C2^3.16D28 - GroupNames

G = C₂³.₁₆D₂₈ order 448 = 2⁶·7

9^th non-split extension by C₂³ of D₂₈ acting via D₂₈/C₁₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₁₆D₂₈, C₂⁴.₁₆D₁₄, (C₂×C₂₈)⋊₇D₄, C₁₄.₃₄C₂²≀C₂, C₂.₇(C₂₈⋊₂D₄), C₂.₈(C₂₈⋊₇D₄), (C₂×Dic₇).₅₇D₄, (C₂²×D₇).₃₂D₄, (C₂²×C₁₄).₇₀D₄, C₂².₂₄₃(D₄×D₇), (C₂²×C₄).₃₆D₁₄, C₁₄.₆₀(C₄⋊D₄), C₂².₁₂₇(C₂×D₂₈), C₇⋊₃(C₂³.₁₀D₄), C₂.₃₅(C₂²⋊D₂₈), C₁₄.C₄²⋊₁₈C₂, C₁₄.₃₆(C₄.₄D₄), (C₂²×C₂₈).₆₂C₂², (C₂³×C₁₄).₄₄C₂², (C₂³×D₇).₁₇C₂², C₂³.₃₇₃(C₂²×D₇), C₂.₁₁(Dic₇⋊D₄), C₂.₂₃(D₁₄.D₄), C₂².₁₀₁(C₄○D₂₈), C₂².₉₈(D₄⋊₂D₇), (C₂²×C₁₄).₃₃₅C₂³, C₂.₂₃(Dic₇.D₄), C₂.₁₇(C₂².D₂₈), C₁₄.₃₅(C₂².D₄), (C₂²×Dic₇).₄₇C₂², (C₂×D₁₄⋊C₄)⋊₉C₂, (C₂×C₄)⋊₄(C₇⋊D₄), (C₂×C₂²⋊C₄)⋊₉D₇, (C₂×C₄⋊Dic₇)⋊₁₃C₂, (C₂×C₂³.D₇)⋊₆C₂, (C₁₄×C₂²⋊C₄)⋊₁₂C₂, (C₂×C₁₄).₃₂₆(C₂×D₄), (C₂²×C₇⋊D₄).₆C₂, (C₂×C₁₄).₈₁(C₄○D₄), C₂².₁₂₉(C₂×C₇⋊D₄), SmallGroup(448,495)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₄ — C₂³.₁₆D₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — C₂³×D₇ — C₂×D₁₄⋊C₄ — C₂³.₁₆D₂₈

Lower central

C₇ — C₂²×C₁₄ — C₂³.₁₆D₂₈

Upper central

C₁ — C₂³ — C₂×C₂²⋊C₄

Generators and relations for C₂³.₁₆D₂₈
G = < a,b,c,d,e | a²=b²=c²=d²⁸=1, e²=c, ab=ba, dad^-1=ac=ca, eae^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=cd^-1 >

Subgroups: 1268 in 238 conjugacy classes, 63 normal (51 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, Dic₇, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×D₄, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, C₂³.₁₀D₄, C₄⋊Dic₇, D₁₄⋊C₄, C₂³.D₇, C₇×C₂²⋊C₄, C₂²×Dic₇, C₂×C₇⋊D₄, C₂²×C₂₈, C₂³×D₇, C₂³×C₁₄, C₁₄.C₄², C₂×C₄⋊Dic₇, C₂×D₁₄⋊C₄, C₂×C₂³.D₇, C₁₄×C₂²⋊C₄, C₂²×C₇⋊D₄, C₂³.₁₆D₂₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₄○D₄, D₁₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₄.₄D₄, D₂₈, C₇⋊D₄, C₂²×D₇, C₂³.₁₀D₄, C₂×D₂₈, C₄○D₂₈, D₄×D₇, D₄⋊₂D₇, C₂×C₇⋊D₄, C₂²⋊D₂₈, D₁₄.D₄, Dic₇.D₄, C₂².D₂₈, C₂₈⋊₇D₄, C₂₈⋊₂D₄, Dic₇⋊D₄, C₂³.₁₆D₂₈

