direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4⋊D4, D4⋊3(C7×D4), Q8⋊3(C7×D4), (C2×D8)⋊2C14, (C7×D4)⋊21D4, (C7×Q8)⋊21D4, (C14×D8)⋊16C2, C4⋊D4⋊2C14, C22⋊C8⋊5C14, C4.23(D4×C14), D4⋊C4⋊8C14, Q8⋊C4⋊4C14, (C2×SD16)⋊8C14, (C2×C28).459D4, C28.384(C2×D4), C23.13(C7×D4), C14.96C22≀C2, (C14×SD16)⋊25C2, C22.79(D4×C14), (C22×C14).31D4, C14.119(C4○D8), (C2×C28).914C23, (C2×C56).298C22, C14.132(C8⋊C22), (D4×C14).294C22, (Q8×C14).259C22, (C22×C28).421C22, C2.6(C7×C4○D8), (C2×C4○D4)⋊1C14, C4⋊C4.2(C2×C14), (C2×C8).1(C2×C14), C2.7(C7×C8⋊C22), (C14×C4○D4)⋊17C2, (C7×C4⋊D4)⋊29C2, (C7×C22⋊C8)⋊22C2, (C2×C4).105(C7×D4), (C7×D4⋊C4)⋊32C2, (C7×Q8⋊C4)⋊27C2, C2.10(C7×C22≀C2), (C2×D4).52(C2×C14), (C2×C14).635(C2×D4), (C2×Q8).44(C2×C14), (C7×C4⋊C4).224C22, (C22×C4).39(C2×C14), (C2×C4).89(C22×C14), SmallGroup(448,857)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4⋊D4
G = < a,b,c,d,e | a7=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >
Subgroups: 322 in 162 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, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, D4⋊D4, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×D8, C7×SD16, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C22⋊C8, C7×D4⋊C4, C7×Q8⋊C4, C7×C4⋊D4, C14×D8, C14×SD16, C14×C4○D4, C7×D4⋊D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C4○D8, C8⋊C22, C7×D4, C22×C14, D4⋊D4, D4×C14, C7×C22≀C2, C7×C4○D8, C7×C8⋊C22, 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 50 77 59)(2 51 71 60)(3 52 72 61)(4 53 73 62)(5 54 74 63)(6 55 75 57)(7 56 76 58)(8 27 39 218)(9 28 40 219)(10 22 41 220)(11 23 42 221)(12 24 36 222)(13 25 37 223)(14 26 38 224)(15 217 30 49)(16 211 31 43)(17 212 32 44)(18 213 33 45)(19 214 34 46)(20 215 35 47)(21 216 29 48)(64 84 96 86)(65 78 97 87)(66 79 98 88)(67 80 92 89)(68 81 93 90)(69 82 94 91)(70 83 95 85)(99 143 121 134)(100 144 122 135)(101 145 123 136)(102 146 124 137)(103 147 125 138)(104 141 126 139)(105 142 120 140)(106 133 115 151)(107 127 116 152)(108 128 117 153)(109 129 118 154)(110 130 119 148)(111 131 113 149)(112 132 114 150)(155 190 177 199)(156 191 178 200)(157 192 179 201)(158 193 180 202)(159 194 181 203)(160 195 182 197)(161 196 176 198)(162 207 171 189)(163 208 172 183)(164 209 173 184)(165 210 174 185)(166 204 175 186)(167 205 169 187)(168 206 170 188)
(1 95)(2 96)(3 97)(4 98)(5 92)(6 93)(7 94)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 40)(16 41)(17 42)(18 36)(19 37)(20 38)(21 39)(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 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(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)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 182)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)(183 198)(184 199)(185 200)(186 201)(187 202)(188 203)(189 197)(190 209)(191 210)(192 204)(193 205)(194 206)(195 207)(196 208)
(1 48 83 27)(2 49 84 28)(3 43 78 22)(4 44 79 23)(5 45 80 24)(6 46 81 25)(7 47 82 26)(8 50 29 95)(9 51 30 96)(10 52 31 97)(11 53 32 98)(12 54 33 92)(13 55 34 93)(14 56 35 94)(15 64 40 60)(16 65 41 61)(17 66 42 62)(18 67 36 63)(19 68 37 57)(20 69 38 58)(21 70 39 59)(71 217 86 219)(72 211 87 220)(73 212 88 221)(74 213 89 222)(75 214 90 223)(76 215 91 224)(77 216 85 218)(99 184 108 199)(100 185 109 200)(101 186 110 201)(102 187 111 202)(103 188 112 203)(104 189 106 197)(105 183 107 198)(113 193 124 205)(114 194 125 206)(115 195 126 207)(116 196 120 208)(117 190 121 209)(118 191 122 210)(119 192 123 204)(127 176 142 172)(128 177 143 173)(129 178 144 174)(130 179 145 175)(131 180 146 169)(132 181 147 170)(133 182 141 171)(134 164 153 155)(135 165 154 156)(136 166 148 157)(137 167 149 158)(138 168 150 159)(139 162 151 160)(140 163 152 161)
