C28:(C4oD4) - GroupNames

G = C₂₈⋊(C₄○D₄) order 448 = 2⁶·7

2^nd semidirect product of C₂₈ and C₄○D₄ acting via C₄○D₄/C₂²=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₈⋊Q₈⋊₁₈C₂, C₂₈⋊₉(C₄○D₄), C₄⋊D₄⋊₂₅D₇, C₂₈⋊D₄⋊₁₄C₂, C₄⋊₃(D₄⋊₂D₇), C₂₈⋊₇D₄⋊₃₁C₂, C₂².₁(D₄×D₇), C₄⋊C₄.₁₇₅D₁₄, (C₂×Dic₇)⋊₁₃D₄, (D₄×Dic₇)⋊₁₄C₂, Dic₇.₅(C₂×D₄), D₂₈⋊C₄⋊₁₉C₂, Dic₇⋊₂(C₄○D₄), Dic₇⋊D₄⋊₇C₂, Dic₇⋊₄D₄⋊₅C₂, (C₂×D₄).₁₅₀D₁₄, (C₂×C₂₈).₃₄C₂³, C₂²⋊C₄.₄₅D₁₄, C₁₄.₅₉(C₂²×D₄), (C₂×C₁₄).₁₄₀C₂⁴, D₁₄⋊C₄.₅₇C₂², (C₂²×C₄).₃₆₆D₁₄, Dic₇.D₄⋊₁₆C₂, (D₄×C₁₄).₁₁₄C₂², (C₂×D₂₈).₁₄₁C₂², Dic₇⋊C₄.₁₂C₂², C₄⋊Dic₇.₂₀₃C₂², (C₂²×C₁₄).₁₁C₂³, C₇⋊₃(C₂².₂₆C₂⁴), (C₂²×D₇).₅₉C₂³, C₂².₁₆₁(C₂³×D₇), C₂³.₁₇₈(C₂²×D₇), C₂³.D₇.₁₈C₂², (C₂²×C₂₈).₂₃₅C₂², (C₄×Dic₇).₂₅₄C₂², (C₂×Dic₇).₂₂₃C₂³, (C₂×Dic₁₄).₁₅₀C₂², (C₂²×Dic₇).₂₂₁C₂², C₂.₃₂(C₂×D₄×D₇), (C₂×C₄×Dic₇)⋊₇C₂, (C₇×C₄⋊D₄)⋊₅C₂, C₂.₃₃(D₇×C₄○D₄), (C₂×C₁₄).₃(C₂×D₄), (C₂×D₄⋊₂D₇)⋊₈C₂, C₁₄.₇₉(C₂×C₄○D₄), (C₂×C₄×D₇).₇₉C₂², C₂.₃₀(C₂×D₄⋊₂D₇), (C₂×C₄).₃₄(C₂²×D₇), (C₇×C₄⋊C₄).₁₃₆C₂², (C₂×C₇⋊D₄).₂₃C₂², (C₇×C₂²⋊C₄).₅C₂², SmallGroup(448,1049)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₂₈⋊(C₄○D₄)

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×Dic₇ — C₂²×Dic₇ — C₂×D₄⋊₂D₇ — C₂₈⋊(C₄○D₄)

Lower central

C₇ — C₂×C₁₄ — C₂₈⋊(C₄○D₄)

Upper central

C₁ — C₂² — C₄⋊D₄

Generators and relations for C₂₈⋊(C₄○D₄)
G = < a,b,c,d | a²⁸=b⁴=d²=1, c²=b², bab^-1=a¹³, cac^-1=a¹⁵, ad=da, bc=cb, bd=db, dcd=b²c >

Subgroups: 1420 in 310 conjugacy classes, 109 normal (43 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, D₇, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₄², C₄×D₄, C₄⋊D₄, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂×C₄○D₄, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, C₂².₂₆C₂⁴, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₂³.D₇, C₇×C₂²⋊C₄, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×D₂₈, D₄⋊₂D₇, C₂²×Dic₇, C₂²×Dic₇, C₂×C₇⋊D₄, C₂²×C₂₈, D₄×C₁₄, D₄×C₁₄, Dic₇⋊₄D₄, Dic₇.D₄, C₂₈⋊Q₈, D₂₈⋊C₄, C₂×C₄×Dic₇, C₂₈⋊₇D₄, D₄×Dic₇, Dic₇⋊D₄, C₂₈⋊D₄, C₇×C₄⋊D₄, C₂×D₄⋊₂D₇, C₂₈⋊(C₄○D₄)
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₄○D₄, C₂⁴, D₁₄, C₂²×D₄, C₂×C₄○D₄, C₂²×D₇, C₂².₂₆C₂⁴, D₄×D₇, D₄⋊₂D₇, C₂³×D₇, C₂×D₄×D₇, C₂×D₄⋊₂D₇, D₇×C₄○D₄, C₂₈⋊(C₄○D₄)

