C2^3xC7:D4 - GroupNames

G = C₂³×C₇⋊D₄ order 448 = 2⁶·7

Direct product of C₂³ and C₇⋊D₄

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

Aliases: C₂³×C₇⋊D₄, C₂⁵⋊₂D₇, D₁₄⋊₃C₂⁴, C₂⁴⋊₁₂D₁₄, C₁₄.₂₀C₂⁵, Dic₇⋊₂C₂⁴, C₇⋊₃(D₄×C₂³), (C₂×C₁₄)⋊₃C₂⁴, (C₂⁴×C₁₄)⋊₄C₂, (D₇×C₂⁴)⋊₆C₂, C₁₄⋊₃(C₂²×D₄), (C₂²×C₁₄)⋊₁₉D₄, C₂.₂₀(D₇×C₂⁴), C₂³⋊₆(C₂²×D₇), C₂²⋊₂(C₂³×D₇), (C₂²×C₁₄)⋊₉C₂³, (C₂²×D₇)⋊₉C₂³, (C₂³×C₁₄)⋊₁₉C₂², (C₂³×Dic₇)⋊₁₂C₂, (C₂×Dic₇)⋊₁₃C₂³, (C₂³×D₇)⋊₂₄C₂², (C₂²×Dic₇)⋊₅₄C₂², (C₂×C₁₄)⋊₁₇(C₂×D₄), SmallGroup(448,1384)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₂³×C₇⋊D₄

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²×D₇ — C₂³×D₇ — D₇×C₂⁴ — C₂³×C₇⋊D₄

Lower central

C₇ — C₁₄ — C₂³×C₇⋊D₄

Upper central

C₁ — C₂⁴ — C₂⁵

Generators and relations for C₂³×C₇⋊D₄
G = < a,b,c,d,e,f | a²=b²=c²=d⁷=e⁴=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, ede^-1=fdf=d^-1, fef=e^-1 >

Subgroups: 5252 in 1362 conjugacy classes, 543 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, D₄, C₂³, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, Dic₇, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂³×C₄, C₂²×D₄, C₂⁵, C₂⁵, C₂×Dic₇, C₇⋊D₄, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, D₄×C₂³, C₂²×Dic₇, C₂×C₇⋊D₄, C₂³×D₇, C₂³×D₇, C₂³×C₁₄, C₂³×C₁₄, C₂³×C₁₄, C₂³×Dic₇, C₂²×C₇⋊D₄, D₇×C₂⁴, C₂⁴×C₁₄, C₂³×C₇⋊D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₂²×D₄, C₂⁵, C₇⋊D₄, C₂²×D₇, D₄×C₂³, C₂×C₇⋊D₄, C₂³×D₇, C₂²×C₇⋊D₄, D₇×C₂⁴, C₂³×C₇⋊D₄

Smallest permutation representation of C₂³×C₇⋊D₄
►On 224 points

Generators in S₂₂₄

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

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

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

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

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

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

136 conjugacy classes

class	1	2A	···	2O	2P	···	2W	2X	···	2AE	4A	···	4H	7A	7B	7C	14A	···	14CO
order	1	2	···	2	2	···	2	2	···	2	4	···	4	7	7	7	14	···	14
size	1	1	···	1	2	···	2	14	···	14	14	···	14	2	2	2	2	···	2

136 irreducible representations

dim	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	D₄	D₇	D₁₄	C₇⋊D₄
kernel	C₂³×C₇⋊D₄	C₂³×Dic₇	C₂²×C₇⋊D₄	D₇×C₂⁴	C₂⁴×C₁₄	C₂²×C₁₄	C₂⁵	C₂⁴	C₂³
# reps	1	1	28	1	1	8	3	45	48

Matrix representation of C₂³×C₇⋊D₄ ►in GL₅(𝔽₂₉)

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

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

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	0	28
0	0	0	1	3

1	0	0	0	0
0	28	0	0	0
0	0	28	0	0
0	0	0	5	2
0	0	0	16	24

28	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	28	26
0	0	0	0	1

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,28,3],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,5,16,0,0,0,2,24],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,26,1] >;

C₂³×C₇⋊D₄ in GAP, Magma, Sage, TeX

C_2^3\times C_7\rtimes D_4

% in TeX

G:=Group("C2^3xC7:D4");

// GroupNames label

G:=SmallGroup(448,1384);

// by ID

G=gap.SmallGroup(448,1384);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,1684,18822]);

// Polycyclic

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

// generators/relations

G = C23×C7⋊D4 order 448 = 26·7

Direct product of C23 and C7⋊D4

G = C₂³×C₇⋊D₄ order 448 = 2⁶·7

Direct product of C₂³ and C₇⋊D₄