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G = C2×C4×C7⋊D4  order 448 = 26·7

Direct product of C2×C4 and C7⋊D4

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

Aliases: C2×C4×C7⋊D4, C24.69D14, C14⋊4(C4×D4), C28⋊16(C2×D4), (C2×C28)⋊39D4, (C23×C4)⋊3D7, C23⋊5(C4×D7), (C23×C28)⋊12C2, D14⋊5(C22×C4), (C22×C4)⋊42D14, D14⋊C4⋊76C22, C14.39(C23×C4), Dic7⋊3(C22×C4), (C2×C14).286C24, (C2×C28).886C23, Dic7⋊C4⋊78C22, (C22×C28)⋊63C22, (C4×Dic7)⋊82C22, C14.132(C22×D4), C23.D7⋊68C22, C22.43(C23×D7), C22.81(C4○D28), C23.336(C22×D7), (C23×C14).108C22, (C22×C14).415C23, (C2×Dic7).279C23, (C22×D7).237C23, (C23×D7).112C22, (C22×Dic7).230C22, C7⋊5(C2×C4×D4), C22⋊3(C2×C4×D7), (C2×C4×Dic7)⋊38C2, C2.6(C2×C4○D28), (C2×C4×D7)⋊71C22, (D7×C22×C4)⋊25C2, (C2×D14⋊C4)⋊46C2, C2.39(D7×C22×C4), (C2×C14)⋊6(C22×C4), C14.61(C2×C4○D4), C2.3(C22×C7⋊D4), (C2×Dic7⋊C4)⋊52C2, (C22×C14)⋊14(C2×C4), (C2×Dic7)⋊18(C2×C4), (C2×C14).573(C2×D4), (C22×D7)⋊13(C2×C4), (C2×C23.D7)⋊33C2, (C2×C4).830(C22×D7), (C22×C7⋊D4).14C2, C22.102(C2×C7⋊D4), (C2×C14).112(C4○D4), (C2×C7⋊D4).148C22, SmallGroup(448,1241)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C2×C4×C7⋊D4
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C22×C7⋊D4 — C2×C4×C7⋊D4
Lower central
C7 — C14 — C2×C4×C7⋊D4
Upper central
C1 — C22×C4 — C23×C4

Generators and relations for C2×C4×C7⋊D4
 G = < a,b,c,d,e | a2=b4=c7=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1604 in 426 conjugacy classes, 183 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C23×C4, C22×D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C4×D4, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C2×C4×D7, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, C22×C28, C22×C28, C23×D7, C23×C14, C2×C4×Dic7, C2×Dic7⋊C4, C2×D14⋊C4, C4×C7⋊D4, C2×C23.D7, D7×C22×C4, C22×C7⋊D4, C23×C28, C2×C4×C7⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, C24, D14, C4×D4, C23×C4, C22×D4, C2×C4○D4, C4×D7, C7⋊D4, C22×D7, C2×C4×D4, C2×C4×D7, C4○D28, C2×C7⋊D4, C23×D7, C4×C7⋊D4, D7×C22×C4, C2×C4○D28, C22×C7⋊D4, C2×C4×C7⋊D4

Smallest permutation representation of C2×C4×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 50 15 36)(2 51 16 37)(3 52 17 38)(4 53 18 39)(5 54 19 40)(6 55 20 41)(7 56 21 42)(8 43 22 29)(9 44 23 30)(10 45 24 31)(11 46 25 32)(12 47 26 33)(13 48 27 34)(14 49 28 35)(57 106 71 92)(58 107 72 93)(59 108 73 94)(60 109 74 95)(61 110 75 96)(62 111 76 97)(63 112 77 98)(64 99 78 85)(65 100 79 86)(66 101 80 87)(67 102 81 88)(68 103 82 89)(69 104 83 90)(70 105 84 91)(113 162 127 148)(114 163 128 149)(115 164 129 150)(116 165 130 151)(117 166 131 152)(118 167 132 153)(119 168 133 154)(120 155 134 141)(121 156 135 142)(122 157 136 143)(123 158 137 144)(124 159 138 145)(125 160 139 146)(126 161 140 147)(169 218 183 204)(170 219 184 205)(171 220 185 206)(172 221 186 207)(173 222 187 208)(174 223 188 209)(175 224 189 210)(176 211 190 197)(177 212 191 198)(178 213 192 199)(179 214 193 200)(180 215 194 201)(181 216 195 202)(182 217 196 203)

(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)

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

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

136 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I4J4K4L4M···4X7A7B7C14A···14AS28A···28AV
order12···2222222224···444444···477714···1428···28
size11···12222141414141···1222214···142222···22···2

136 irreducible representations

dim111111111122222222
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4D7C4○D4D14D14C7⋊D4C4×D7C4○D28
kernelC2×C4×C7⋊D4C2×C4×Dic7C2×Dic7⋊C4C2×D14⋊C4C4×C7⋊D4C2×C23.D7D7×C22×C4C22×C7⋊D4C23×C28C2×C7⋊D4C2×C28C23×C4C2×C14C22×C4C24C2×C4C23C22
# reps11118111116434183242424

Matrix representation of C2×C4×C7⋊D4 ►in GL6(𝔽29)

2800000
0280000
0028000
0002800
0000280
0000028
,
1700000
0170000
0028000
0002800
000010
000001
,
0280000
130000
00212800
0062600
0000028
000013
,
130000
0280000
0001800
0021000
00002427
0000135
,
28260000
010000
0001100
008000
000013
0000028

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,28,3,0,0,0,0,0,0,21,6,0,0,0,0,28,26,0,0,0,0,0,0,0,1,0,0,0,0,28,3],[1,0,0,0,0,0,3,28,0,0,0,0,0,0,0,21,0,0,0,0,18,0,0,0,0,0,0,0,24,13,0,0,0,0,27,5],[28,0,0,0,0,0,26,1,0,0,0,0,0,0,0,8,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,3,28] >;
 

C2×C4×C7⋊D4 in GAP, Magma, Sage, TeX 

C_2\times C_4\times C_7\rtimes D_4
 
% in TeX
 
G:=Group("C2xC4xC7:D4");
 
// GroupNames label
 
G:=SmallGroup(448,1241);
 
// by ID
 
G=gap.SmallGroup(448,1241);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,80,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^7=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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