C₈	Q₈⋊Dic₇	C₇×Q₈⋊C₄	C₄⋊C₄⋊₇D₇	D₁₄⋊₃Q₈	C₂×Dic₇	C₂²×D₇	Q₈⋊C₄	C₂₈	C₄⋊C₄	C₂×C₈	C₂×Q₈	C₁₄	C₄	C₁₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	1	1	3	4	3	3	3	4	12	1	3	3	6	6

Matrix representation of (C₂×C₈).D₁₄ ►in GL₆(𝔽₁₁₃)

112	0	0	0	0	0
0	112	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

76	108	0	0	0	0
25	37	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	69	0
0	0	0	0	0	18

98	46	0	0	0	0
0	15	0	0	0	0
0	0	10	10	0	0
0	0	103	24	0	0
0	0	0	0	0	95
0	0	0	0	69	0

98	0	0	0	0	0
0	98	0	0	0	0
0	0	10	10	0	0
0	0	24	103	0	0
0	0	0	0	0	18
0	0	0	0	69	0

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[76,25,0,0,0,0,108,37,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,69,0,0,0,0,0,0,18],[98,0,0,0,0,0,46,15,0,0,0,0,0,0,10,103,0,0,0,0,10,24,0,0,0,0,0,0,0,69,0,0,0,0,95,0],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,10,24,0,0,0,0,10,103,0,0,0,0,0,0,0,69,0,0,0,0,18,0] >;

(C₂×C₈).D₁₄ in GAP, Magma, Sage, TeX

(C_2\times C_8).D_{14}

% in TeX

G:=Group("(C2xC8).D14");

// GroupNames label

G:=SmallGroup(448,349);

// by ID

G=gap.SmallGroup(448,349);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,64,926,219,184,851,438,102,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^14=a,d^2=a*b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=b^3,d*c*d^-1=b^4*c^13>;

// generators/relations

G = (C2×C8).D14 order 448 = 26·7

125th non-split extension by C2×C8 of D14 acting via D14/C7=C22

G = (C₂×C₈).D₁₄ order 448 = 2⁶·7

125^th non-split extension by C₂×C₈ of D₁₄ acting via D₁₄/C₇=C₂²