(1 207)(2 208)(3 209)(4 210)(5 204)(6 205)(7 206)(8 151)(9 152)(10 153)(11 154)(12 148)(13 149)(14 150)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 141)(22 117)(23 118)(24 119)(25 113)(26 114)(27 115)(28 116)(29 139)(30 140)(31 134)(32 135)(33 136)(34 137)(35 138)(36 130)(37 131)(38 132)(39 133)(40 127)(41 128)(42 129)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 120)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 157)(93 158)(94 159)(95 160)(96 161)(97 155)(98 156)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
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), (1,50,77,59)(2,51,71,60)(3,52,72,61)(4,53,73,62)(5,54,74,63)(6,55,75,57)(7,56,76,58)(8,27,39,218)(9,28,40,219)(10,22,41,220)(11,23,42,221)(12,24,36,222)(13,25,37,223)(14,26,38,224)(15,217,30,49)(16,211,31,43)(17,212,32,44)(18,213,33,45)(19,214,34,46)(20,215,35,47)(21,216,29,48)(64,84,96,86)(65,78,97,87)(66,79,98,88)(67,80,92,89)(68,81,93,90)(69,82,94,91)(70,83,95,85)(99,143,121,134)(100,144,122,135)(101,145,123,136)(102,146,124,137)(103,147,125,138)(104,141,126,139)(105,142,120,140)(106,133,115,151)(107,127,116,152)(108,128,117,153)(109,129,118,154)(110,130,119,148)(111,131,113,149)(112,132,114,150)(155,190,177,199)(156,191,178,200)(157,192,179,201)(158,193,180,202)(159,194,181,203)(160,195,182,197)(161,196,176,198)(162,207,171,189)(163,208,172,183)(164,209,173,184)(165,210,174,185)(166,204,175,186)(167,205,169,187)(168,206,170,188), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,40)(16,41)(17,42)(18,36)(19,37)(20,38)(21,39)(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)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(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)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(183,198)(184,199)(185,200)(186,201)(187,202)(188,203)(189,197)(190,209)(191,210)(192,204)(193,205)(194,206)(195,207)(196,208), (1,48,83,27)(2,49,84,28)(3,43,78,22)(4,44,79,23)(5,45,80,24)(6,46,81,25)(7,47,82,26)(8,50,29,95)(9,51,30,96)(10,52,31,97)(11,53,32,98)(12,54,33,92)(13,55,34,93)(14,56,35,94)(15,64,40,60)(16,65,41,61)(17,66,42,62)(18,67,36,63)(19,68,37,57)(20,69,38,58)(21,70,39,59)(71,217,86,219)(72,211,87,220)(73,212,88,221)(74,213,89,222)(75,214,90,223)(76,215,91,224)(77,216,85,218)(99,184,108,199)(100,185,109,200)(101,186,110,201)(102,187,111,202)(103,188,112,203)(104,189,106,197)(105,183,107,198)(113,193,124,205)(114,194,125,206)(115,195,126,207)(116,196,120,208)(117,190,121,209)(118,191,122,210)(119,192,123,204)(127,176,142,172)(128,177,143,173)(129,178,144,174)(130,179,145,175)(131,180,146,169)(132,181,147,170)(133,182,141,171)(134,164,153,155)(135,165,154,156)(136,166,148,157)(137,167,149,158)(138,168,150,159)(139,162,151,160)(140,163,152,161), (1,207)(2,208)(3,209)(4,210)(5,204)(6,205)(7,206)(8,151)(9,152)(10,153)(11,154)(12,148)(13,149)(14,150)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,141)(22,117)(23,118)(24,119)(25,113)(26,114)(27,115)(28,116)(29,139)(30,140)(31,134)(32,135)(33,136)(34,137)(35,138)(36,130)(37,131)(38,132)(39,133)(40,127)(41,128)(42,129)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,120)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,157)(93,158)(94,159)(95,160)(96,161)(97,155)(98,156)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224)>;
