C2xD4.10D14 - GroupNames

G = C₂×D₄.₁₀D₁₄ order 448 = 2⁶·7

Direct product of C₂ and D₄.₁₀D₁₄

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×D₄.₁₀D₁₄, C₁₄.₁₃C₂⁵, C₂₈.₄₈C₂⁴, D₁₄.₇C₂⁴, D₂₈.₄₀C₂³, C₁₄⋊₂2_-¹⁺⁴, Dic₇.₈C₂⁴, Dic₁₄.₃₇C₂³, C₄○D₄⋊₁₈D₁₄, (C₂×C₁₄).₄C₂⁴, (Q₈×D₇)⋊₁₄C₂², C₄.₆₃(C₂³×D₇), C₂.₁₄(D₇×C₂⁴), C₇⋊D₄.₁C₂³, (C₂×D₄).₂₅₄D₁₄, C₇⋊₂(C₂×2_-¹⁺⁴), C₄○D₂₈⋊₂₆C₂², (C₂×Q₈).₂₁₂D₁₄, D₄.₂₉(C₂²×D₇), (C₄×D₇).₁₉C₂³, (C₇×D₄).₂₉C₂³, Q₈.₃₀(C₂²×D₇), (C₇×Q₈).₃₀C₂³, D₄⋊₂D₇⋊₁₃C₂², (C₂×C₂₈).₅₆₇C₂³, (C₂²×C₄).₂₉₂D₁₄, C₂².₅₇(C₂³×D₇), (C₂×Dic₁₄)⋊₇₅C₂², (C₂²×Dic₁₄)⋊₂₅C₂, (D₄×C₁₄).₂₇₉C₂², (C₂×D₂₈).₂₉₀C₂², (Q₈×C₁₄).₂₄₇C₂², C₂³.₂₁₅(C₂²×D₇), (C₂²×C₁₄).₂₄₉C₂³, (C₂²×C₂₈).₃₀₃C₂², (C₂×Dic₇).₁₆₇C₂³, (C₂²×D₇).₂₅₀C₂³, (C₂²×Dic₇).₁₆₉C₂², (C₂×Q₈×D₇)⋊₂₁C₂, (C₂×C₄○D₄)⋊₁₄D₇, (C₁₄×C₄○D₄)⋊₁₅C₂, (C₂×C₄○D₂₈)⋊₃₈C₂, (C₂×D₄⋊₂D₇)⋊₃₀C₂, (C₇×C₄○D₄)⋊₂₁C₂², (C₂×C₄×D₇).₁₇₃C₂², (C₂×C₄).₂₅₃(C₂²×D₇), (C₂×C₇⋊D₄).₁₄₄C₂², SmallGroup(448,1377)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₂×D₄.₁₀D₁₄

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²×D₇ — C₂×C₄×D₇ — C₂×Q₈×D₇ — C₂×D₄.₁₀D₁₄

Lower central

C₇ — C₁₄ — C₂×D₄.₁₀D₁₄

Upper central

C₁ — C₂² — C₂×C₄○D₄

Generators and relations for C₂×D₄.₁₀D₁₄
G = < a,b,c,d,e | a²=b⁴=c²=1, d¹⁴=e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd^-1=ebe^-1=b^-1, dcd^-1=b²c, ce=ec, ede^-1=d¹³ >

Subgroups: 2772 in 794 conjugacy classes, 447 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₇, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, 2_-¹⁺⁴, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, C₂×2_-¹⁺⁴, C₂×Dic₁₄, C₂×C₄×D₇, C₂×D₂₈, C₄○D₂₈, D₄⋊₂D₇, Q₈×D₇, C₂²×Dic₇, C₂×C₇⋊D₄, C₂²×C₂₈, D₄×C₁₄, Q₈×C₁₄, C₇×C₄○D₄, C₂²×Dic₁₄, C₂×C₄○D₂₈, C₂×D₄⋊₂D₇, C₂×Q₈×D₇, D₄.₁₀D₁₄, C₁₄×C₄○D₄, C₂×D₄.₁₀D₁₄
Quotients: C₁, C₂, C₂², C₂³, D₇, C₂⁴, D₁₄, 2_-¹⁺⁴, C₂⁵, C₂²×D₇, C₂×2_-¹⁺⁴, C₂³×D₇, D₄.₁₀D₁₄, D₇×C₂⁴, C₂×D₄.₁₀D₁₄

