C8:(C4xD7) - GroupNames

G = C₈⋊(C₄×D₇) order 448 = 2⁶·7

3^rd semidirect product of C₈ and C₄×D₇ acting via C₄×D₇/C₁₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₈⋊₃(C₄×D₇), C₅₆⋊₇(C₂×C₄), C₄.Q₈⋊₂D₇, C₈⋊D₇⋊₁C₄, (C₄×D₇).₁Q₈, C₄.₂₅(Q₈×D₇), C₅₆⋊₁C₄⋊₂₅C₂, (C₂×C₈).₆₀D₁₄, C₂₈.₁₄(C₂×Q₈), D₁₄.₄(C₄⋊C₄), C₄⋊C₄.₁₆₂D₁₄, C₂².₈₅(D₄×D₇), C₂₈.Q₈⋊₁₅C₂, C₂.₅(D₅₆⋊C₂), Dic₇.₅(C₄⋊C₄), C₂₈.₄₄(C₂²×C₄), C₄.Dic₁₄⋊₁₅C₂, (C₂²×D₇).₈₁D₄, C₁₄.₆₈(C₈⋊C₂²), C₇⋊₁(M₄(2)⋊C₄), (C₂×C₂₈).₂₇₇C₂³, (C₂×C₅₆).₁₀₉C₂², (C₂×Dic₇).₁₆₂D₄, C₂.₆(SD₁₆⋊D₇), C₁₄.₄₁(C₈.C₂²), C₄⋊Dic₇.₁₀₉C₂², C₇⋊C₈⋊₄(C₂×C₄), C₄.₇₈(C₂×C₄×D₇), (D₇×C₄⋊C₄).₅C₂, C₂.₁₃(D₇×C₄⋊C₄), (C₇×C₄.Q₈)⋊₂C₂, C₁₄.₁₂(C₂×C₄⋊C₄), (C₄×D₇).₆(C₂×C₄), C₄⋊C₄⋊₇D₇.₅C₂, (C₂×C₈⋊D₇).₂C₂, (C₂×C₇⋊C₈).₅₅C₂², (C₂×C₄×D₇).₃₀C₂², (C₂×C₁₄).₂₈₂(C₂×D₄), (C₇×C₄⋊C₄).₇₀C₂², (C₂×C₄).₃₈₀(C₂²×D₇), SmallGroup(448,395)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — C₈⋊(C₄×D₇)

Chief series

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

Lower central

C₇ — C₁₄ — C₂₈ — C₈⋊(C₄×D₇)

Upper central

C₁ — C₂² — C₂×C₄ — C₄.Q₈

Generators and relations for C₈⋊(C₄×D₇)
G = < a,b,c,d | a⁸=b⁴=c⁷=d²=1, bab^-1=a³, ac=ca, dad=a⁵, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 556 in 118 conjugacy classes, 55 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, D₇, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₄.Q₈, C₄.Q₈, C₂.D₈, C₂×C₄⋊C₄, C₄²⋊C₂, C₂×M₄(2), C₇⋊C₈, C₅₆, C₄×D₇, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×D₇, M₄(2)⋊C₄, C₈⋊D₇, C₂×C₇⋊C₈, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₇×C₄⋊C₄, C₂×C₅₆, C₂×C₄×D₇, C₂×C₄×D₇, C₂₈.Q₈, C₄.Dic₁₄, C₅₆⋊₁C₄, C₇×C₄.Q₈, D₇×C₄⋊C₄, C₄⋊C₄⋊₇D₇, C₂×C₈⋊D₇, C₈⋊(C₄×D₇)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, D₇, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, D₁₄, C₂×C₄⋊C₄, C₈⋊C₂², C₈.C₂², C₄×D₇, C₂²×D₇, M₄(2)⋊C₄, C₂×C₄×D₇, D₄×D₇, Q₈×D₇, D₇×C₄⋊C₄, D₅₆⋊C₂, SD₁₆⋊D₇, C₈⋊(C₄×D₇)

Smallest permutation representation of C₈⋊(C₄×D₇)
►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)

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

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

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

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

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

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

64 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	28A	···	28F	28G	···	28R	56A	···	56L
order	1	2	2	2	2	2	4	4	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	14	14	2	2	4	4	4	4	14	14	28	28	28	28	2	2	2	4	4	28	28	2	···	2	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	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+		-	+	+	+	+	+		+	-	-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	Q₈	D₄	D₄	D₇	D₁₄	D₁₄	C₄×D₇	C₈⋊C₂²	C₈.C₂²	Q₈×D₇	D₄×D₇	D₅₆⋊C₂	SD₁₆⋊D₇
kernel	C₈⋊(C₄×D₇)	C₂₈.Q₈	C₄.Dic₁₄	C₅₆⋊₁C₄	C₇×C₄.Q₈	D₇×C₄⋊C₄	C₄⋊C₄⋊₇D₇	C₂×C₈⋊D₇	C₈⋊D₇	C₄×D₇	C₂×Dic₇	C₂²×D₇	C₄.Q₈	C₄⋊C₄	C₂×C₈	C₈	C₁₄	C₁₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	8	2	1	1	3	6	3	12	1	1	3	3	6	6

Matrix representation of C₈⋊(C₄×D₇) ►in GL₆(𝔽₁₁₃)

16	30	0	0	0	0
63	97	0	0	0	0
0	0	0	0	106	96
0	0	0	0	3	7
0	0	60	65	106	96
0	0	55	53	3	7

98	13	0	0	0	0
0	15	0	0	0	0
0	0	110	0	31	0
0	0	0	110	0	31
0	0	69	0	3	0
0	0	0	69	0	3

1	0	0	0	0	0
0	1	0	0	0	0
0	0	34	112	0	0
0	0	60	88	0	0
0	0	0	0	34	112
0	0	0	0	60	88

112	0	0	0	0	0
0	112	0	0	0	0
0	0	88	33	0	0
0	0	53	25	0	0
0	0	0	0	88	33
0	0	0	0	53	25

G:=sub<GL(6,GF(113))| [16,63,0,0,0,0,30,97,0,0,0,0,0,0,0,0,60,55,0,0,0,0,65,53,0,0,106,3,106,3,0,0,96,7,96,7],[98,0,0,0,0,0,13,15,0,0,0,0,0,0,110,0,69,0,0,0,0,110,0,69,0,0,31,0,3,0,0,0,0,31,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,60,0,0,0,0,112,88,0,0,0,0,0,0,34,60,0,0,0,0,112,88],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,88,53,0,0,0,0,33,25,0,0,0,0,0,0,88,53,0,0,0,0,33,25] >;

C₈⋊(C₄×D₇) in GAP, Magma, Sage, TeX

C_8\rtimes (C_4\times D_7)

% in TeX

G:=Group("C8:(C4xD7)");

// GroupNames label

G:=SmallGroup(448,395);

// by ID

G=gap.SmallGroup(448,395);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,555,58,438,102,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^4=c^7=d^2=1,b*a*b^-1=a^3,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = C8⋊(C4×D7) order 448 = 26·7

3rd semidirect product of C8 and C4×D7 acting via C4×D7/C14=C22

G = C₈⋊(C₄×D₇) order 448 = 2⁶·7

3^rd semidirect product of C₈ and C₄×D₇ acting via C₄×D₇/C₁₄=C₂²