direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic7⋊4D4, C24.53D14, C14⋊2(C4×D4), C23⋊4(C4×D7), Dic7⋊9(C2×D4), C22⋊C4⋊50D14, (C2×Dic7)⋊23D4, D14⋊2(C22×C4), C14.8(C23×C4), D14⋊C4⋊56C22, (C2×C14).29C24, Dic7⋊1(C22×C4), (C23×Dic7)⋊3C2, C22.124(D4×D7), C14.34(C22×D4), (C2×C28).570C23, Dic7⋊C4⋊57C22, (C4×Dic7)⋊73C22, (C22×C4).312D14, C22.18(C23×D7), (C23×C14).55C22, (C23×D7).94C22, C23.319(C22×D7), C22.66(D4⋊2D7), (C22×C14).121C23, (C22×C28).351C22, (C2×Dic7).303C23, (C22×Dic7)⋊41C22, (C22×D7).149C23, C7⋊2(C2×C4×D4), C2.2(C2×D4×D7), C22⋊2(C2×C4×D7), C7⋊D4⋊5(C2×C4), (C2×C7⋊D4)⋊8C4, (C2×C4×Dic7)⋊30C2, (C2×C4×D7)⋊64C22, (D7×C22×C4)⋊16C2, (C2×D14⋊C4)⋊30C2, C2.10(D7×C22×C4), (C22×C14)⋊9(C2×C4), (C2×C14)⋊3(C22×C4), (C2×C22⋊C4)⋊19D7, C14.67(C2×C4○D4), C2.2(C2×D4⋊2D7), (C14×C22⋊C4)⋊25C2, (C2×Dic7⋊C4)⋊35C2, (C2×Dic7)⋊16(C2×C4), (C2×C14).380(C2×D4), (C22×D7)⋊10(C2×C4), (C22×C7⋊D4).8C2, (C7×C22⋊C4)⋊60C22, (C2×C4).256(C22×D7), (C2×C7⋊D4).87C22, (C2×C14).167(C4○D4), SmallGroup(448,938)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic7⋊4D4
G = < a,b,c,d,e | a2=b14=d4=e2=1, c2=b7, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1748 in 426 conjugacy classes, 175 normal (31 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C4×D4, C4×Dic7, Dic7⋊C4, D14⋊C4, C7×C22⋊C4, C2×C4×D7, C2×C4×D7, C22×Dic7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, Dic7⋊4D4, C2×C4×Dic7, C2×Dic7⋊C4, C2×D14⋊C4, C14×C22⋊C4, D7×C22×C4, C23×Dic7, C22×C7⋊D4, C2×Dic7⋊4D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, C24, D14, C4×D4, C23×C4, C22×D4, C2×C4○D4, C4×D7, C22×D7, C2×C4×D4, C2×C4×D7, D4×D7, D4⋊2D7, C23×D7, Dic7⋊4D4, D7×C22×C4, C2×D4×D7, C2×D4⋊2D7, C2×Dic7⋊4D4
►On 224 points
Generators in S224
(1 211)(2 212)(3 213)(4 214)(5 215)(6 216)(7 217)(8 218)(9 219)(10 220)(11 221)(12 222)(13 223)(14 224)(15 119)(16 120)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 188)(30 189)(31 190)(32 191)(33 192)(34 193)(35 194)(36 195)(37 196)(38 183)(39 184)(40 185)(41 186)(42 187)(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)(57 162)(58 163)(59 164)(60 165)(61 166)(62 167)(63 168)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 161)(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)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 127)(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 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)
(1 172 8 179)(2 171 9 178)(3 170 10 177)(4 169 11 176)(5 182 12 175)(6 181 13 174)(7 180 14 173)(15 112 22 105)(16 111 23 104)(17 110 24 103)(18 109 25 102)(19 108 26 101)(20 107 27 100)(21 106 28 99)(29 48 36 55)(30 47 37 54)(31 46 38 53)(32 45 39 52)(33 44 40 51)(34 43 41 50)(35 56 42 49)(57 85 64 92)(58 98 65 91)(59 97 66 90)(60 96 67 89)(61 95 68 88)(62 94 69 87)(63 93 70 86)(71 214 78 221)(72 213 79 220)(73 212 80 219)(74 211 81 218)(75 224 82 217)(76 223 83 216)(77 222 84 215)(113 209 120 202)(114 208 121 201)(115 207 122 200)(116 206 123 199)(117 205 124 198)(118 204 125 197)(119 203 126 210)(127 156 134 163)(128 155 135 162)(129 168 136 161)(130 167 137 160)(131 166 138 159)(132 165 139 158)(133 164 140 157)(141 186 148 193)(142 185 149 192)(143 184 150 191)(144 183 151 190)(145 196 152 189)(146 195 153 188)(147 194 154 187)
