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G = (C2×C42)⋊D7  order 448 = 26·7

2nd semidirect product of C2×C42 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D14⋊C4⋊1C4, (C2×C42)⋊2D7, C2.20(C4×D28), (C2×C4).66D28, C14.63(C4×D4), (C2×C28).448D4, C2.3(C28⋊7D4), C22.40(C2×D28), C14.57(C4⋊D4), C14.C42⋊8C2, C2.4(C4.D28), (C22×C4).400D14, C2.3(C42⋊2D7), C14.5(C42⋊2C2), C14.13(C4.4D4), C22.49(C4○D28), C2.15(C42⋊D7), C7⋊3(C24.C22), (C23×D7).12C22, C23.273(C22×D7), C14.15(C42⋊C2), (C22×C14).315C23, (C22×C28).479C22, C2.2(C23.23D14), C14.56(C22.D4), (C22×Dic7).33C22, (C2×C4×C28)⋊1C2, C2.6(C4×C7⋊D4), (C2×C4).92(C4×D7), (C2×Dic7⋊C4)⋊5C2, C22.120(C2×C4×D7), (C2×C28).208(C2×C4), (C2×D14⋊C4).10C2, (C2×C14).148(C2×D4), C22.44(C2×C7⋊D4), (C2×C14).74(C4○D4), (C2×C4).213(C7⋊D4), (C2×Dic7).27(C2×C4), (C22×D7).19(C2×C4), (C2×C14).101(C22×C4), SmallGroup(448,474)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — (C2×C42)⋊D7
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — C2×D14⋊C4 — (C2×C42)⋊D7
Lower central
C7 — C2×C14 — (C2×C42)⋊D7
Upper central
C1 — C23 — C2×C42

Generators and relations for (C2×C42)⋊D7
 G = < a,b,c,d,e | a2=b4=c4=d7=e2=1, ebe=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 964 in 190 conjugacy classes, 71 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, C28, D14, C2×C14, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C24.C22, Dic7⋊C4, D14⋊C4, D14⋊C4, C4×C28, C22×Dic7, C22×C28, C23×D7, C14.C42, C2×Dic7⋊C4, C2×D14⋊C4, C2×C4×C28, (C2×C42)⋊D7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C4×D7, D28, C7⋊D4, C22×D7, C24.C22, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C42⋊D7, C4×D28, C4.D28, C42⋊2D7, C4×C7⋊D4, C23.23D14, C28⋊7D4, (C2×C42)⋊D7

Smallest permutation representation of (C2×C42)⋊D7
►On 224 points
Generators in S224
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)

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

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

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

124 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4R7A7B7C14A···14U28A···28BT
order12···2224···44···477714···1428···28
size11···128282···228···282222···22···2

124 irreducible representations

dim11111122222222
type+++++++++
imageC1C2C2C2C2C4D4D7C4○D4D14C4×D7D28C7⋊D4C4○D28
kernel(C2×C42)⋊D7C14.C42C2×Dic7⋊C4C2×D14⋊C4C2×C4×C28D14⋊C4C2×C28C2×C42C2×C14C22×C4C2×C4C2×C4C2×C4C22
# reps121318438912121248

Matrix representation of (C2×C42)⋊D7 ►in GL6(𝔽29)

2800000
0280000
001000
000100
0000280
0000028
,
5160000
13240000
0028000
0002800
0000726
0000722
,
1200000
0120000
0012000
0001200
00002622
0000263
,
010000
2830000
000100
0028300
000010
000001
,
0280000
2800000
0002800
0028000
000010
00002428

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[5,13,0,0,0,0,16,24,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,7,7,0,0,0,0,26,22],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,26,26,0,0,0,0,22,3],[0,28,0,0,0,0,1,3,0,0,0,0,0,0,0,28,0,0,0,0,1,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,24,0,0,0,0,0,28] >;
 

(C2×C42)⋊D7 in GAP, Magma, Sage, TeX 

(C_2\times C_4^2)\rtimes D_7
 
% in TeX
 
G:=Group("(C2xC4^2):D7");
 
// GroupNames label
 
G:=SmallGroup(448,474);
 
// by ID
 
G=gap.SmallGroup(448,474);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,232,758,58,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^7=e^2=1,e*b*e=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
 
// generators/relations
 

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