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## G = C7×D4.D4order 448 = 26·7

### Direct product of C7 and D4.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×D4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — Q8×C14 — C7×C4⋊Q8 — C7×D4.D4
 Lower central C1 — C2 — C2×C4 — C7×D4.D4
 Upper central C1 — C2×C14 — C4×C28 — C7×D4.D4

Generators and relations for C7×D4.D4
G = < a,b,c,d,e | a7=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 218 in 120 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C28, C28, C28, C2×C14, C2×C14, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4.D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×SD16, C22×C28, D4×C14, Q8×C14, C7×Q8⋊C4, C7×C4⋊C8, D4×C28, C7×C4⋊Q8, C14×SD16, C7×D4.D4
Quotients: C1, C2, C22, C7, D4, C23, C14, SD16, C2×D4, C4○D4, C2×C14, C4⋊D4, C2×SD16, C8.C22, C7×D4, C22×C14, D4.D4, C7×SD16, D4×C14, C7×C4○D4, C7×C4⋊D4, C14×SD16, C7×C8.C22, C7×D4.D4

Smallest permutation representation of C7×D4.D4
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 83 69 85)(2 84 70 86)(3 78 64 87)(4 79 65 88)(5 80 66 89)(6 81 67 90)(7 82 68 91)(8 41 31 18)(9 42 32 19)(10 36 33 20)(11 37 34 21)(12 38 35 15)(13 39 29 16)(14 40 30 17)(22 211 220 43)(23 212 221 44)(24 213 222 45)(25 214 223 46)(26 215 224 47)(27 216 218 48)(28 217 219 49)(50 95 59 71)(51 96 60 72)(52 97 61 73)(53 98 62 74)(54 92 63 75)(55 93 57 76)(56 94 58 77)(99 117 129 108)(100 118 130 109)(101 119 131 110)(102 113 132 111)(103 114 133 112)(104 115 127 106)(105 116 128 107)(120 143 153 134)(121 144 154 135)(122 145 148 136)(123 146 149 137)(124 147 150 138)(125 141 151 139)(126 142 152 140)(155 164 185 173)(156 165 186 174)(157 166 187 175)(158 167 188 169)(159 168 189 170)(160 162 183 171)(161 163 184 172)(176 190 209 199)(177 191 210 200)(178 192 204 201)(179 193 205 202)(180 194 206 203)(181 195 207 197)(182 196 208 198)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(15 38)(16 39)(17 40)(18 41)(19 42)(20 36)(21 37)(22 220)(23 221)(24 222)(25 223)(26 224)(27 218)(28 219)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 92)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(162 171)(163 172)(164 173)(165 174)(166 175)(167 169)(168 170)(176 209)(177 210)(178 204)(179 205)(180 206)(181 207)(182 208)
(1 115 59 151)(2 116 60 152)(3 117 61 153)(4 118 62 154)(5 119 63 148)(6 113 57 149)(7 114 58 150)(8 155 211 199)(9 156 212 200)(10 157 213 201)(11 158 214 202)(12 159 215 203)(13 160 216 197)(14 161 217 198)(15 170 26 206)(16 171 27 207)(17 172 28 208)(18 173 22 209)(19 174 23 210)(20 175 24 204)(21 169 25 205)(29 183 48 195)(30 184 49 196)(31 185 43 190)(32 186 44 191)(33 187 45 192)(34 188 46 193)(35 189 47 194)(36 166 222 178)(37 167 223 179)(38 168 224 180)(39 162 218 181)(40 163 219 182)(41 164 220 176)(42 165 221 177)(50 125 69 106)(51 126 70 107)(52 120 64 108)(53 121 65 109)(54 122 66 110)(55 123 67 111)(56 124 68 112)(71 139 83 127)(72 140 84 128)(73 134 78 129)(74 135 79 130)(75 136 80 131)(76 137 81 132)(77 138 82 133)(85 104 95 141)(86 105 96 142)(87 99 97 143)(88 100 98 144)(89 101 92 145)(90 102 93 146)(91 103 94 147)
(1 218 69 27)(2 219 70 28)(3 220 64 22)(4 221 65 23)(5 222 66 24)(6 223 67 25)(7 224 68 26)(8 97 31 73)(9 98 32 74)(10 92 33 75)(11 93 34 76)(12 94 35 77)(13 95 29 71)(14 96 30 72)(15 58 38 56)(16 59 39 50)(17 60 40 51)(18 61 41 52)(19 62 42 53)(20 63 36 54)(21 57 37 55)(43 78 211 87)(44 79 212 88)(45 80 213 89)(46 81 214 90)(47 82 215 91)(48 83 216 85)(49 84 217 86)(99 185 129 155)(100 186 130 156)(101 187 131 157)(102 188 132 158)(103 189 133 159)(104 183 127 160)(105 184 128 161)(106 171 115 162)(107 172 116 163)(108 173 117 164)(109 174 118 165)(110 175 119 166)(111 169 113 167)(112 170 114 168)(120 209 153 176)(121 210 154 177)(122 204 148 178)(123 205 149 179)(124 206 150 180)(125 207 151 181)(126 208 152 182)(134 199 143 190)(135 200 144 191)(136 201 145 192)(137 202 146 193)(138 203 147 194)(139 197 141 195)(140 198 142 196)

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

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

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

133 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14AD 28A ··· 28X 28Y ··· 28AP 28AQ ··· 28BB 56A ··· 56X order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 4 4 2 2 2 2 4 4 4 8 8 1 ··· 1 4 4 4 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

133 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 SD16 C4○D4 C7×D4 C7×D4 C7×SD16 C7×C4○D4 C8.C22 C7×C8.C22 kernel C7×D4.D4 C7×Q8⋊C4 C7×C4⋊C8 D4×C28 C7×C4⋊Q8 C14×SD16 D4.D4 Q8⋊C4 C4⋊C8 C4×D4 C4⋊Q8 C2×SD16 C2×C28 C7×D4 C28 C28 C2×C4 D4 C4 C4 C14 C2 # reps 1 2 1 1 1 2 6 12 6 6 6 12 2 2 4 2 12 12 24 12 1 6

Matrix representation of C7×D4.D4 in GL4(𝔽113) generated by

 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 112 1 0 0 111 1 0 0 0 0 112 0 0 0 0 112
,
 112 1 0 0 0 1 0 0 0 0 112 91 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 15 104 0 0 0 98
,
 26 100 0 0 26 87 0 0 0 0 95 60 0 0 53 18
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[112,111,0,0,1,1,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,1,1,0,0,0,0,112,0,0,0,91,1],[1,0,0,0,0,1,0,0,0,0,15,0,0,0,104,98],[26,26,0,0,100,87,0,0,0,0,95,53,0,0,60,18] >;

C7×D4.D4 in GAP, Magma, Sage, TeX

C_7\times D_4.D_4
% in TeX

G:=Group("C7xD4.D4");
// GroupNames label

G:=SmallGroup(448,869);
// by ID

G=gap.SmallGroup(448,869);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,400,2438,14117,3547,124]);
// Polycyclic

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

׿
×
𝔽