C2^2xD4:D7 - GroupNames

G = C₂²×D₄⋊D₇ order 448 = 2⁶·7

Direct product of C₂² and D₄⋊D₇

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

Aliases: C₂²×D₄⋊D₇, D₂₈⋊₆C₂³, C₂₈.₂₇C₂⁴, (C₂×C₁₄)⋊₉D₈, C₁₄⋊₃(C₂×D₈), C₇⋊C₈⋊₈C₂³, C₇⋊₃(C₂²×D₈), (C₂×D₄)⋊₃₃D₁₄, (C₇×D₄)⋊₄C₂³, (C₂²×D₄)⋊₃D₇, D₄⋊₄(C₂²×D₇), C₂₈.₂₄₈(C₂×D₄), (C₂×C₂₈).₂₀₆D₄, C₄.₂₇(C₂³×D₇), (C₂×D₂₈)⋊₅₄C₂², (D₄×C₁₄)⋊₄₁C₂², (C₂²×D₂₈)⋊₁₈C₂, (C₂×C₂₈).₅₃₆C₂³, (C₂²×C₁₄).₂₀₆D₄, (C₂²×C₄).₃₇₅D₁₄, C₁₄.₁₃₆(C₂²×D₄), C₂³.₁₀₄(C₇⋊D₄), (C₂²×C₂₈).₂₆₉C₂², (D₄×C₂×C₁₄)⋊₂C₂, (C₂²×C₇⋊C₈)⋊₁₁C₂, (C₂×C₇⋊C₈)⋊₃₈C₂², C₄.₂₀(C₂×C₇⋊D₄), C₂.₉(C₂²×C₇⋊D₄), (C₂×C₁₄).₅₇₆(C₂×D₄), (C₂×C₄).₁₅₁(C₇⋊D₄), (C₂×C₄).₆₂₀(C₂²×D₇), C₂².₁₀₅(C₂×C₇⋊D₄), SmallGroup(448,1245)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — C₂²×D₄⋊D₇

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — D₂₈ — C₂×D₂₈ — C₂²×D₂₈ — C₂²×D₄⋊D₇

Lower central

C₇ — C₁₄ — C₂₈ — C₂²×D₄⋊D₇

Upper central

C₁ — C₂³ — C₂²×C₄ — C₂²×D₄

Generators and relations for C₂²×D₄⋊D₇
G = < a,b,c,d,e,f | a²=b²=c⁴=d²=e⁷=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf=c^-1, ce=ec, de=ed, fdf=cd, fef=e^-1 >

Subgroups: 1684 in 338 conjugacy classes, 127 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, D₄, D₄, C₂³, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₂×C₈, D₈, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂²×C₈, C₂×D₈, C₂²×D₄, C₂²×D₄, C₇⋊C₈, D₂₈, D₂₈, C₂×C₂₈, C₇×D₄, C₇×D₄, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, C₂²×D₈, C₂×C₇⋊C₈, D₄⋊D₇, C₂×D₂₈, C₂×D₂₈, C₂²×C₂₈, D₄×C₁₄, D₄×C₁₄, C₂³×D₇, C₂³×C₁₄, C₂²×C₇⋊C₈, C₂×D₄⋊D₇, C₂²×D₂₈, D₄×C₂×C₁₄, C₂²×D₄⋊D₇
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, D₈, C₂×D₄, C₂⁴, D₁₄, C₂×D₈, C₂²×D₄, C₇⋊D₄, C₂²×D₇, C₂²×D₈, D₄⋊D₇, C₂×C₇⋊D₄, C₂³×D₇, C₂×D₄⋊D₇, C₂²×C₇⋊D₄, C₂²×D₄⋊D₇

Smallest permutation representation of C₂²×D₄⋊D₇
►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 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(113 169)(114 170)(115 171)(116 172)(117 173)(118 174)(119 175)(120 176)(121 177)(122 178)(123 179)(124 180)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 187)(132 188)(133 189)(134 190)(135 191)(136 192)(137 193)(138 194)(139 195)(140 196)(141 197)(142 198)(143 199)(144 200)(145 201)(146 202)(147 203)(148 204)(149 205)(150 206)(151 207)(152 208)(153 209)(154 210)(155 211)(156 212)(157 213)(158 214)(159 215)(160 216)(161 217)(162 218)(163 219)(164 220)(165 221)(166 222)(167 223)(168 224)

(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(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 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(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)(113 134 120 127)(114 135 121 128)(115 136 122 129)(116 137 123 130)(117 138 124 131)(118 139 125 132)(119 140 126 133)(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)(169 190 176 183)(170 191 177 184)(171 192 178 185)(172 193 179 186)(173 194 180 187)(174 195 181 188)(175 196 182 189)(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)

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

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

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

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

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

88 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	2O	4A	4B	4C	4D	7A	7B	7C	8A	···	8H	14A	···	14U	14V	···	14AS	28A	···	28L
order	1	2	···	2	2	2	2	2	2	2	2	2	4	4	4	4	7	7	7	8	···	8	14	···	14	14	···	14	28	···	28
size	1	1	···	1	4	4	4	4	28	28	28	28	2	2	2	2	2	2	2	14	···	14	2	···	2	4	···	4	4	···	4

88 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₈

D₁₄

C₇⋊D₄

D₄⋊D₇

kernel

C₂²×D₄⋊D₇

C₂²×C₇⋊C₈

C₂×D₄⋊D₇

C₂²×D₂₈

D₄×C₂×C₁₄

C₂×C₂₈

C₂²×C₁₄

C₂²×D₄

C₂×C₁₄

C₂²×C₄

C₂×D₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂²×D₄⋊D₇ ►in GL₆(𝔽₁₁₃)

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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	36
0	0	0	0	69	112

86	5	0	0	0	0
103	27	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	14
0	0	0	0	105	0

89	112	0	0	0	0
2	33	0	0	0	0
0	0	79	112	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

81	33	0	0	0	0
82	32	0	0	0	0
0	0	34	1	0	0
0	0	88	79	0	0
0	0	0	0	1	0
0	0	0	0	69	112

G:=sub<GL(6,GF(113))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,69,0,0,0,0,36,112],[86,103,0,0,0,0,5,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,105,0,0,0,0,14,0],[89,2,0,0,0,0,112,33,0,0,0,0,0,0,79,1,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[81,82,0,0,0,0,33,32,0,0,0,0,0,0,34,88,0,0,0,0,1,79,0,0,0,0,0,0,1,69,0,0,0,0,0,112] >;

C₂²×D₄⋊D₇ in GAP, Magma, Sage, TeX

C_2^2\times D_4\rtimes D_7

% in TeX

G:=Group("C2^2xD4:D7");

// GroupNames label

G:=SmallGroup(448,1245);

// by ID

G=gap.SmallGroup(448,1245);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,675,1684,235,102,18822]);

// Polycyclic

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

// generators/relations

G = C22×D4⋊D7 order 448 = 26·7

Direct product of C22 and D4⋊D7

G = C₂²×D₄⋊D₇ order 448 = 2⁶·7

Direct product of C₂² and D₄⋊D₇