(1 205 89 30)(2 204 90 29)(3 203 91 42)(4 202 92 41)(5 201 93 40)(6 200 94 39)(7 199 95 38)(8 198 96 37)(9 197 97 36)(10 210 98 35)(11 209 85 34)(12 208 86 33)(13 207 87 32)(14 206 88 31)(15 156 154 72)(16 155 141 71)(17 168 142 84)(18 167 143 83)(19 166 144 82)(20 165 145 81)(21 164 146 80)(22 163 147 79)(23 162 148 78)(24 161 149 77)(25 160 150 76)(26 159 151 75)(27 158 152 74)(28 157 153 73)(43 176 120 64)(44 175 121 63)(45 174 122 62)(46 173 123 61)(47 172 124 60)(48 171 125 59)(49 170 126 58)(50 169 113 57)(51 182 114 70)(52 181 115 69)(53 180 116 68)(54 179 117 67)(55 178 118 66)(56 177 119 65)(99 133 188 212)(100 132 189 211)(101 131 190 224)(102 130 191 223)(103 129 192 222)(104 128 193 221)(105 127 194 220)(106 140 195 219)(107 139 196 218)(108 138 183 217)(109 137 184 216)(110 136 185 215)(111 135 186 214)(112 134 187 213)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 147)(16 148)(17 149)(18 150)(19 151)(20 152)(21 153)(22 154)(23 141)(24 142)(25 143)(26 144)(27 145)(28 146)(29 197)(30 198)(31 199)(32 200)(33 201)(34 202)(35 203)(36 204)(37 205)(38 206)(39 207)(40 208)(41 209)(42 210)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 195)(100 196)(101 183)(102 184)(103 185)(104 186)(105 187)(106 188)(107 189)(108 190)(109 191)(110 192)(111 193)(112 194)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
G:=sub<Sym(224)| (1,211)(2,212)(3,213)(4,214)(5,215)(6,216)(7,217)(8,218)(9,219)(10,220)(11,221)(12,222)(13,223)(14,224)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,188)(30,189)(31,190)(32,191)(33,192)(34,193)(35,194)(36,195)(37,196)(38,183)(39,184)(40,185)(41,186)(42,187)(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)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(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)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,127)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(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), (1,172,8,179)(2,171,9,178)(3,170,10,177)(4,169,11,176)(5,182,12,175)(6,181,13,174)(7,180,14,173)(15,112,22,105)(16,111,23,104)(17,110,24,103)(18,109,25,102)(19,108,26,101)(20,107,27,100)(21,106,28,99)(29,48,36,55)(30,47,37,54)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,56,42,49)(57,85,64,92)(58,98,65,91)(59,97,66,90)(60,96,67,89)(61,95,68,88)(62,94,69,87)(63,93,70,86)(71,214,78,221)(72,213,79,220)(73,212,80,219)(74,211,81,218)(75,224,82,217)(76,223,83,216)(77,222,84,215)(113,209,120,202)(114,208,121,201)(115,207,122,200)(116,206,123,199)(117,205,124,198)(118,204,125,197)(119,203,126,210)(127,156,134,163)(128,155,135,162)(129,168,136,161)(130,167,137,160)(131,166,138,159)(132,165,139,158)(133,164,140,157)(141,186,148,193)(142,185,149,192)(143,184,150,191)(144,183,151,190)(145,196,152,189)(146,195,153,188)(147,194,154,187), (1,205,89,30)(2,204,90,29)(3,203,91,42)(4,202,92,41)(5,201,93,40)(6,200,94,39)(7,199,95,38)(8,198,96,37)(9,197,97,36)(10,210,98,35)(11,209,85,34)(12,208,86,33)(13,207,87,32)(14,206,88,31)(15,156,154,72)(16,155,141,71)(17,168,142,84)(18,167,143,83)(19,166,144,82)(20,165,145,81)(21,164,146,80)(22,163,147,79)(23,162,148,78)(24,161,149,77)(25,160,150,76)(26,159,151,75)(27,158,152,74)(28,157,153,73)(43,176,120,64)(44,175,121,63)(45,174,122,62)(46,173,123,61)(47,172,124,60)(48,171,125,59)(49,170,126,58)(50,169,113,57)(51,182,114,70)(52,181,115,69)(53,180,116,68)(54,179,117,67)(55,178,118,66)(56,177,119,65)(99,133,188,212)(100,132,189,211)(101,131,190,224)(102,130,191,223)(103,129,192,222)(104,128,193,221)(105,127,194,220)(106,140,195,219)(107,139,196,218)(108,138,183,217)(109,137,184,216)(110,136,185,215)(111,135,186,214)(112,134,187,213), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,153)(22,154)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,197)(30,198)(31,199)(32,200)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(41,209)(42,210)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,195)(100,196)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,193)(112,194)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224)>;
G:=Group( (1,211)(2,212)(3,213)(4,214)(5,215)(6,216)(7,217)(8,218)(9,219)(10,220)(11,221)(12,222)(13,223)(14,224)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,188)(30,189)(31,190)(32,191)(33,192)(34,193)(35,194)(36,195)(37,196)(38,183)(39,184)(40,185)(41,186)(42,187)(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)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(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)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,127)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(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), (1,172,8,179)(2,171,9,178)(3,170,10,177)(4,169,11,176)(5,182,12,175)(6,181,13,174)(7,180,14,173)(15,112,22,105)(16,111,23,104)(17,110,24,103)(18,109,25,102)(19,108,26,101)(20,107,27,100)(21,106,28,99)(29,48,36,55)(30,47,37,54)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,56,42,49)(57,85,64,92)(58,98,65,91)(59,97,66,90)(60,96,67,89)(61,95,68,88)(62,94,69,87)(63,93,70,86)(71,214,78,221)(72,213,79,220)(73,212,80,219)(74,211,81,218)(75,224,82,217)(76,223,83,216)(77,222,84,215)(113,209,120,202)(114,208,121,201)(115,207,122,200)(116,206,123,199)(117,205,124,198)(118,204,125,197)(119,203,126,210)(127,156,134,163)(128,155,135,162)(129,168,136,161)(130,167,137,160)(131,166,138,159)(132,165,139,158)(133,164,140,157)(141,186,148,193)(142,185,149,192)(143,184,150,191)(144,183,151,190)(145,196,152,189)(146,195,153,188)(147,194,154,187), (1,205,89,30)(2,204,90,29)(3,203,91,42)(4,202,92,41)(5,201,93,40)(6,200,94,39)(7,199,95,38)(8,198,96,37)(9,197,97,36)(10,210,98,35)(11,209,85,34)(12,208,86,33)(13,207,87,32)(14,206,88,31)(15,156,154,72)(16,155,141,71)(17,168,142,84)(18,167,143,83)(19,166,144,82)(20,165,145,81)(21,164,146,80)(22,163,147,79)(23,162,148,78)(24,161,149,77)(25,160,150,76)(26,159,151,75)(27,158,152,74)(28,157,153,73)(43,176,120,64)(44,175,121,63)(45,174,122,62)(46,173,123,61)(47,172,124,60)(48,171,125,59)(49,170,126,58)(50,169,113,57)(51,182,114,70)(52,181,115,69)(53,180,116,68)(54,179,117,67)(55,178,118,66)(56,177,119,65)(99,133,188,212)(100,132,189,211)(101,131,190,224)(102,130,191,223)(103,129,192,222)(104,128,193,221)(105,127,194,220)(106,140,195,219)(107,139,196,218)(108,138,183,217)(109,137,184,216)(110,136,185,215)(111,135,186,214)(112,134,187,213), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,153)(22,154)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,197)(30,198)(31,199)(32,200)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(41,209)(42,210)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,195)(100,196)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,193)(112,194)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224) );
G=PermutationGroup([[(1,211),(2,212),(3,213),(4,214),(5,215),(6,216),(7,217),(8,218),(9,219),(10,220),(11,221),(12,222),(13,223),(14,224),(15,119),(16,120),(17,121),(18,122),(19,123),(20,124),(21,125),(22,126),(23,113),(24,114),(25,115),(26,116),(27,117),(28,118),(29,188),(30,189),(31,190),(32,191),(33,192),(34,193),(35,194),(36,195),(37,196),(38,183),(39,184),(40,185),(41,186),(42,187),(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),(57,162),(58,163),(59,164),(60,165),(61,166),(62,167),(63,168),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,161),(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),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,134),(92,135),(93,136),(94,137),(95,138),(96,139),(97,140),(98,127),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(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)], [(1,172,8,179),(2,171,9,178),(3,170,10,177),(4,169,11,176),(5,182,12,175),(6,181,13,174),(7,180,14,173),(15,112,22,105),(16,111,23,104),(17,110,24,103),(18,109,25,102),(19,108,26,101),(20,107,27,100),(21,106,28,99),(29,48,36,55),(30,47,37,54),(31,46,38,53),(32,45,39,52),(33,44,40,51),(34,43,41,50),(35,56,42,49),(57,85,64,92),(58,98,65,91),(59,97,66,90),(60,96,67,89),(61,95,68,88),(62,94,69,87),(63,93,70,86),(71,214,78,221),(72,213,79,220),(73,212,80,219),(74,211,81,218),(75,224,82,217),(76,223,83,216),(77,222,84,215),(113,209,120,202),(114,208,121,201),(115,207,122,200),(116,206,123,199),(117,205,124,198),(118,204,125,197),(119,203,126,210),(127,156,134,163),(128,155,135,162),(129,168,136,161),(130,167,137,160),(131,166,138,159),(132,165,139,158),(133,164,140,157),(141,186,148,193),(142,185,149,192),(143,184,150,191),(144,183,151,190),(145,196,152,189),(146,195,153,188),(147,194,154,187)], [(1,205,89,30),(2,204,90,29),(3,203,91,42),(4,202,92,41),(5,201,93,40),(6,200,94,39),(7,199,95,38),(8,198,96,37),(9,197,97,36),(10,210,98,35),(11,209,85,34),(12,208,86,33),(13,207,87,32),(14,206,88,31),(15,156,154,72),(16,155,141,71),(17,168,142,84),(18,167,143,83),(19,166,144,82),(20,165,145,81),(21,164,146,80),(22,163,147,79),(23,162,148,78),(24,161,149,77),(25,160,150,76),(26,159,151,75),(27,158,152,74),(28,157,153,73),(43,176,120,64),(44,175,121,63),(45,174,122,62),(46,173,123,61),(47,172,124,60),(48,171,125,59),(49,170,126,58),(50,169,113,57),(51,182,114,70),(52,181,115,69),(53,180,116,68),(54,179,117,67),(55,178,118,66),(56,177,119,65),(99,133,188,212),(100,132,189,211),(101,131,190,224),(102,130,191,223),(103,129,192,222),(104,128,193,221),(105,127,194,220),(106,140,195,219),(107,139,196,218),(108,138,183,217),(109,137,184,216),(110,136,185,215),(111,135,186,214),(112,134,187,213)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,147),(16,148),(17,149),(18,150),(19,151),(20,152),(21,153),(22,154),(23,141),(24,142),(25,143),(26,144),(27,145),(28,146),(29,197),(30,198),(31,199),(32,200),(33,201),(34,202),(35,203),(36,204),(37,205),(38,206),(39,207),(40,208),(41,209),(42,210),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,195),(100,196),(101,183),(102,184),(103,185),(104,186),(105,187),(106,188),(107,189),(108,190),(109,191),(110,192),(111,193),(112,194),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(133,140),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(161,168),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4×D7 | D4×D7 | D4⋊2D7 |
| kernel | C2×Dic7⋊4D4 | Dic7⋊4D4 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×D14⋊C4 | C14×C22⋊C4 | D7×C22×C4 | C23×Dic7 | C22×C7⋊D4 | C2×C7⋊D4 | C2×Dic7 | C2×C22⋊C4 | C2×C14 | C22⋊C4 | C22×C4 | C24 | C23 | C22 | C22 |
| # reps | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 16 | 4 | 3 | 4 | 12 | 6 | 3 | 24 | 6 | 6 |
Matrix representation of C2×Dic7⋊4D4 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 27 | 25 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 4 | 22 | 0 | 0 |
| 0 | 27 | 25 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 22 | 0 | 0 |
| 0 | 27 | 25 | 0 | 0 |
| 0 | 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 1 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 28 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,7,27,0,0,0,0,25,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,4,27,0,0,0,22,25,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,4,27,0,0,0,22,25,0,0,0,0,0,1,1,0,0,0,27,28],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,28] >;
C2×Dic7⋊4D4 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_7\rtimes_4D_4 % in TeX
G:=Group("C2xDic7:4D4"); // GroupNames label
G:=SmallGroup(448,938);
// by ID
G=gap.SmallGroup(448,938);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,297,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^14=d^4=e^2=1,c^2=b^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations