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## G = D4⋊(C4×D7)  order 448 = 26·7

### 2nd semidirect product of D4 and C4×D7 acting via C4×D7/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D4⋊(C4×D7)
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×C4×D7 — C2×D4⋊2D7 — D4⋊(C4×D7)
 Lower central C7 — C14 — C28 — D4⋊(C4×D7)
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D4⋊(C4×D7)
G = < a,b,c,d,e | a4=b2=c4=d7=e2=1, bab=cac-1=a-1, ad=da, ae=ea, cbc-1=a-1b, bd=db, ebe=a2b, cd=dc, ce=ec, ede=d-1 >

Subgroups: 820 in 162 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, D4⋊C4, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C23.36D4, C8⋊D7, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×C4×D7, D42D7, D42D7, C22×Dic7, C2×C7⋊D4, D4×C14, C14.Q16, C28.44D4, D4⋊Dic7, C7×D4⋊C4, D7×C4⋊C4, C2×C8⋊D7, C2×D42D7, D4⋊(C4×D7)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C8⋊C22, C8.C22, C4×D7, C22×D7, C23.36D4, C2×C4×D7, D4×D7, D7×C22⋊C4, D8⋊D7, SD16⋊D7, D4⋊(C4×D7)

Smallest permutation representation of D4⋊(C4×D7)
On 224 points
Generators in S224
(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 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 71 64 78)(58 72 65 79)(59 73 66 80)(60 74 67 81)(61 75 68 82)(62 76 69 83)(63 77 70 84)(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)(113 127 120 134)(114 128 121 135)(115 129 122 136)(116 130 123 137)(117 131 124 138)(118 132 125 139)(119 133 126 140)(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)(197 218 204 211)(198 219 205 212)(199 220 206 213)(200 221 207 214)(201 222 208 215)(202 223 209 216)(203 224 210 217)
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 113)(16 114)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 121)(24 122)(25 123)(26 124)(27 125)(28 126)(29 162)(30 163)(31 164)(32 165)(33 166)(34 167)(35 168)(36 155)(37 156)(38 157)(39 158)(40 159)(41 160)(42 161)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 183)(58 184)(59 185)(60 186)(61 187)(62 188)(63 189)(64 190)(65 191)(66 192)(67 193)(68 194)(69 195)(70 196)(71 176)(72 177)(73 178)(74 179)(75 180)(76 181)(77 182)(78 169)(79 170)(80 171)(81 172)(82 173)(83 174)(84 175)(85 211)(86 212)(87 213)(88 214)(89 215)(90 216)(91 217)(92 218)(93 219)(94 220)(95 221)(96 222)(97 223)(98 224)(99 204)(100 205)(101 206)(102 207)(103 208)(104 209)(105 210)(106 197)(107 198)(108 199)(109 200)(110 201)(111 202)(112 203)
(1 85 29 57)(2 86 30 58)(3 87 31 59)(4 88 32 60)(5 89 33 61)(6 90 34 62)(7 91 35 63)(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)(113 211 141 183)(114 212 142 184)(115 213 143 185)(116 214 144 186)(117 215 145 187)(118 216 146 188)(119 217 147 189)(120 218 148 190)(121 219 149 191)(122 220 150 192)(123 221 151 193)(124 222 152 194)(125 223 153 195)(126 224 154 196)(127 204 155 176)(128 205 156 177)(129 206 157 178)(130 207 158 179)(131 208 159 180)(132 209 160 181)(133 210 161 182)(134 197 162 169)(135 198 163 170)(136 199 164 171)(137 200 165 172)(138 201 166 173)(139 202 167 174)(140 203 168 175)
(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 7)(2 6)(3 5)(8 14)(9 13)(10 12)(15 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 35)(30 34)(31 33)(36 42)(37 41)(38 40)(43 49)(44 48)(45 47)(50 56)(51 55)(52 54)(57 63)(58 62)(59 61)(64 70)(65 69)(66 68)(71 77)(72 76)(73 75)(78 84)(79 83)(80 82)(85 91)(86 90)(87 89)(92 98)(93 97)(94 96)(99 105)(100 104)(101 103)(106 112)(107 111)(108 110)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 168)(156 167)(157 166)(158 165)(159 164)(160 163)(161 162)(169 182)(170 181)(171 180)(172 179)(173 178)(174 177)(175 176)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(197 210)(198 209)(199 208)(200 207)(201 206)(202 205)(203 204)(211 224)(212 223)(213 222)(214 221)(215 220)(216 219)(217 218)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 4 4 14 14 2 2 4 4 14 14 28 28 28 28 2 2 2 4 4 28 28 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D7 D14 D14 D14 C4×D7 C8⋊C22 C8.C22 D4×D7 D4×D7 D8⋊D7 SD16⋊D7 kernel D4⋊(C4×D7) C14.Q16 C28.44D4 D4⋊Dic7 C7×D4⋊C4 D7×C4⋊C4 C2×C8⋊D7 C2×D4⋊2D7 D4⋊2D7 C4×D7 C2×Dic7 C22×D7 D4⋊C4 C4⋊C4 C2×C8 C2×D4 D4 C14 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 1 1 3 3 3 3 12 1 1 3 3 6 6

Matrix representation of D4⋊(C4×D7) in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 0 79 86 87 0 0 0 112 69 69 0 0 1 2 1 1 0 0 112 34 0 0
,
 73 88 0 0 0 0 73 40 0 0 0 0 0 0 24 30 8 105 0 0 75 43 65 10 0 0 46 0 81 8 0 0 46 110 38 78
,
 77 87 0 0 0 0 89 36 0 0 0 0 0 0 109 47 5 23 0 0 0 14 50 50 0 0 95 77 99 95 0 0 18 66 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 32 112 33 0 0 1 80 0 1 0 0 0 0 0 112 0 0 0 0 1 79
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 9 9 1 88 0 0 79 104 0 79 0 0 0 0 79 25 0 0 0 0 112 34

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,112,0,0,79,112,2,34,0,0,86,69,1,0,0,0,87,69,1,0],[73,73,0,0,0,0,88,40,0,0,0,0,0,0,24,75,46,46,0,0,30,43,0,110,0,0,8,65,81,38,0,0,105,10,8,78],[77,89,0,0,0,0,87,36,0,0,0,0,0,0,109,0,95,18,0,0,47,14,77,66,0,0,5,50,99,0,0,0,23,50,95,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,1,0,0,0,0,32,80,0,0,0,0,112,0,0,1,0,0,33,1,112,79],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,9,79,0,0,0,0,9,104,0,0,0,0,1,0,79,112,0,0,88,79,25,34] >;

D4⋊(C4×D7) in GAP, Magma, Sage, TeX

D_4\rtimes (C_4\times D_7)
% in TeX

G:=Group("D4:(C4xD7)");
// GroupNames label

G:=SmallGroup(448,305);
// by ID

G=gap.SmallGroup(448,305);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,219,58,851,438,102,18822]);
// Polycyclic

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

׿
×
𝔽