direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4.2D4, C4⋊C8⋊5C14, (C4×D4)⋊4C14, D4.2(C7×D4), (D4×C28)⋊33C2, (C7×D4).27D4, (C14×D8).9C2, (C2×D8).2C14, C4.35(D4×C14), Q8⋊C4⋊6C14, (C2×C28).325D4, C4.4D4⋊3C14, C28.396(C2×D4), D4⋊C4⋊11C14, (C2×SD16)⋊12C14, (C14×SD16)⋊29C2, C42.18(C2×C14), C22.87(D4×C14), C28.345(C4○D4), C14.121(C4○D8), (C4×C28).260C22, (C2×C56).301C22, (C2×C28).922C23, C14.146(C4⋊D4), C14.136(C8⋊C22), (D4×C14).187C22, (Q8×C14).161C22, (C7×C4⋊C8)⋊24C2, C2.8(C7×C4○D8), C4.44(C7×C4○D4), (C2×C4).30(C7×D4), C4⋊C4.55(C2×C14), (C2×C8).38(C2×C14), C2.15(C7×C4⋊D4), C2.11(C7×C8⋊C22), (C2×Q8).6(C2×C14), (C7×D4⋊C4)⋊35C2, (C7×Q8⋊C4)⋊29C2, (C2×D4).57(C2×C14), (C2×C14).643(C2×D4), (C7×C4.4D4)⋊23C2, (C7×C4⋊C4).376C22, (C2×C4).97(C22×C14), SmallGroup(448,871)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4.2D4
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=b2d-1 >
Subgroups: 250 in 124 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4.2D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×D8, C7×SD16, C22×C28, D4×C14, Q8×C14, C7×D4⋊C4, C7×Q8⋊C4, C7×C4⋊C8, D4×C28, C7×C4.4D4, C14×D8, C14×SD16, C7×D4.2D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C4○D8, C8⋊C22, C7×D4, C22×C14, D4.2D4, D4×C14, C7×C4○D4, C7×C4⋊D4, C7×C4○D8, C7×C8⋊C22, C7×D4.2D4
►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 75 78 68)(2 76 79 69)(3 77 80 70)(4 71 81 64)(5 72 82 65)(6 73 83 66)(7 74 84 67)(8 35 25 42)(9 29 26 36)(10 30 27 37)(11 31 28 38)(12 32 22 39)(13 33 23 40)(14 34 24 41)(15 212 221 44)(16 213 222 45)(17 214 223 46)(18 215 224 47)(19 216 218 48)(20 217 219 49)(21 211 220 43)(50 95 59 87)(51 96 60 88)(52 97 61 89)(53 98 62 90)(54 92 63 91)(55 93 57 85)(56 94 58 86)(99 117 145 108)(100 118 146 109)(101 119 147 110)(102 113 141 111)(103 114 142 112)(104 115 143 106)(105 116 144 107)(120 154 127 137)(121 148 128 138)(122 149 129 139)(123 150 130 140)(124 151 131 134)(125 152 132 135)(126 153 133 136)(155 164 201 173)(156 165 202 174)(157 166 203 175)(158 167 197 169)(159 168 198 170)(160 162 199 171)(161 163 200 172)(176 193 183 210)(177 194 184 204)(178 195 185 205)(179 196 186 206)(180 190 187 207)(181 191 188 208)(182 192 189 209)
(1 95)(2 96)(3 97)(4 98)(5 92)(6 93)(7 94)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 43)(15 42)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 216)(23 217)(24 211)(25 212)(26 213)(27 214)(28 215)(29 222)(30 223)(31 224)(32 218)(33 219)(34 220)(35 221)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 66)(58 67)(59 68)(60 69)(61 70)(62 64)(63 65)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(99 153)(100 154)(101 148)(102 149)(103 150)(104 151)(105 152)(106 131)(107 132)(108 133)(109 127)(110 128)(111 129)(112 130)(113 122)(114 123)(115 124)(116 125)(117 126)(118 120)(119 121)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(155 182)(156 176)(157 177)(158 178)(159 179)(160 180)(161 181)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(169 195)(170 196)(171 190)(172 191)(173 192)(174 193)(175 194)(183 202)(184 203)(185 197)(186 198)(187 199)(188 200)(189 201)
(1 106 50 151)(2 107 51 152)(3 108 52 153)(4 109 53 154)(5 110 54 148)(6 111 55 149)(7 112 56 150)(8 202 212 176)(9 203 213 177)(10 197 214 178)(11 198 215 179)(12 199 216 180)(13 200 217 181)(14 201 211 182)(15 210 42 165)(16 204 36 166)(17 205 37 167)(18 206 38 168)(19 207 39 162)(20 208 40 163)(21 209 41 164)(22 160 48 187)(23 161 49 188)(24 155 43 189)(25 156 44 183)(26 157 45 184)(27 158 46 185)(28 159 47 186)(29 175 222 194)(30 169 223 195)(31 170 224 196)(32 171 218 190)(33 172 219 191)(34 173 220 192)(35 174 221 193)(57 139 83 113)(58 140 84 114)(59 134 78 115)(60 135 79 116)(61 136 80 117)(62 137 81 118)(63 138 82 119)(64 146 90 120)(65 147 91 121)(66 141 85 122)(67 142 86 123)(68 143 87 124)(69 144 88 125)(70 145 89 126)(71 100 98 127)(72 101 92 128)(73 102 93 129)(74 103 94 130)(75 104 95 131)(76 105 96 132)(77 99 97 133)
(1 218 78 19)(2 219 79 20)(3 220 80 21)(4 221 81 15)(5 222 82 16)(6 223 83 17)(7 224 84 18)(8 90 25 98)(9 91 26 92)(10 85 27 93)(11 86 28 94)(12 87 22 95)(13 88 23 96)(14 89 24 97)(29 63 36 54)(30 57 37 55)(31 58 38 56)(32 59 39 50)(33 60 40 51)(34 61 41 52)(35 62 42 53)(43 77 211 70)(44 71 212 64)(45 72 213 65)(46 73 214 66)(47 74 215 67)(48 75 216 68)(49 76 217 69)(99 155 145 201)(100 156 146 202)(101 157 147 203)(102 158 141 197)(103 159 142 198)(104 160 143 199)(105 161 144 200)(106 162 115 171)(107 163 116 172)(108 164 117 173)(109 165 118 174)(110 166 119 175)(111 167 113 169)(112 168 114 170)(120 176 127 183)(121 177 128 184)(122 178 129 185)(123 179 130 186)(124 180 131 187)(125 181 132 188)(126 182 133 189)(134 190 151 207)(135 191 152 208)(136 192 153 209)(137 193 154 210)(138 194 148 204)(139 195 149 205)(140 196 150 206)
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), (1,75,78,68)(2,76,79,69)(3,77,80,70)(4,71,81,64)(5,72,82,65)(6,73,83,66)(7,74,84,67)(8,35,25,42)(9,29,26,36)(10,30,27,37)(11,31,28,38)(12,32,22,39)(13,33,23,40)(14,34,24,41)(15,212,221,44)(16,213,222,45)(17,214,223,46)(18,215,224,47)(19,216,218,48)(20,217,219,49)(21,211,220,43)(50,95,59,87)(51,96,60,88)(52,97,61,89)(53,98,62,90)(54,92,63,91)(55,93,57,85)(56,94,58,86)(99,117,145,108)(100,118,146,109)(101,119,147,110)(102,113,141,111)(103,114,142,112)(104,115,143,106)(105,116,144,107)(120,154,127,137)(121,148,128,138)(122,149,129,139)(123,150,130,140)(124,151,131,134)(125,152,132,135)(126,153,133,136)(155,164,201,173)(156,165,202,174)(157,166,203,175)(158,167,197,169)(159,168,198,170)(160,162,199,171)(161,163,200,172)(176,193,183,210)(177,194,184,204)(178,195,185,205)(179,196,186,206)(180,190,187,207)(181,191,188,208)(182,192,189,209), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,43)(15,42)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,216)(23,217)(24,211)(25,212)(26,213)(27,214)(28,215)(29,222)(30,223)(31,224)(32,218)(33,219)(34,220)(35,221)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,66)(58,67)(59,68)(60,69)(61,70)(62,64)(63,65)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(99,153)(100,154)(101,148)(102,149)(103,150)(104,151)(105,152)(106,131)(107,132)(108,133)(109,127)(110,128)(111,129)(112,130)(113,122)(114,123)(115,124)(116,125)(117,126)(118,120)(119,121)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,182)(156,176)(157,177)(158,178)(159,179)(160,180)(161,181)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(169,195)(170,196)(171,190)(172,191)(173,192)(174,193)(175,194)(183,202)(184,203)(185,197)(186,198)(187,199)(188,200)(189,201), (1,106,50,151)(2,107,51,152)(3,108,52,153)(4,109,53,154)(5,110,54,148)(6,111,55,149)(7,112,56,150)(8,202,212,176)(9,203,213,177)(10,197,214,178)(11,198,215,179)(12,199,216,180)(13,200,217,181)(14,201,211,182)(15,210,42,165)(16,204,36,166)(17,205,37,167)(18,206,38,168)(19