D28:22D4 - GroupNames

G = D₂₈⋊₂₂D₄ order 448 = 2⁶·7

10^th semidirect product of D₂₈ and D₄ acting via D₄/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₈⋊₂₂D₄, C₁₄.₁₈2_-¹⁺⁴, C₂²⋊Q₈⋊₈D₇, C₇⋊₅(D₄⋊₆D₄), C₄.₁₁₂(D₄×D₇), C₄⋊C₄.₁₈₉D₁₄, D₁₄.₂₁(C₂×D₄), C₂₈.₂₃₅(C₂×D₄), D₂₈⋊C₄⋊₂₆C₂, D₁₄⋊D₄⋊₂₅C₂, D₁₄⋊₂Q₈⋊₂₅C₂, Dic₇⋊₅(C₄○D₄), (C₂×Q₈).₁₂₆D₁₄, C₂²⋊C₄.₁₆D₁₄, C₁₄.₇₇(C₂²×D₄), Dic₇⋊Q₈⋊₁₄C₂, D₁₄.D₄⋊₂₅C₂, D₁₄.₅D₄⋊₁₈C₂, (C₂×C₂₈).₅₀₃C₂³, (C₂×C₁₄).₁₇₅C₂⁴, D₁₄⋊C₄.₂₃C₂², (C₂²×C₄).₂₃₇D₁₄, (C₂×D₂₈).₁₄₉C₂², Dic₇⋊C₄.₂₇C₂², C₄⋊Dic₇.₂₁₅C₂², (Q₈×C₁₄).₁₀₇C₂², C₂².₁₉₆(C₂³×D₇), C₂³.₁₁₉(C₂²×D₇), (C₂²×C₂₈).₂₅₅C₂², (C₂²×C₁₄).₂₀₃C₂³, (C₄×Dic₇).₁₀₅C₂², (C₂×Dic₇).₂₃₄C₂³, (C₂²×D₇).₁₉₇C₂³, C₂³.D₇.₁₁₆C₂², C₂.₁₉(Q₈.₁₀D₁₄), (C₂×Dic₁₄).₂₉₄C₂², C₂.₅₀(C₂×D₄×D₇), (D₇×C₄⋊C₄)⋊₂₆C₂, (C₄×C₇⋊D₄)⋊₂₃C₂, C₂.₄₉(D₇×C₄○D₄), (C₂×C₄○D₂₈)⋊₂₄C₂, (C₂×Q₈⋊₂D₇)⋊₈C₂, (C₇×C₂²⋊Q₈)⋊₁₁C₂, (C₂×C₄×D₇).₉₅C₂², C₁₄.₁₆₁(C₂×C₄○D₄), (C₂×C₄).₄₈(C₂²×D₇), (C₇×C₄⋊C₄).₁₅₈C₂², (C₂×C₇⋊D₄).₁₂₃C₂², (C₇×C₂²⋊C₄).₃₀C₂², SmallGroup(448,1084)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₂₈⋊₂₂D₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — D₇×C₄⋊C₄ — D₂₈⋊₂₂D₄

Lower central

C₇ — C₂×C₁₄ — D₂₈⋊₂₂D₄

Upper central

C₁ — C₂² — C₂²⋊Q₈

Generators and relations for D₂₈⋊₂₂D₄
G = < a,b,c,d | a²⁸=b²=c⁴=d²=1, bab=a^-1, cac^-1=dad=a¹³, cbc^-1=a¹²b, dbd=a²⁶b, dcd=c^-1 >

Subgroups: 1404 in 292 conjugacy classes, 105 normal (43 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₄⋊C₄, C₄×D₄, C₄⋊D₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂².D₄, C₄⋊Q₈, C₂×C₄○D₄, Dic₁₄, C₄×D₇, D₂₈, D₂₈, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, D₄⋊₆D₄, C₄×Dic₇, Dic₇⋊C₄, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, D₁₄⋊C₄, C₂³.D₇, C₇×C₂²⋊C₄, C₇×C₄⋊C₄, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×C₄×D₇, C₂×D₂₈, C₂×D₂₈, C₄○D₂₈, Q₈⋊₂D₇, C₂×C₇⋊D₄, C₂×C₇⋊D₄, C₂²×C₂₈, Q₈×C₁₄, D₁₄.D₄, D₁₄⋊D₄, D₇×C₄⋊C₄, D₂₈⋊C₄, D₁₄.₅D₄, D₁₄⋊₂Q₈, C₄×C₇⋊D₄, Dic₇⋊Q₈, C₇×C₂²⋊Q₈, C₂×C₄○D₂₈, C₂×Q₈⋊₂D₇, D₂₈⋊₂₂D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₄○D₄, C₂⁴, D₁₄, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂²×D₇, D₄⋊₆D₄, D₄×D₇, C₂³×D₇, C₂×D₄×D₇, Q₈.₁₀D₁₄, D₇×C₄○D₄, D₂₈⋊₂₂D₄

Smallest permutation representation of D₂₈⋊₂₂D₄
►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)

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

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

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

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

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

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

67 conjugacy classes

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

67 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₇	C₄○D₄	D₁₄	D₁₄	D₁₄	D₁₄	2_-¹⁺⁴	D₄×D₇	Q₈.₁₀D₁₄	D₇×C₄○D₄
kernel	D₂₈⋊₂₂D₄	D₁₄.D₄	D₁₄⋊D₄	D₇×C₄⋊C₄	D₂₈⋊C₄	D₁₄.₅D₄	D₁₄⋊₂Q₈	C₄×C₇⋊D₄	Dic₇⋊Q₈	C₇×C₂²⋊Q₈	C₂×C₄○D₂₈	C₂×Q₈⋊₂D₇	D₂₈	C₂²⋊Q₈	Dic₇	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×Q₈	C₁₄	C₄	C₂	C₂
# reps	1	2	2	2	1	2	1	1	1	1	1	1	4	3	4	6	9	3	3	1	6	6	6

Matrix representation of D₂₈⋊₂₂D₄ ►in GL₆(𝔽₂₉)

0	12	0	0	0	0
12	0	0	0	0	0
0	0	8	26	0	0
0	0	3	28	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	28	0	0	0	0
0	0	8	26	0	0
0	0	21	21	0	0
0	0	0	0	28	0
0	0	0	0	0	28

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	3	28	0	0
0	0	0	0	14	9
0	0	0	0	20	15

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	3	28	0	0
0	0	0	0	20	15
0	0	0	0	14	9

G:=sub<GL(6,GF(29))| [0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,3,0,0,0,0,26,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,8,21,0,0,0,0,26,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,0,14,20,0,0,0,0,9,15],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,0,20,14,0,0,0,0,15,9] >;

D₂₈⋊₂₂D₄ in GAP, Magma, Sage, TeX

D_{28}\rtimes_{22}D_4

% in TeX

G:=Group("D28:22D4");

// GroupNames label

G:=SmallGroup(448,1084);

// by ID

G=gap.SmallGroup(448,1084);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,1571,570,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^13,c*b*c^-1=a^12*b,d*b*d=a^26*b,d*c*d=c^-1>;

// generators/relations

G = D28⋊22D4 order 448 = 26·7

10th semidirect product of D28 and D4 acting via D4/C22=C2

G = D₂₈⋊₂₂D₄ order 448 = 2⁶·7

10^th semidirect product of D₂₈ and D₄ acting via D₄/C₂²=C₂