Smallest permutation representation of C₂×D₄.₁₀D₁₄
►On 224 points

Generators in S₂₂₄

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

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

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

(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)

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

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

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

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

94 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	4A	···	4H	4I	···	4T	7A	7B	7C	14A	···	14I	14J	···	14AA	28A	···	28L	28M	···	28AD
order	1	2	2	2	2	···	2	2	2	2	2	4	···	4	4	···	4	7	7	7	14	···	14	14	···	14	28	···	28	28	···	28
size	1	1	1	1	2	···	2	14	14	14	14	2	···	2	14	···	14	2	2	2	2	···	2	4	···	4	2	···	2	4	···	4

94 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₇	D₁₄	D₁₄	D₁₄	D₁₄	2_-¹⁺⁴	D₄.₁₀D₁₄
kernel	C₂×D₄.₁₀D₁₄	C₂²×Dic₁₄	C₂×C₄○D₂₈	C₂×D₄⋊₂D₇	C₂×Q₈×D₇	D₄.₁₀D₁₄	C₁₄×C₄○D₄	C₂×C₄○D₄	C₂²×C₄	C₂×D₄	C₂×Q₈	C₄○D₄	C₁₄	C₂
# reps	1	3	3	6	2	16	1	3	9	9	3	24	2	12

Matrix representation of C₂×D₄.₁₀D₁₄ ►in GL₆(𝔽₂₉)

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	28	0
0	0	0	0	0	28

1	0	0	0	0	0
0	1	0	0	0	0
0	0	7	24	0	0
0	0	10	22	0	0
0	0	0	8	8	5
0	0	13	10	16	21

28	0	0	0	0	0
0	28	0	0	0	0
0	0	28	0	8	10
0	0	0	28	27	6
0	0	0	0	1	0
0	0	0	0	0	1

6	0	0	0	0	0
0	5	0	0	0	0
0	0	5	25	10	3
0	0	8	17	28	21
0	0	19	25	7	4
0	0	9	14	7	0

0	5	0	0	0	0
6	0	0	0	0	0
0	0	27	7	0	0
0	0	20	2	0	0
0	0	0	0	0	10
0	0	0	0	26	0

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,10,0,13,0,0,24,22,8,10,0,0,0,0,8,16,0,0,0,0,5,21],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,8,27,1,0,0,0,10,6,0,1],[6,0,0,0,0,0,0,5,0,0,0,0,0,0,5,8,19,9,0,0,25,17,25,14,0,0,10,28,7,7,0,0,3,21,4,0],[0,6,0,0,0,0,5,0,0,0,0,0,0,0,27,20,0,0,0,0,7,2,0,0,0,0,0,0,0,26,0,0,0,0,10,0] >;

C₂×D₄.₁₀D₁₄ in GAP, Magma, Sage, TeX

C_2\times D_4._{10}D_{14}

% in TeX

G:=Group("C2xD4.10D14");

// GroupNames label

G:=SmallGroup(448,1377);

// by ID

G=gap.SmallGroup(448,1377);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,297,136,1684,102,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^14=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^2*c,c*e=e*c,e*d*e^-1=d^13>;

// generators/relations

G = C2×D4.10D14 order 448 = 26·7

Direct product of C2 and D4.10D14

G = C₂×D₄.₁₀D₁₄ order 448 = 2⁶·7

Direct product of C₂ and D₄.₁₀D₁₄