direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊Dic7, (D4×C14)⋊5C4, (C2×D4)⋊3Dic7, D4⋊3(C2×Dic7), (C2×C14).47D8, C14.71(C2×D8), C28.202(C2×D4), (C2×C28).188D4, C14⋊3(D4⋊C4), (C22×D4).1D7, (C2×D4).195D14, C28.79(C22×C4), C4⋊Dic7⋊67C22, (C2×C14).34SD16, C14.63(C2×SD16), C4.7(C23.D7), C4.9(C22×Dic7), C28.30(C22⋊C4), (C2×C28).469C23, C22.25(D4⋊D7), (C22×C4).352D14, (C22×C14).194D4, (D4×C14).237C22, C23.100(C7⋊D4), C22.12(D4.D7), (C22×C28).194C22, C22.33(C23.D7), C7⋊4(C2×D4⋊C4), (C22×C7⋊C8)⋊7C2, C2.4(C2×D4⋊D7), (D4×C2×C14).1C2, (C7×D4)⋊16(C2×C4), (C2×C7⋊C8)⋊32C22, C4.88(C2×C7⋊D4), C2.4(C2×D4.D7), (C2×C4⋊Dic7)⋊34C2, (C2×C28).115(C2×C4), C2.7(C2×C23.D7), (C2×C14).551(C2×D4), C14.71(C2×C22⋊C4), (C2×C4).49(C2×Dic7), C22.89(C2×C7⋊D4), (C2×C4).147(C7⋊D4), (C2×C4).556(C22×D7), (C2×C14).110(C22⋊C4), SmallGroup(448,748)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4⋊Dic7
G = < a,b,c,d,e | a2=b4=c2=d14=1, e2=d7, 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: 660 in 202 conjugacy classes, 87 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C14, C14, C14, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C22×C14, C2×D4⋊C4, C2×C7⋊C8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C22×Dic7, C22×C28, D4×C14, D4×C14, C23×C14, D4⋊Dic7, C22×C7⋊C8, C2×C4⋊Dic7, D4×C2×C14, C2×D4⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, D8, SD16, C22×C4, C2×D4, Dic7, D14, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C2×Dic7, C7⋊D4, C22×D7, C2×D4⋊C4, D4⋊D7, D4.D7, C23.D7, C22×Dic7, C2×C7⋊D4, D4⋊Dic7, C2×D4⋊D7, C2×D4.D7, C2×C23.D7, C2×D4⋊Dic7
►On 224 points
Generators in S224
(1 216)(2 217)(3 218)(4 219)(5 220)(6 221)(7 222)(8 223)(9 224)(10 211)(11 212)(12 213)(13 214)(14 215)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 50)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 155)(78 156)(79 157)(80 158)(81 159)(82 160)(83 161)(84 162)(85 180)(86 181)(87 182)(88 169)(89 170)(90 171)(91 172)(92 173)(93 174)(94 175)(95 176)(96 177)(97 178)(98 179)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 141)(110 142)(111 143)(112 144)(113 207)(114 208)(115 209)(116 210)(117 197)(118 198)(119 199)(120 200)(121 201)(122 202)(123 203)(124 204)(125 205)(126 206)(127 196)(128 183)(129 184)(130 185)(131 186)(132 187)(133 188)(134 189)(135 190)(136 191)(137 192)(138 193)(139 194)(140 195)
(1 58 118 34)(2 59 119 35)(3 60 120 36)(4 61 121 37)(5 62 122 38)(6 63 123 39)(7 64 124 40)(8 65 125 41)(9 66 126 42)(10 67 113 29)(11 68 114 30)(12 69 115 31)(13 70 116 32)(14 57 117 33)(15 204 48 222)(16 205 49 223)(17 206 50 224)(18 207 51 211)(19 208 52 212)(20 209 53 213)(21 210 54 214)(22 197 55 215)(23 198 56 216)(24 199 43 217)(25 200 44 218)(26 201 45 219)(27 202 46 220)(28 203 47 221)(71 108 133 91)(72 109 134 92)(73 110 135 93)(74 111 136 94)(75 112 137 95)(76 99 138 96)(77 100 139 97)(78 101 140 98)(79 102 127 85)(80 103 128 86)(81 104 129 87)(82 105 