C2^3.35D28 - GroupNames

G = C₂³.₃₅D₂₈ order 448 = 2⁶·7

6^th non-split extension by C₂³ of D₂₈ acting via D₂₈/D₁₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₃₅D₂₈, C₂².₃Dic₂₈, C₅₆⋊₁C₄⋊₂C₂, (C₂×C₈).₁D₁₄, (C₂×C₂₈).₄₁D₄, (C₂×C₄).₃₀D₂₈, C₂²⋊C₈.₃D₇, (C₂×C₁₄).₄Q₁₆, C₁₄.₅(C₂×Q₁₆), (C₂×C₅₆).₁C₂², C₂.₇(C₂×Dic₂₈), C₂₈.₄₄D₄⋊₄C₂, C₁₄.₇(C₈⋊C₂²), (C₂²×C₁₄).₄₉D₄, (C₂²×C₄).₇₆D₁₄, C₂₈.₂₈₀(C₄○D₄), C₂.₁₀(C₈⋊D₁₄), (C₂×C₂₈).₇₃₉C₂³, C₂₈.₄₈D₄.₃C₂, C₂².₁₀₂(C₂×D₂₈), C₇⋊₁(C₂³.₄₈D₄), C₄⋊Dic₇.₁₁C₂², C₄.₁₀₄(D₄⋊₂D₇), (C₂²×C₂₈).₄₉C₂², (C₂×Dic₁₄).₁₁C₂², C₁₄.₁₅(C₂².D₄), C₂.₁₁(C₂².D₂₈), (C₇×C₂²⋊C₈).₅C₂, (C₂×C₁₄).₁₂₂(C₂×D₄), (C₂×C₄⋊Dic₇).₁₂C₂, (C₂×C₄).₆₈₄(C₂²×D₇), SmallGroup(448,256)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — C₂³.₃₅D₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₄⋊Dic₇ — C₂×C₄⋊Dic₇ — C₂³.₃₅D₂₈

Lower central

C₇ — C₁₄ — C₂×C₂₈ — C₂³.₃₅D₂₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊C₈

Generators and relations for C₂³.₃₅D₂₈
G = < a,b,c,d,e | a²=b²=c²=1, d²⁸=e²=c, ab=ba, ac=ca, dad^-1=eae^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=bd²⁷ >

Subgroups: 508 in 104 conjugacy classes, 43 normal (25 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₄, C₁₄, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, Dic₇, C₂₈, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂²⋊C₈, Q₈⋊C₄, C₂.D₈, C₂×C₄⋊C₄, C₂²⋊Q₈, C₅₆, Dic₁₄, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×C₁₄, C₂³.₄₈D₄, Dic₇⋊C₄, C₄⋊Dic₇, C₄⋊Dic₇, C₄⋊Dic₇, C₂³.D₇, C₂×C₅₆, C₂×Dic₁₄, C₂²×Dic₇, C₂²×C₂₈, C₂₈.₄₄D₄, C₅₆⋊₁C₄, C₇×C₂²⋊C₈, C₂₈.₄₈D₄, C₂×C₄⋊Dic₇, C₂³.₃₅D₂₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, Q₁₆, C₂×D₄, C₄○D₄, D₁₄, C₂².D₄, C₂×Q₁₆, C₈⋊C₂², D₂₈, C₂²×D₇, C₂³.₄₈D₄, Dic₂₈, C₂×D₂₈, D₄⋊₂D₇, C₂².D₂₈, C₂×Dic₂₈, C₈⋊D₁₄, C₂³.₃₅D₂₈

Smallest permutation representation of C₂³.₃₅D₂₈
►On 224 points

Generators in S₂₂₄

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

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

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

(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)

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

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

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

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

79 conjugacy classes

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

79 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+		-	+	+	+	+	-	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₇	C₄○D₄	Q₁₆	D₁₄	D₁₄	D₂₈	D₂₈	Dic₂₈	C₈⋊C₂²	D₄⋊₂D₇	C₈⋊D₁₄
kernel	C₂³.₃₅D₂₈	C₂₈.₄₄D₄	C₅₆⋊₁C₄	C₇×C₂²⋊C₈	C₂₈.₄₈D₄	C₂×C₄⋊Dic₇	C₂×C₂₈	C₂²×C₁₄	C₂²⋊C₈	C₂₈	C₂×C₁₄	C₂×C₈	C₂²×C₄	C₂×C₄	C₂³	C₂²	C₁₄	C₄	C₂
# reps	1	2	2	1	1	1	1	1	3	4	4	6	3	6	6	24	1	6	6

Matrix representation of C₂³.₃₅D₂₈ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	1	0
0	0	0	0	51	112

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

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

64	0	0	0	0	0
26	83	0	0	0	0
0	0	87	49	0	0
0	0	34	88	0	0
0	0	0	0	62	2
0	0	0	0	56	51

17	95	0	0	0	0
16	96	0	0	0	0
0	0	23	18	0	0
0	0	71	90	0	0
0	0	0	0	87	83
0	0	0	0	79	26

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,51,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,26,0,0,0,0,0,83,0,0,0,0,0,0,87,34,0,0,0,0,49,88,0,0,0,0,0,0,62,56,0,0,0,0,2,51],[17,16,0,0,0,0,95,96,0,0,0,0,0,0,23,71,0,0,0,0,18,90,0,0,0,0,0,0,87,79,0,0,0,0,83,26] >;

C₂³.₃₅D₂₈ in GAP, Magma, Sage, TeX

C_2^3._{35}D_{28}

% in TeX

G:=Group("C2^3.35D28");

// GroupNames label

G:=SmallGroup(448,256);

// by ID

G=gap.SmallGroup(448,256);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,336,254,219,310,1123,136,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=e^2=c,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^27>;

// generators/relations

G = C23.35D28 order 448 = 26·7

6th non-split extension by C23 of D28 acting via D28/D14=C2

G = C₂³.₃₅D₂₈ order 448 = 2⁶·7

6^th non-split extension by C₂³ of D₂₈ acting via D₂₈/D₁₄=C₂