,207,39,162)(20,208,40,163)(21,209,41,164)(22,160,48,187)(23,161,49,188)(24,155,43,189)(25,156,44,183)(26,157,45,184)(27,158,46,185)(28,159,47,186)(29,175,222,194)(30,169,223,195)(31,170,224,196)(32,171,218,190)(33,172,219,191)(34,173,220,192)(35,174,221,193)(57,139,83,113)(58,140,84,114)(59,134,78,115)(60,135,79,116)(61,136,80,117)(62,137,81,118)(63,138,82,119)(64,146,90,120)(65,147,91,121)(66,141,85,122)(67,142,86,123)(68,143,87,124)(69,144,88,125)(70,145,89,126)(71,100,98,127)(72,101,92,128)(73,102,93,129)(74,103,94,130)(75,104,95,131)(76,105,96,132)(77,99,97,133), (1,218,78,19)(2,219,79,20)(3,220,80,21)(4,221,81,15)(5,222,82,16)(6,223,83,17)(7,224,84,18)(8,90,25,98)(9,91,26,92)(10,85,27,93)(11,86,28,94)(12,87,22,95)(13,88,23,96)(14,89,24,97)(29,63,36,54)(30,57,37,55)(31,58,38,56)(32,59,39,50)(33,60,40,51)(34,61,41,52)(35,62,42,53)(43,77,211,70)(44,71,212,64)(45,72,213,65)(46,73,214,66)(47,74,215,67)(48,75,216,68)(49,76,217,69)(99,155,145,201)(100,156,146,202)(101,157,147,203)(102,158,141,197)(103,159,142,198)(104,160,143,199)(105,161,144,200)(106,162,115,171)(107,163,116,172)(108,164,117,173)(109,165,118,174)(110,166,119,175)(111,167,113,169)(112,168,114,170)(120,176,127,183)(121,177,128,184)(122,178,129,185)(123,179,130,186)(124,180,131,187)(125,181,132,188)(126,182,133,189)(134,190,151,207)(135,191,152,208)(136,192,153,209)(137,193,154,210)(138,194,148,204)(139,195,149,205)(140,196,150,206)>;
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), (1,75,78,68)(2,76,79,69)(3,77,80,70)(4,71,81,64)(5,72,82,65)(6,73,83,66)(7,74,84,67)(8,35,25,42)(9,29,26,36)(10,30,27,37)(11,31,28,38)(12,32,22,39)(13,33,23,40)(14,34,24,41)(15,212,221,44)(16,213,222,45)(17,214,223,46)(18,215,224,47)(19,216,218,48)(20,217,219,49)(21,211,220,43)(50,95,59,87)(51,96,60,88)(52,97,61,89)(53,98,62,90)(54,92,63,91)(55,93,57,85)(56,94,58,86)(99,117,145,108)(100,118,146,109)(101,119,147,110)(102,113,141,111)(103,114,142,112)(104,115,143,106)(105,116,144,107)(120,154,127,137)(121,148,128,138)(122,149,129,139)(123,150,130,140)(124,151,131,134)(125,152,132,135)(126,153,133,136)(155,164,201,173)(156,165,202,174)(157,166,203,175)(158,167,197,169)(159,168,198,170)(160,162,199,171)(161,163,200,172)(176,193,183,210)(177,194,184,204)(178,195,185,205)(179,196,186,206)(180,190,187,207)(181,191,188,208)(182,192,189,209), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,43)(15,42)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,216)(23,217)(24,211)(25,212)(26,213)(27,214)(28,215)(29,222)(30,223)(31,224)(32,218)(33,219)(34,220)(35,221)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,66)(58,67)(59,68)(60,69)(61,70)(62,64)(63,65)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(99,153)(100,154)(101,148)(102,149)(103,150)(104,151)(105,152)(106,131)(107,132)(108,133)(109,127)(110,128)(111,129)(112,130)(113,122)(114,123)(115,124)(116,125)(117,126)(118,120)(119,121)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,182)(156,176)(157,177)(158,178)(159,179)(160,180)(161,181)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(169,195)(170,196)(171,190)(172,191)(173,192)(174,193)(175,194)(183,202)(184,203)(185,197)(186,198)(187,199)(188,200)(189,201), (1,106,50,151)(2,107,51,152)(3,108,52,153)(4,109,53,154)(5,110,54,148)(6,111,55,149)(7,112,56,150)(8,202,212,176)(9,203,213,177)(10,197,214,178)(11,198,215,179)(12,199,216,180)(13,200,217,181)(14,201,211,182)(15,210,42,165)(16,204,36,166)(17,205,37,167)(18,206,38,168)(19,207,39,162)(20,208,40,163)(21,209,41,164)(22,160,48,187)(23,161,49,188)(24,155,43,189)(25,156,44,183)(26,157,45,184)(27,158,46,185)(28,159,47,186)(29,175,222,194)(30,169,223,195)(31,170,224,196)(32,171,218,190)(33,172,219,191)(34,173,220,192)(35,174,221,193)(57,139,83,113)(58,140,84,114)(59,134,78,115)(60,135,79,116)(61,136,80,117)(62,137,81,118)(63,138,82,119)(64,146,90,120)(65,147,91,121)(66,141,85,122)(67,142,86,123)(68,143,87,124)(69,144,88,125)(70,145,89,126)(71,100,98,127)(72,101,92,128)(73,102,93,129)(74,103,94,130)(75,104,95,131)(76,105,96,132)(77,99,97,133), (1,218,78,19)(2,219,79,20)(3,220,80,21)(4,221,81,15)(5,222,82,16)(6,223,83,17)(7,224,84,18)(8,90,25,98)(9,91,26,92)(10,85,27,93)(11,86,28,94)(12,87,22,95)(13,88,23,96)(14,89,24,97)(29,63,36,54)(30,57,37,55)(31,58,38,56)(32,59,39,50)(33,60,40,51)(34,61,41,52)(35,62,42,53)(43,77,211,70)(44,71,212,64)(45,72,213,65)(46,73,214,66)(47,74,215,67)(48,75,216,68)(49,76,217,69)(99,155,145,201)(100,156,146,202)(101,157,147,203)(102,158,141,197)(103,159,142,198)(104,160,143,199)(105,161,144,200)(106,162,115,171)(107,163,116,172)(108,164,117,173)(109,165,118,174)(110,166,119,175)(111,167,113,169)(112,168,114,170)(120,176,127,183)(121,177,128,184)(122,178,129,185)(123,179,130,186)(124,180,131,187)(125,181,132,188)(126,182,133,189)(134,190,151,207)(135,191,152,208)(136,192,153,209)(137,193,154,210)(138,194,148,204)(139,195,149,205)(140,196,150,206) );
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)], [(1,75,78,68),(2,76,79,69),(3,77,80,70),(4,71,81,64),(5,72,82,65),(6,73,83,66),(7,74,84,67),(8,35,25,42),(9,29,26,36),(10,30,27,37),(11,31,28,38),(12,32,22,39),(13,33,23,40),(14,34,24,41),(15,212,221,44),(16,213,222,45),(17,214,223,46),(18,215,224,47),(19,216,218,48),(20,217,219,49),(21,211,220,43),(50,95,59,87),(51,96,60,88),(52,97,61,89),(53,98,62,90),(54,92,63,91),(55,93,57,85),(56,94,58,86),(99,117,145,108),(100,118,146,109),(101,119,147,110),(102,113,141,111),(103,114,142,112),(104,115,143,106),(105,116,144,107),(120,154,127,137),(121,148,128,138),(122,149,129,139),(123,150,130,140),(124,151,131,134),(125,152,132,135),(126,153,133,136),(155,164,201,173),(156,165,202,174),(157,166,203,175),(158,167,197,169),(159,168,198,170),(160,162,199,171),(161,163,200,172),(176,193,183,210),(177,194,184,204),(178,195,185,205),(179,196,186,206),(180,190,187,207),(181,191,188,208),(182,192,189,209)], [(1,95),(2,96),(3,97),(4,98),(5,92),(6,93),(7,94),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,43),(15,42),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,216),(23,217),(24,211),(25,212),(26,213),(27,214),(28,215),(29,222),(30,223),(31,224),(32,218),(33,219),(34,220),(35,221),(50,75),(51,76),(52,77),(53,71),(54,72),(55,73),(56,74),(57,66),(58,67),(59,68),(60,69),(61,70),(62,64),(63,65),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(99,153),(100,154),(101,148),(102,149),(103,150),(104,151),(105,152),(106,131),(107,132),(108,133),(109,127),(110,128),(111,129),(112,130),(113,122),(114,123),(115,124),(116,125),(117,126),(118,120),(119,121),(134,143),(135,144),(136,145),(137,146),(138,147),(139,141),(140,142),(155,182),(156,176),(157,177),(158,178),(159,179),(160,180),(161,181),(162,207),(163,208),(164,209),(165,210),(166,204),(167,205),(168,206),(169,195),(170,196),(171,190),(172,191),(173,192),(174,193),(175,194),(183,202),(184,203),(185,197),(186,198),(187,199),(188,200),(189,201)], [(1,106,50,151),(2,107,51,152),(3,108,52,153),(4,109,53,154),(5,110,54,148),(6,111,55,149),(7,112,56,150),(8,202,212,176),(9,203,213,177),(10,197,214,178),(11,198,215,179),(12,199,216,180),(13,200,217,181),(14,201,211,182),(15,210,42,165),(16,204,36,166),(17,205,37,167),(18,206,38,168),(19,207,39,162),(20,208,40,163),(21,209,41,164),(22,160,48,187),(23,161,49,188),(24,155,43,189),(25,156,44,183),(26,157,45,184),(27,158,46,185),(28,159,47,186),(29,175,222,194),(30,169,223,195),(31,170,224,196),(32,171,218,190),(33,172,219,191),(34,173,220,192),(35,174,221,193),(57,139,83,113),(58,140,84,114),(59,134,78,115),(60,135,79,116),(61,136,80,117),(62,137,81,118),(63,138,82,119),(64,146,90,120),(65,147,91,121),(66,141,85,122),(67,142,86,123),(68,143,87,124),(69,144,88,125),(70,145,89,126),(71,100,98,127),(72,101,92,128),(73,102,93,129),(74,103,94,130),(75,104,95,131),(76,105,96,132),(77,99,97,133)], [(1,218,78,19),(2,219,79,20),(3,220,80,21),(4,221,81,15),(5,222,82,16),(6,223,83,17),(7,224,84,18),(8,90,25,98),(9,91,26,92),(10,85,27,93),(11,86,28,94),(12,87,22,95),(13,88,23,96),(14,89,24,97),(29,63,36,54),(30,57,37,55),(31,58,38,56),(32,59,39,50),(33,60,40,51),(34,61,41,52),(35,62,42,53),(43,77,211,70),(44,71,212,64),(45,72,213,65),(46,73,214,66),(47,74,215,67),(48,75,216,68),(49,76,217,69),(99,155,145,201),(100,156,146,202),(101,157,147,203),(102,158,141,197),(103,159,142,198),(104,160,143,199),(105,161,144,200),(106,162,115,171),(107,163,116,172),(108,164,117,173),(109,165,118,174),(110,166,119,175),(111,167,113,169),(112,168,114,170),(120,176,127,183),(121,177,128,184),(122,178,129,185),(123,179,130,186),(124,180,131,187),(125,181,132,188),(126,182,133,189),(134,190,151,207),(135,191,152,208),(136,192,153,209),(137,193,154,210),(138,194,148,204),(139,195,149,205),(140,196,150,206)]])
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 | ··· | 14AD | 14AE | ··· | 14AJ | 28A | ··· | 28X | 28Y | ··· | 28AP | 28AQ | ··· | 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 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C4○D4 | C4○D8 | C7×D4 | C7×D4 | C7×C4○D4 | C7×C4○D8 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×D4.2D4 | C7×D4⋊C4 | C7×Q8⋊C4 | C7×C4⋊C8 | D4×C28 | C7×C4.4D4 | C14×D8 | C14×SD16 | D4.2D4 | D4⋊C4 | Q8⋊C4 | C4⋊C8 | C4×D4 | C4.4D4 | C2×D8 | C2×SD16 | C2×C28 | C7×D4 | C28 | C14 | C2×C4 | D4 | C4 | C2 | C14 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 24 | 1 | 6 |
Matrix representation of C7×D4.2D4 ►in GL4(𝔽113) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 111 |
| 0 | 0 | 1 | 112 |
| 1 | 101 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 111 |
| 0 | 0 | 0 | 112 |
| 98 | 67 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 21 | 68 | 0 | 0 |
| 60 | 92 | 0 | 0 |
| 0 | 0 | 0 | 87 |
| 0 | 0 | 100 | 0 |
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,30,0,0,0,0,30],[112,0,0,0,0,112,0,0,0,0,1,1,0,0,111,112],[1,0,0,0,101,112,0,0,0,0,1,0,0,0,111,112],[98,0,0,0,67,15,0,0,0,0,15,0,0,0,0,15],[21,60,0,0,68,92,0,0,0,0,0,100,0,0,87,0] >;
C7×D4.2D4 in GAP, Magma, Sage, TeX
C_7\times D_4._2D_4
% in TeX
G:=Group("C7xD4.2D4"); // GroupNames label
G:=SmallGroup(448,871);
// by ID
G=gap.SmallGroup(448,871);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1968,2438,14117,3547,124]);
// 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=b^2*d^-1>;
// generators/relations