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G = D7×C2.C42  order 448 = 26·7

Direct product of D7 and C2.C42

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

Aliases: D7×C2.C42, D14.5C42, C2.5(D7×C42), D14.8(C4⋊C4), C14.3(C2×C42), C22.57(D4×D7), C22.15(Q8×D7), (C22×D7).11Q8, (C22×D7).103D4, (C22×C4).296D14, C14.C42⋊34C2, D14.13(C22⋊C4), C23.253(C22×D7), (C22×C28).330C22, (C22×C14).288C23, (C23×D7).120C22, (C22×Dic7).175C22, (C2×C4×D7)⋊8C4, C2.2(D7×C4⋊C4), C14.4(C2×C4⋊C4), (C2×C4)⋊14(C4×D7), (C2×C28)⋊18(C2×C4), C2.1(D7×C22⋊C4), C22.31(C2×C4×D7), C14.2(C2×C22⋊C4), C7⋊1(C2×C2.C42), (C2×C14).66(C2×Q8), (D7×C22×C4).11C2, (C2×Dic7)⋊19(C2×C4), (C2×C14).197(C2×D4), (C2×C14).47(C22×C4), (C22×D7).69(C2×C4), (C7×C2.C42)⋊17C2, SmallGroup(448,197)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — D7×C2.C42
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — D7×C22×C4 — D7×C2.C42
Lower central
C7 — C14 — D7×C2.C42
Upper central
C1 — C23 — C2.C42

Generators and relations for D7×C2.C42
 G = < a,b,c,d,e | a7=b2=c2=d4=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ce=ec >

Subgroups: 1500 in 330 conjugacy classes, 131 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C14, C22×C4, C22×C4, C24, Dic7, C28, D14, C2×C14, C2×C14, C2.C42, C2.C42, C23×C4, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×C14, C2×C2.C42, C2×C4×D7, C2×C4×D7, C22×Dic7, C22×C28, C23×D7, C14.C42, C7×C2.C42, D7×C22×C4, D7×C2.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D7, C22×D7, C2×C2.C42, C2×C4×D7, D4×D7, Q8×D7, D7×C42, D7×C22⋊C4, D7×C4⋊C4, D7×C2.C42

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

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

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

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

100 conjugacy classes

class 1 2A···2G2H···2O4A···4L4M···4X7A7B7C14A···14U28A···28AJ
order12···22···24···44···477714···1428···28
size11···17···72···214···142222···24···4

100 irreducible representations

dim111112222244
type+++++-+++-
imageC1C2C2C2C4D4Q8D7D14C4×D7D4×D7Q8×D7
kernelD7×C2.C42C14.C42C7×C2.C42D7×C22×C4C2×C4×D7C22×D7C22×D7C2.C42C22×C4C2×C4C22C22
# reps13132462393693

Matrix representation of D7×C2.C42 ►in GL5(𝔽29)

10000
00100
028300
00010
00001
,
280000
002800
028000
00010
00001
,
10000
01000
00100
000280
000028
,
10000
012000
001200
0001610
0001813
,
120000
028000
002800
0002827
00011

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,1,3,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,16,18,0,0,0,10,13],[12,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,1,0,0,0,27,1] >;
 

D7×C2.C42 in GAP, Magma, Sage, TeX 

D_7\times C_2.C_4^2
 
% in TeX
 
G:=Group("D7xC2.C4^2");
 
// GroupNames label
 
G:=SmallGroup(448,197);
 
// by ID
 
G=gap.SmallGroup(448,197);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,387,58,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^4=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,c*e=e*c>;
 
// generators/relations
 

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×
⋊
ℤ
𝔽
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