C2^3.10D28 - GroupNames

G = C₂³.₁₀D₂₈ order 448 = 2⁶·7

3^rd non-split extension by C₂³ of D₂₈ acting via D₂₈/C₁₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₁₀D₂₈, C₈⋊Dic₇⋊₇C₂, C₅₆⋊₁C₄⋊₃C₂, (C₂×C₈).₂D₁₄, C₂²⋊C₈.₅D₇, C₁₄.₇(C₄○D₈), (C₂×C₂₈).₂₃₉D₄, (C₂×C₄).₁₁₇D₂₈, (C₂×C₅₆).₂C₂², C₂₈.₄₄D₄⋊₅C₂, (C₂²×C₄).₇₇D₁₄, (C₂²×C₁₄).₅₀D₄, C₂₈.₂₈₁(C₄○D₄), C₂.₉(D₅₆⋊₇C₂), (C₂×C₂₈).₇₄₀C₂³, C₂₈.₄₈D₄.₈C₂, C₂².₁₀₃(C₂×D₂₈), C₁₄.₈(C₈.C₂²), C₇⋊₁(C₂³.₂₀D₄), C₄.₁₀₅(D₄⋊₂D₇), C₂.₁₁(C₈.D₁₄), C₄⋊Dic₇.₂₆₉C₂², (C₂²×C₂₈).₉₂C₂², (C₂×Dic₁₄).₁₂C₂², C₂³.₂₁D₁₄.₃C₂, C₁₄.₁₆(C₂².D₄), C₂.₁₂(C₂².D₂₈), (C₇×C₂²⋊C₈).₇C₂, (C₂×C₁₄).₁₂₃(C₂×D₄), (C₂×C₄).₆₈₅(C₂²×D₇), SmallGroup(448,257)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — C₂³.₁₀D₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₄⋊Dic₇ — C₂³.₂₁D₁₄ — C₂³.₁₀D₂₈

Lower central

C₇ — C₁₄ — C₂×C₂₈ — C₂³.₁₀D₂₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊C₈

Generators and relations for C₂³.₁₀D₂₈
G = < a,b,c,d,e | a²=b²=c²=1, d²⁸=e²=c, dad^-1=ab=ba, ac=ca, eae^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=bd²⁷ >

Subgroups: 444 in 96 conjugacy classes, 39 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂×Q₈, Dic₇, C₂₈, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂²⋊C₈, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₂²⋊Q₈, C₅₆, Dic₁₄, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×C₁₄, C₂³.₂₀D₄, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₂³.D₇, C₂×C₅₆, C₂×Dic₁₄, C₂²×C₂₈, C₂₈.₄₄D₄, C₈⋊Dic₇, C₅₆⋊₁C₄, C₇×C₂²⋊C₈, C₂₈.₄₈D₄, C₂³.₂₁D₁₄, C₂³.₁₀D₂₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₄○D₄, D₁₄, C₂².D₄, C₄○D₈, C₈.C₂², D₂₈, C₂²×D₇, C₂³.₂₀D₄, C₂×D₂₈, D₄⋊₂D₇, C₂².D₂₈, D₅₆⋊₇C₂, C₈.D₁₄, C₂³.₁₀D₂₈

Smallest permutation representation of C₂³.₁₀D₂₈
►On 224 points

Generators in S₂₂₄

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

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

(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)

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

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

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

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

79 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	14J	···	14O	28A	···	28L	28M	···	28R	56A	···	56X
order	1	2	2	2	2	4	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	4	2	2	2	2	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	1	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+		+	+		+	+		-	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₇	C₄○D₄	D₁₄	D₁₄	C₄○D₈	D₂₈	D₂₈	D₅₆⋊₇C₂	C₈.C₂²	D₄⋊₂D₇	C₈.D₁₄
kernel	C₂³.₁₀D₂₈	C₂₈.₄₄D₄	C₈⋊Dic₇	C₅₆⋊₁C₄	C₇×C₂²⋊C₈	C₂₈.₄₈D₄	C₂³.₂₁D₁₄	C₂×C₂₈	C₂²×C₁₄	C₂²⋊C₈	C₂₈	C₂×C₈	C₂²×C₄	C₁₄	C₂×C₄	C₂³	C₂	C₁₄	C₄	C₂
# reps	1	2	1	1	1	1	1	1	1	3	4	6	3	4	6	6	24	1	6	6

Matrix representation of C₂³.₁₀D₂₈ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
53	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	82	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

95	0	0	0	0	0
52	44	0	0	0	0
0	0	103	24	0	0
0	0	89	1	0	0
0	0	0	0	19	96
0	0	0	0	8	94

110	79	0	0	0	0
100	3	0	0	0	0
0	0	37	47	0	0
0	0	31	76	0	0
0	0	0	0	59	84
0	0	0	0	7	54

G:=sub<GL(6,GF(113))| [1,53,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,82,0,0,0,0,0,112],[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],[95,52,0,0,0,0,0,44,0,0,0,0,0,0,103,89,0,0,0,0,24,1,0,0,0,0,0,0,19,8,0,0,0,0,96,94],[110,100,0,0,0,0,79,3,0,0,0,0,0,0,37,31,0,0,0,0,47,76,0,0,0,0,0,0,59,7,0,0,0,0,84,54] >;

C₂³.₁₀D₂₈ in GAP, Magma, Sage, TeX

C_2^3._{10}D_{28}

% in TeX

G:=Group("C2^3.10D28");

// GroupNames label

G:=SmallGroup(448,257);

// by ID

G=gap.SmallGroup(448,257);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,254,219,142,1123,136,18822]);

// Polycyclic

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

// generators/relations

G = C23.10D28 order 448 = 26·7

3rd non-split extension by C23 of D28 acting via D28/C14=C22

G = C₂³.₁₀D₂₈ order 448 = 2⁶·7

3^rd non-split extension by C₂³ of D₂₈ acting via D₂₈/C₁₄=C₂²