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), (1,50,77,59)(2,51,71,60)(3,52,72,61)(4,53,73,62)(5,54,74,63)(6,55,75,57)(7,56,76,58)(8,27,39,218)(9,28,40,219)(10,22,41,220)(11,23,42,221)(12,24,36,222)(13,25,37,223)(14,26,38,224)(15,217,30,49)(16,211,31,43)(17,212,32,44)(18,213,33,45)(19,214,34,46)(20,215,35,47)(21,216,29,48)(64,84,96,86)(65,78,97,87)(66,79,98,88)(67,80,92,89)(68,81,93,90)(69,82,94,91)(70,83,95,85)(99,143,121,134)(100,144,122,135)(101,145,123,136)(102,146,124,137)(103,147,125,138)(104,141,126,139)(105,142,120,140)(106,133,115,151)(107,127,116,152)(108,128,117,153)(109,129,118,154)(110,130,119,148)(111,131,113,149)(112,132,114,150)(155,190,177,199)(156,191,178,200)(157,192,179,201)(158,193,180,202)(159,194,181,203)(160,195,182,197)(161,196,176,198)(162,207,171,189)(163,208,172,183)(164,209,173,184)(165,210,174,185)(166,204,175,186)(167,205,169,187)(168,206,170,188), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,40)(16,41)(17,42)(18,36)(19,37)(20,38)(21,39)(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)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(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)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(183,198)(184,199)(185,200)(186,201)(187,202)(188,203)(189,197)(190,209)(191,210)(192,204)(193,205)(194,206)(195,207)(196,208), (1,48,83,27)(2,49,84,28)(3,43,78,22)(4,44,79,23)(5,45,80,24)(6,46,81,25)(7,47,82,26)(8,50,29,95)(9,51,30,96)(10,52,31,97)(11,53,32,98)(12,54,33,92)(13,55,34,93)(14,56,35,94)(15,64,40,60)(16,65,41,61)(17,66,42,62)(18,67,36,63)(19,68,37,57)(20,69,38,58)(21,70,39,59)(71,217,86,219)(72,211,87,220)(73,212,88,221)(74,213,89,222)(75,214,90,223)(76,215,91,224)(77,216,85,218)(99,184,108,199)(100,185,109,200)(101,186,110,201)(102,187,111,202)(103,188,112,203)(104,189,106,197)(105,183,107,198)(113,193,124,205)(114,194,125,206)(115,195,126,207)(116,196,120,208)(117,190,121,209)(118,191,122,210)(119,192,123,204)(127,176,142,172)(128,177,143,173)(129,178,144,174)(130,179,145,175)(131,180,146,169)(132,181,147,170)(133,182,141,171)(134,164,153,155)(135,165,154,156)(136,166,148,157)(137,167,149,158)(138,168,150,159)(139,162,151,160)(140,163,152,161), (1,207)(2,208)(3,209)(4,210)(5,204)(6,205)(7,206)(8,151)(9,152)(10,153)(11,154)(12,148)(13,149)(14,150)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,141)(22,117)(23,118)(24,119)(25,113)(26,114)(27,115)(28,116)(29,139)(30,140)(31,134)(32,135)(33,136)(34,137)(35,138)(36,130)(37,131)(38,132)(39,133)(40,127)(41,128)(42,129)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,120)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,157)(93,158)(94,159)(95,160)(96,161)(97,155)(98,156)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224) );
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)], [(1,50,77,59),(2,51,71,60),(3,52,72,61),(4,53,73,62),(5,54,74,63),(6,55,75,57),(7,56,76,58),(8,27,39,218),(9,28,40,219),(10,22,41,220),(11,23,42,221),(12,24,36,222),(13,25,37,223),(14,26,38,224),(15,217,30,49),(16,211,31,43),(17,212,32,44),(18,213,33,45),(19,214,34,46),(20,215,35,47),(21,216,29,48),(64,84,96,86),(65,78,97,87),(66,79,98,88),(67,80,92,89),(68,81,93,90),(69,82,94,91),(70,83,95,85),(99,143,121,134),(100,144,122,135),(101,145,123,136),(102,146,124,137),(103,147,125,138),(104,141,126,139),(105,142,120,140),(106,133,115,151),(107,127,116,152),(108,128,117,153),(109,129,118,154),(110,130,119,148),(111,131,113,149),(112,132,114,150),(155,190,177,199),(156,191,178,200),(157,192,179,201),(158,193,180,202),(159,194,181,203),(160,195,182,197),(161,196,176,198),(162,207,171,189),(