D7xC2^3xC4 - GroupNames

G = D₇×C₂³×C₄ order 448 = 2⁶·7

Direct product of C₂³×C₄ and D₇

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

Aliases: D₇×C₂³×C₄, C₂₈⋊₃C₂⁴, C₁₄.₂C₂⁵, Dic₇⋊₃C₂⁴, C₂⁴.₈₁D₁₄, D₁₄.₁₁C₂⁴, C₇⋊₁(C₂⁴×C₄), C₁₄⋊₁(C₂³×C₄), C₂.₁(D₇×C₂⁴), (C₂³×C₂₈)⋊₁₃C₂, (C₂×C₂₈)⋊₁₆C₂³, (D₇×C₂⁴).₄C₂, (C₂×C₁₄).₃₂₄C₂⁴, (C₂²×C₂₈)⋊₆₄C₂², (C₂³×Dic₇)⋊₁₃C₂, (C₂×Dic₇)⋊₁₄C₂³, C₂².₅₂(C₂³×D₇), C₂³.₃₄₅(C₂²×D₇), (C₂²×C₁₄).₄₃₁C₂³, (C₂³×C₁₄).₁₁₄C₂², (C₂²×Dic₇)⋊₅₅C₂², (C₂²×D₇).₂₆₁C₂³, (C₂³×D₇).₁₂₃C₂², (C₂×C₁₄)⋊₇(C₂²×C₄), (C₂²×C₁₄)⋊₁₅(C₂×C₄), SmallGroup(448,1366)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₇ — D₇×C₂³×C₄

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²×D₇ — C₂³×D₇ — D₇×C₂⁴ — D₇×C₂³×C₄

Lower central

C₇ — D₇×C₂³×C₄

Upper central

C₁ — C₂³×C₄

Generators and relations for D₇×C₂³×C₄
G = < a,b,c,d,e,f | a²=b²=c²=d⁴=e⁷=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 4740 in 1362 conjugacy classes, 799 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂⁴, Dic₇, C₂₈, D₁₄, C₂×C₁₄, C₂³×C₄, C₂³×C₄, C₂⁵, C₄×D₇, C₂×Dic₇, C₂×C₂₈, C₂²×D₇, C₂²×C₁₄, C₂⁴×C₄, C₂×C₄×D₇, C₂²×Dic₇, C₂²×C₂₈, C₂³×D₇, C₂³×C₁₄, D₇×C₂²×C₄, C₂³×Dic₇, C₂³×C₂₈, D₇×C₂⁴, D₇×C₂³×C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, D₇, C₂²×C₄, C₂⁴, D₁₄, C₂³×C₄, C₂⁵, C₄×D₇, C₂²×D₇, C₂⁴×C₄, C₂×C₄×D₇, C₂³×D₇, D₇×C₂²×C₄, D₇×C₂⁴, D₇×C₂³×C₄

Smallest permutation representation of D₇×C₂³×C₄
►On 224 points

Generators in S₂₂₄

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

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

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

(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)

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

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

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

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

160 conjugacy classes

class	1	2A	···	2O	2P	···	2AE	4A	···	4P	4Q	···	4AF	7A	7B	7C	14A	···	14AS	28A	···	28AV
order	1	2	···	2	2	···	2	4	···	4	4	···	4	7	7	7	14	···	14	28	···	28
size	1	1	···	1	7	···	7	1	···	1	7	···	7	2	2	2	2	···	2	2	···	2

160 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+		+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	D₇	D₁₄	D₁₄	C₄×D₇
kernel	D₇×C₂³×C₄	D₇×C₂²×C₄	C₂³×Dic₇	C₂³×C₂₈	D₇×C₂⁴	C₂³×D₇	C₂³×C₄	C₂²×C₄	C₂⁴	C₂³
# reps	1	28	1	1	1	32	3	42	3	48

Matrix representation of D₇×C₂³×C₄ ►in GL₅(𝔽₂₉)

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

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

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

28	0	0	0	0
0	28	0	0	0
0	0	28	0	0
0	0	0	17	0
0	0	0	0	17

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	10	13
0	0	0	28	22

1	0	0	0	0
0	28	0	0	0
0	0	28	0	0
0	0	0	19	17
0	0	0	1	10

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,17,0,0,0,0,0,17],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,28,0,0,0,13,22],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,19,1,0,0,0,17,10] >;

D₇×C₂³×C₄ in GAP, Magma, Sage, TeX

D_7\times C_2^3\times C_4

% in TeX

G:=Group("D7xC2^3xC4");

// GroupNames label

G:=SmallGroup(448,1366);

// by ID

G=gap.SmallGroup(448,1366);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,102,18822]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^4=e^7=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = D7×C23×C4 order 448 = 26·7

Direct product of C23×C4 and D7

G = D₇×C₂³×C₄ order 448 = 2⁶·7

Direct product of C₂³×C₄ and D₇