(C2xDic7):C8 - GroupNames

G = (C₂×Dic₇)⋊C₈ order 448 = 2⁶·7

The semidirect product of C₂×Dic₇ and C₈ acting via C₈/C₂=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×Dic₇)⋊C₈, C₂²⋊C₈.₂D₇, C₂².₄(C₈×D₇), C₂.₆(D₁₄⋊C₈), (C₂×C₄).₁₀₈D₂₈, (C₂×C₂₈).₄₃₉D₄, C₂³.₄₁(C₄×D₇), (C₂²×C₄).₂D₁₄, C₁₄.₄(C₂²⋊C₈), C₁₄.₅(C₂³⋊C₄), (C₂×C₁₄).₂M₄(2), C₂².₄(C₈⋊D₇), (C₂²×Dic₇).₂C₄, C₂².₃₃(D₁₄⋊C₄), C₂₈.₅₅D₄.₁₁C₂, C₂.₁(C₄.₁₂D₂₈), C₁₄.₂(C₄.₁₀D₄), (C₂²×C₂₈).₃₂₃C₂², C₂.₂(C₂³.₁D₁₄), C₇⋊₁(C₂².M₄(2)), (C₂×C₁₄).₂(C₂×C₈), (C₇×C₂²⋊C₈).₂C₂, (C₂×C₄).₂₁₀(C₇⋊D₄), (C₂×Dic₇⋊C₄).₂₄C₂, (C₂²×C₁₄).₂₇(C₂×C₄), (C₂×C₁₄).₄₁(C₂²⋊C₄), SmallGroup(448,26)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — (C₂×Dic₇)⋊C₈

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — C₂²×C₂₈ — C₂×Dic₇⋊C₄ — (C₂×Dic₇)⋊C₈

Lower central

C₇ — C₁₄ — C₂×C₁₄ — (C₂×Dic₇)⋊C₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊C₈

Generators and relations for (C₂×Dic₇)⋊C₈
G = < a,b,c,d | a²=b¹⁴=d⁸=1, c²=b⁷, ab=ba, ac=ca, dad^-1=ab⁷, cbc^-1=b^-1, bd=db, dcd^-1=ab⁷c >

Subgroups: 348 in 78 conjugacy classes, 31 normal (29 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, C₂³, C₁₄, C₁₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, Dic₇, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂²⋊C₈, C₂²⋊C₈, C₂×C₄⋊C₄, C₇⋊C₈, C₅₆, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×C₁₄, C₂².M₄(2), C₂×C₇⋊C₈, Dic₇⋊C₄, C₂×C₅₆, C₂²×Dic₇, C₂²×C₂₈, C₂₈.₅₅D₄, C₇×C₂²⋊C₈, C₂×Dic₇⋊C₄, (C₂×Dic₇)⋊C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, D₇, C₂²⋊C₄, C₂×C₈, M₄(2), D₁₄, C₂²⋊C₈, C₂³⋊C₄, C₄.₁₀D₄, C₄×D₇, D₂₈, C₇⋊D₄, C₂².M₄(2), C₈×D₇, C₈⋊D₇, D₁₄⋊C₄, C₂³.₁D₁₄, D₁₄⋊C₈, C₄.₁₂D₂₈, (C₂×Dic₇)⋊C₈

Smallest permutation representation of (C₂×Dic₇)⋊C₈
►On 224 points

Generators in S₂₂₄

(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)

(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)

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

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

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

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

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

82 conjugacy classes

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

82 irreducible representations

dim	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	C₄	C₈	D₄	D₇	M₄(2)	D₁₄	D₂₈	C₇⋊D₄	C₄×D₇	C₈×D₇	C₈⋊D₇	C₂³⋊C₄	C₄.₁₀D₄	C₂³.₁D₁₄	C₄.₁₂D₂₈
kernel	(C₂×Dic₇)⋊C₈	C₂₈.₅₅D₄	C₇×C₂²⋊C₈	C₂×Dic₇⋊C₄	C₂²×Dic₇	C₂×Dic₇	C₂×C₂₈	C₂²⋊C₈	C₂×C₁₄	C₂²×C₄	C₂×C₄	C₂×C₄	C₂³	C₂²	C₂²	C₁₄	C₁₄	C₂	C₂
# reps	1	1	1	1	4	8	2	3	2	3	6	6	6	12	12	1	1	6	6

Matrix representation of (C₂×Dic₇)⋊C₈ ►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	112	0
0	0	0	0	0	112

96	66	0	0	0	0
66	96	0	0	0	0
0	0	104	1	0	0
0	0	112	0	0	0
0	0	0	0	104	1
0	0	0	0	112	0

74	27	0	0	0	0
86	39	0	0	0	0
0	0	50	42	0	0
0	0	40	63	0	0
0	0	0	0	50	42
0	0	0	0	40	63

0	18	0	0	0	0
18	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	34	5	0	0
0	0	108	79	0	0

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[96,66,0,0,0,0,66,96,0,0,0,0,0,0,104,112,0,0,0,0,1,0,0,0,0,0,0,0,104,112,0,0,0,0,1,0],[74,86,0,0,0,0,27,39,0,0,0,0,0,0,50,40,0,0,0,0,42,63,0,0,0,0,0,0,50,40,0,0,0,0,42,63],[0,18,0,0,0,0,18,0,0,0,0,0,0,0,0,0,34,108,0,0,0,0,5,79,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C₂×Dic₇)⋊C₈ in GAP, Magma, Sage, TeX

(C_2\times {\rm Dic}_7)\rtimes C_8

% in TeX

G:=Group("(C2xDic7):C8");

// GroupNames label

G:=SmallGroup(448,26);

// by ID

G=gap.SmallGroup(448,26);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,758,100,570,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=d^8=1,c^2=b^7,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^7,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^7*c>;

// generators/relations

G = (C2×Dic7)⋊C8 order 448 = 26·7

The semidirect product of C2×Dic7 and C8 acting via C8/C2=C4

G = (C₂×Dic₇)⋊C₈ order 448 = 2⁶·7

The semidirect product of C₂×Dic₇ and C₈ acting via C₈/C₂=C₄