163,208,172,183),(164,209,173,184),(165,210,174,185),(166,204,175,186),(167,205,169,187),(168,206,170,188)], [(1,95),(2,96),(3,97),(4,98),(5,92),(6,93),(7,94),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,40),(16,41),(17,42),(18,36),(19,37),(20,38),(21,39),(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),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(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),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133),(155,173),(156,174),(157,175),(158,169),(159,170),(160,171),(161,172),(162,182),(163,176),(164,177),(165,178),(166,179),(167,180),(168,181),(183,198),(184,199),(185,200),(186,201),(187,202),(188,203),(189,197),(190,209),(191,210),(192,204),(193,205),(194,206),(195,207),(196,208)], [(1,48,83,27),(2,49,84,28),(3,43,78,22),(4,44,79,23),(5,45,80,24),(6,46,81,25),(7,47,82,26),(8,50,29,95),(9,51,30,96),(10,52,31,97),(11,53,32,98),(12,54,33,92),(13,55,34,93),(14,56,35,94),(15,64,40,60),(16,65,41,61),(17,66,42,62),(18,67,36,63),(19,68,37,57),(20,69,38,58),(21,70,39,59),(71,217,86,219),(72,211,87,220),(73,212,88,221),(74,213,89,222),(75,214,90,223),(76,215,91,224),(77,216,85,218),(99,184,108,199),(100,185,109,200),(101,186,110,201),(102,187,111,202),(103,188,112,203),(104,189,106,197),(105,183,107,198),(113,193,124,205),(114,194,125,206),(115,195,126,207),(116,196,120,208),(117,190,121,209),(118,191,122,210),(119,192,123,204),(127,176,142,172),(128,177,143,173),(129,178,144,174),(130,179,145,175),(131,180,146,169),(132,181,147,170),(133,182,141,171),(134,164,153,155),(135,165,154,156),(136,166,148,157),(137,167,149,158),(138,168,150,159),(139,162,151,160),(140,163,152,161)], [(1,207),(2,208),(3,209),(4,210),(5,204),(6,205),(7,206),(8,151),(9,152),(10,153),(11,154),(12,148),(13,149),(14,150),(15,142),(16,143),(17,144),(18,145),(19,146),(20,147),(21,141),(22,117),(23,118),(24,119),(25,113),(26,114),(27,115),(28,116),(29,139),(30,140),(31,134),(32,135),(33,136),(34,137),(35,138),(36,130),(37,131),(38,132),(39,133),(40,127),(41,128),(42,129),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,120),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,157),(93,158),(94,159),(95,160),(96,161),(97,155),(98,156),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AJ | 14AK | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28AJ | 28AK | ··· | 28AP | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 8 | ··· | 8 | 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⋊D4 | C7×C22⋊C8 | C7×D4⋊C4 | C7×Q8⋊C4 | C7×C4⋊D4 | C14×D8 | C14×SD16 | C14×C4○D4 | D4⋊D4 | C22⋊C8 | D4⋊C4 | Q8⋊C4 | C4⋊D4 | C2×D8 | C2×SD16 | 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⋊D4 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 100 | 100 | 0 | 0 |
| 100 | 13 | 0 | 0 |
| 0 | 0 | 30 | 30 |
| 0 | 0 | 34 | 83 |
| 82 | 31 | 0 | 0 |
| 31 | 31 | 0 | 0 |
| 0 | 0 | 30 | 30 |
| 0 | 0 | 19 | 83 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,30,0,0,0,0,30],[0,1,0,0,112,0,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[100,100,0,0,100,13,0,0,0,0,30,34,0,0,30,83],[82,31,0,0,31,31,0,0,0,0,30,19,0,0,30,83] >;
C7×D4⋊D4 in GAP, Magma, Sage, TeX
C_7\times D_4\rtimes D_4
% in TeX
G:=Group("C7xD4:D4"); // GroupNames label
G:=SmallGroup(448,857);
// by ID
G=gap.SmallGroup(448,857);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,2438,1192,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=e^2=1,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=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations