C2^3.34D28 - GroupNames

G = C₂³.₃₄D₂₈ order 448 = 2⁶·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₃₄D₂₈, C₈⋊Dic₇⋊₆C₂, (C₂×C₂₈).₄₀D₄, (C₂×C₄).₂₉D₂₈, C₂²⋊C₈.₆D₇, (C₂×C₈).₁₀₇D₁₄, C₁₄.₆(C₂×SD₁₆), C₂₈.₄₄D₄⋊₉C₂, (C₂×C₁₄).₁₃SD₁₆, (C₂²×C₄).₇₅D₁₄, (C₂²×C₁₄).₄₈D₄, C₂₈.₂₇₉(C₄○D₄), (C₂×C₂₈).₇₃₈C₂³, (C₂×C₅₆).₁₁₈C₂², C₂₈.₄₈D₄.₂C₂, C₂².₁₀₁(C₂×D₂₈), C₂².₈(C₅₆⋊C₂), C₁₄.₇(C₈.C₂²), C₇⋊₁(C₂³.₄₇D₄), C₄.₁₀₃(D₄⋊₂D₇), C₂.₁₀(C₈.D₁₄), C₄⋊Dic₇.₂₆₈C₂², (C₂²×C₂₈).₄₈C₂², (C₂×Dic₁₄).₁₀C₂², C₁₄.₁₄(C₂².D₄), C₂.₁₀(C₂².D₂₈), C₂.₉(C₂×C₅₆⋊C₂), (C₇×C₂²⋊C₈).₈C₂, (C₂×C₁₄).₁₂₁(C₂×D₄), (C₂×C₄⋊Dic₇).₁₁C₂, (C₂×C₄).₆₈₃(C₂²×D₇), SmallGroup(448,255)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — C₂³.₃₄D₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₄⋊Dic₇ — C₂×C₄⋊Dic₇ — C₂³.₃₄D₂₈

Lower central

C₇ — C₁₄ — C₂×C₂₈ — C₂³.₃₄D₂₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊C₈

Generators and relations for C₂³.₃₄D₂₈
G = < a,b,c,d,e | a²=b²=c²=1, d²⁸=c, e²=cb=bc, dad^-1=eae^-1=ab=ba, ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d²⁷ >

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, C₄.Q₈, C₂×C₄⋊C₄, C₂²⋊Q₈, C₅₆, Dic₁₄, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×C₁₄, C₂³.₄₇D₄, Dic₇⋊C₄, C₄⋊Dic₇, C₄⋊Dic₇, C₄⋊Dic₇, C₂³.D₇, C₂×C₅₆, C₂×Dic₁₄, C₂²×Dic₇, C₂²×C₂₈, C₂₈.₄₄D₄, C₈⋊Dic₇, C₇×C₂²⋊C₈, C₂₈.₄₈D₄, C₂×C₄⋊Dic₇, C₂³.₃₄D₂₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, SD₁₆, C₂×D₄, C₄○D₄, D₁₄, C₂².D₄, C₂×SD₁₆, C₈.C₂², D₂₈, C₂²×D₇, C₂³.₄₇D₄, C₅₆⋊C₂, C₂×D₂₈, D₄⋊₂D₇, C₂².D₂₈, C₂×C₅₆⋊C₂, C₈.D₁₄, C₂³.₃₄D₂₈

Smallest permutation representation of C₂³.₃₄D₂₈
►On 224 points

Generators in S₂₂₄

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

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

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

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

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

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

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	SD₁₆	D₁₄	D₁₄	D₂₈	D₂₈	C₅₆⋊C₂	C₈.C₂²	D₄⋊₂D₇	C₈.D₁₄
kernel	C₂³.₃₄D₂₈	C₂₈.₄₄D₄	C₈⋊Dic₇	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 C₂³.₃₄D₂₈ ►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	1	0
0	0	0	0	87	112

1	0	0	0	0	0
0	1	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

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	1	0
0	0	0	0	0	1

3	49	0	0	0	0
66	23	0	0	0	0
0	0	89	104	0	0
0	0	99	103	0	0
0	0	0	0	1	87
0	0	0	0	87	112

10	25	0	0	0	0
5	103	0	0	0	0
0	0	106	72	0	0
0	0	37	7	0	0
0	0	0	0	98	51
0	0	0	0	0	15

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,87,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,66,0,0,0,0,49,23,0,0,0,0,0,0,89,99,0,0,0,0,104,103,0,0,0,0,0,0,1,87,0,0,0,0,87,112],[10,5,0,0,0,0,25,103,0,0,0,0,0,0,106,37,0,0,0,0,72,7,0,0,0,0,0,0,98,0,0,0,0,0,51,15] >;

C₂³.₃₄D₂₈ in GAP, Magma, Sage, TeX

C_2^3._{34}D_{28}

% in TeX

G:=Group("C2^3.34D28");

// GroupNames label

G:=SmallGroup(448,255);

// by ID

G=gap.SmallGroup(448,255);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,254,219,58,1123,136,18822]);

// Polycyclic

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

// generators/relations

G = C23.34D28 order 448 = 26·7

5th non-split extension by C23 of D28 acting via D28/D14=C2

G = C₂³.₃₄D₂₈ order 448 = 2⁶·7

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