Smallest permutation representation of C₂₈⋊(C₄○D₄)
►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 88 191 38)(2 101 192 51)(3 86 193 36)(4 99 194 49)(5 112 195 34)(6 97 196 47)(7 110 169 32)(8 95 170 45)(9 108 171 30)(10 93 172 43)(11 106 173 56)(12 91 174 41)(13 104 175 54)(14 89 176 39)(15 102 177 52)(16 87 178 37)(17 100 179 50)(18 85 180 35)(19 98 181 48)(20 111 182 33)(21 96 183 46)(22 109 184 31)(23 94 185 44)(24 107 186 29)(25 92 187 42)(26 105 188 55)(27 90 189 40)(28 103 190 53)(57 204 142 121)(58 217 143 134)(59 202 144 119)(60 215 145 132)(61 200 146 117)(62 213 147 130)(63 198 148 115)(64 211 149 128)(65 224 150 113)(66 209 151 126)(67 222 152 139)(68 207 153 124)(69 220 154 137)(70 205 155 122)(71 218 156 135)(72 203 157 120)(73 216 158 133)(74 201 159 118)(75 214 160 131)(76 199 161 116)(77 212 162 129)(78 197 163 114)(79 210 164 127)(80 223 165 140)(81 208 166 125)(82 221 167 138)(83 206 168 123)(84 219 141 136)

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

(57 142)(58 143)(59 144)(60 145)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 141)(113 224)(114 197)(115 198)(116 199)(117 200)(118 201)(119 202)(120 203)(121 204)(122 205)(123 206)(124 207)(125 208)(126 209)(127 210)(128 211)(129 212)(130 213)(131 214)(132 215)(133 216)(134 217)(135 218)(136 219)(137 220)(138 221)(139 222)(140 223)

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

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

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

70 conjugacy classes

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

70 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+			+	+	+	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₇	C₄○D₄	C₄○D₄	D₁₄	D₁₄	D₁₄	D₁₄	D₄⋊₂D₇	D₄×D₇	D₇×C₄○D₄
kernel	C₂₈⋊(C₄○D₄)	Dic₇⋊₄D₄	Dic₇.D₄	C₂₈⋊Q₈	D₂₈⋊C₄	C₂×C₄×Dic₇	C₂₈⋊₇D₄	D₄×Dic₇	Dic₇⋊D₄	C₂₈⋊D₄	C₇×C₄⋊D₄	C₂×D₄⋊₂D₇	C₂×Dic₇	C₄⋊D₄	Dic₇	C₂₈	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×D₄	C₄	C₂²	C₂
# reps	1	2	2	1	1	1	1	1	2	1	1	2	4	3	4	4	6	3	3	9	6	6	6

Matrix representation of C₂₈⋊(C₄○D₄) ►in GL₆(𝔽₂₉)

28	0	0	0	0	0
0	28	0	0	0	0
0	0	17	0	0	0
0	0	0	12	0	0
0	0	0	0	18	7
0	0	0	0	4	0

12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	25	19
0	0	0	0	16	4

0	1	0	0	0	0
28	0	0	0	0	0
0	0	0	28	0	0
0	0	1	0	0	0
0	0	0	0	28	0
0	0	0	0	0	28

28	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	28	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,12,0,0,0,0,0,0,18,4,0,0,0,0,7,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,25,16,0,0,0,0,19,4],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂₈⋊(C₄○D₄) in GAP, Magma, Sage, TeX

C_{28}\rtimes (C_4\circ D_4)

% in TeX

G:=Group("C28:(C4oD4)");

// GroupNames label

G:=SmallGroup(448,1049);

// by ID

G=gap.SmallGroup(448,1049);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,387,570,185,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^28=b^4=d^2=1,c^2=b^2,b*a*b^-1=a^13,c*a*c^-1=a^15,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;

// generators/relations

G = C28⋊(C4○D4) order 448 = 26·7

2nd semidirect product of C28 and C4○D4 acting via C4○D4/C22=C22

G = C₂₈⋊(C₄○D₄) order 448 = 2⁶·7

2^nd semidirect product of C₂₈ and C₄○D₄ acting via C₄○D₄/C₂²=C₂²