Smallest permutation representation of C₂³.₁₆D₂₈
►On 224 points

Generators in S₂₂₄

(2 136)(4 138)(6 140)(8 114)(10 116)(12 118)(14 120)(16 122)(18 124)(20 126)(22 128)(24 130)(26 132)(28 134)(29 112)(30 74)(31 86)(32 76)(33 88)(34 78)(35 90)(36 80)(37 92)(38 82)(39 94)(40 84)(41 96)(42 58)(43 98)(44 60)(45 100)(46 62)(47 102)(48 64)(49 104)(50 66)(51 106)(52 68)(53 108)(54 70)(55 110)(56 72)(57 196)(59 170)(61 172)(63 174)(65 176)(67 178)(69 180)(71 182)(73 184)(75 186)(77 188)(79 190)(81 192)(83 194)(85 185)(87 187)(89 189)(91 191)(93 193)(95 195)(97 169)(99 171)(101 173)(103 175)(105 177)(107 179)(109 181)(111 183)(141 198)(143 200)(145 202)(147 204)(149 206)(151 208)(153 210)(155 212)(157 214)(159 216)(161 218)(163 220)(165 222)(167 224)

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

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

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

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

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

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

82 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	···	4J	7A	7B	7C	14A	···	14U	14V	···	14AG	28A	···	28X
order	1	2	···	2	2	2	2	2	4	4	4	4	4	···	4	7	7	7	14	···	14	14	···	14	28	···	28
size	1	1	···	1	4	4	28	28	4	4	4	4	28	···	28	2	2	2	2	···	2	4	···	4	4	···	4

82 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		+	+		+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₄	D₇	C₄○D₄	D₁₄	D₁₄	C₇⋊D₄	D₂₈	C₄○D₂₈	D₄×D₇	D₄⋊₂D₇
kernel	C₂³.₁₆D₂₈	C₁₄.C₄²	C₂×C₄⋊Dic₇	C₂×D₁₄⋊C₄	C₂×C₂³.D₇	C₁₄×C₂²⋊C₄	C₂²×C₇⋊D₄	C₂×Dic₇	C₂×C₂₈	C₂²×D₇	C₂²×C₁₄	C₂×C₂²⋊C₄	C₂×C₁₄	C₂²×C₄	C₂⁴	C₂×C₄	C₂³	C₂²	C₂²	C₂²
# reps	1	1	1	2	1	1	1	2	2	2	2	3	6	6	3	12	12	12	6	6

Matrix representation of C₂³.₁₆D₂₈ ►in GL₆(𝔽₂₉)

1	0	0	0	0	0
0	28	0	0	0	0
0	0	1	0	0	0
0	0	16	28	0	0
0	0	0	0	1	0
0	0	0	0	22	28

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

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	28	0
0	0	0	0	0	28

24	0	0	0	0	0
0	23	0	0	0	0
0	0	18	0	0	0
0	0	5	21	0	0
0	0	0	0	9	15
0	0	0	0	14	20

0	26	0	0	0	0
19	0	0	0	0	0
0	0	8	28	0	0
0	0	5	21	0	0
0	0	0	0	20	14
0	0	0	0	19	9

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,16,0,0,0,0,0,28,0,0,0,0,0,0,1,22,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[24,0,0,0,0,0,0,23,0,0,0,0,0,0,18,5,0,0,0,0,0,21,0,0,0,0,0,0,9,14,0,0,0,0,15,20],[0,19,0,0,0,0,26,0,0,0,0,0,0,0,8,5,0,0,0,0,28,21,0,0,0,0,0,0,20,19,0,0,0,0,14,9] >;

C₂³.₁₆D₂₈ in GAP, Magma, Sage, TeX

C_2^3._{16}D_{28}

% in TeX

G:=Group("C2^3.16D28");

// GroupNames label

G:=SmallGroup(448,495);

// by ID

G=gap.SmallGroup(448,495);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,387,18822]);

// Polycyclic

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

// generators/relations

G = C23.16D28 order 448 = 26·7

9th non-split extension by C23 of D28 acting via D28/C14=C22

G = C₂³.₁₆D₂₈ order 448 = 2⁶·7

9^th non-split extension by C₂³ of D₂₈ acting via D₂₈/C₁₄=C₂²