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## G = C7×D4.7D4order 448 = 26·7

### Direct product of C7 and D4.7D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×D4.7D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — Q8×C14 — C14×SD16 — C7×D4.7D4
 Lower central C1 — C2 — C2×C4 — C7×D4.7D4
 Upper central C1 — C2×C14 — C22×C28 — C7×D4.7D4

Generators and relations for C7×D4.7D4
G = < a,b,c,d,e | a7=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=ece-1=bc, ede-1=d-1 >

Subgroups: 274 in 152 conjugacy classes, 58 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, D4.7D4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×SD16, C7×Q16, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C22⋊C8, C7×D4⋊C4, C7×Q8⋊C4, C7×C22⋊Q8, C14×SD16, C14×Q16, C14×C4○D4, C7×D4.7D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C4○D8, C8.C22, C7×D4, C22×C14, D4.7D4, D4×C14, C7×C22≀C2, C7×C4○D8, C7×C8.C22, C7×D4.7D4

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

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

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

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

133 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14AJ 28A ··· 28X 28Y ··· 28AJ 28AK ··· 28AV 56A ··· 56X order 1 2 2 2 2 2 2 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 4 2 2 2 2 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 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 C14 D4 D4 D4 D4 C4○D8 C7×D4 C7×D4 C7×D4 C7×D4 C7×C4○D8 C8.C22 C7×C8.C22 kernel C7×D4.7D4 C7×C22⋊C8 C7×D4⋊C4 C7×Q8⋊C4 C7×C22⋊Q8 C14×SD16 C14×Q16 C14×C4○D4 D4.7D4 C22⋊C8 D4⋊C4 Q8⋊C4 C22⋊Q8 C2×SD16 C2×Q16 C2×C4○D4 C2×C28 C7×D4 C7×Q8 C22×C14 C14 C2×C4 D4 Q8 C23 C2 C14 C2 # reps 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 1 2 2 1 4 6 12 12 6 24 1 6

Matrix representation of C7×D4.7D4 in GL4(𝔽113) generated by

 30 0 0 0 0 30 0 0 0 0 106 0 0 0 0 106
,
 112 0 0 0 0 112 0 0 0 0 1 111 0 0 1 112
,
 1 0 0 0 71 112 0 0 0 0 1 111 0 0 0 112
,
 71 111 0 0 35 42 0 0 0 0 0 62 0 0 31 0
,
 71 111 0 0 34 42 0 0 0 0 0 87 0 0 100 0
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,106,0,0,0,0,106],[112,0,0,0,0,112,0,0,0,0,1,1,0,0,111,112],[1,71,0,0,0,112,0,0,0,0,1,0,0,0,111,112],[71,35,0,0,111,42,0,0,0,0,0,31,0,0,62,0],[71,34,0,0,111,42,0,0,0,0,0,100,0,0,87,0] >;

C7×D4.7D4 in GAP, Magma, Sage, TeX

C_7\times D_4._7D_4
% in TeX

G:=Group("C7xD4.7D4");
// GroupNames label

G:=SmallGroup(448,860);
// by ID

G=gap.SmallGroup(448,860);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2438,1192,9804,4911,172]);
// 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=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