130 88)(83 106 131 89)(84 107 132 90)(141 189 173 164)(142 190 174 165)(143 191 175 166)(144 192 176 167)(145 193 177 168)(146 194 178 155)(147 195 179 156)(148 196 180 157)(149 183 181 158)(150 184 182 159)(151 185 169 160)(152 186 170 161)(153 187 171 162)(154 188 172 163)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 117)(16 118)(17 119)(18 120)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 113)(26 114)(27 115)(28 116)(29 218)(30 219)(31 220)(32 221)(33 222)(34 223)(35 224)(36 211)(37 212)(38 213)(39 214)(40 215)(41 216)(42 217)(57 204)(58 205)(59 206)(60 207)(61 208)(62 209)(63 210)(64 197)(65 198)(66 199)(67 200)(68 201)(69 202)(70 203)(71 156)(72 157)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 155)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 170)(100 171)(101 172)(102 173)(103 174)(104 175)(105 176)(106 177)(107 178)(108 179)(109 180)(110 181)(111 182)(112 169)(127 189)(128 190)(129 191)(130 192)(131 193)(132 194)(133 195)(134 196)(135 183)(136 184)(137 185)(138 186)(139 187)(140 188)
(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)
(1 110 8 103)(2 109 9 102)(3 108 10 101)(4 107 11 100)(5 106 12 99)(6 105 13 112)(7 104 14 111)(15 159 22 166)(16 158 23 165)(17 157 24 164)(18 156 25 163)(19 155 26 162)(20 168 27 161)(21 167 28 160)(29 140 36 133)(30 139 37 132)(31 138 38 131)(32 137 39 130)(33 136 40 129)(34 135 41 128)(35 134 42 127)(43 189 50 196)(44 188 51 195)(45 187 52 194)(46 186 53 193)(47 185 54 192)(48 184 55 191)(49 183 56 190)(57 74 64 81)(58 73 65 80)(59 72 66 79)(60 71 67 78)(61 84 68 77)(62 83 69 76)(63 82 70 75)(85 119 92 126)(86 118 93 125)(87 117 94 124)(88 116 95 123)(89 115 96 122)(90 114 97 121)(91 113 98 120)(141 224 148 217)(142 223 149 216)(143 222 150 215)(144 221 151 214)(145 220 152 213)(146 219 153 212)(147 218 154 211)(169 210 176 203)(170 209 177 202)(171 208 178 201)(172 207 179 200)(173 206 180 199)(174 205 181 198)(175 204 182 197)
G:=sub<Sym(224)| (1,216)(2,217)(3,218)(4,219)(5,220)(6,221)(7,222)(8,223)(9,224)(10,211)(11,212)(12,213)(13,214)(14,215)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,180)(86,181)(87,182)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,141)(110,142)(111,143)(112,144)(113,207)(114,208)(115,209)(116,210)(117,197)(118,198)(119,199)(120,200)(121,201)(122,202)(123,203)(124,204)(125,205)(126,206)(127,196)(128,183)(129,184)(130,185)(131,186)(132,187)(133,188)(134,189)(135,190)(136,191)(137,192)(138,193)(139,194)(140,195), (1,58,118,34)(2,59,119,35)(3,60,120,36)(4,61,121,37)(5,62,122,38)(6,63,123,39)(7,64,124,40)(8,65,125,41)(9,66,126,42)(10,67,113,29)(11,68,114,30)(12,69,115,31)(13,70,116,32)(14,57,117,33)(15,204,48,222)(16,205,49,223)(17,206,50,224)(18,207,51,211)(19,208,52,212)(20,209,53,213)(21,210,54,214)(22,197,55,215)(23,198,56,216)(24,199,43,217)(25,200,44,218)(26,201,45,219)(27,202,46,220)(28,203,47,221)(71,108,133,91)(72,109,134,92)(73,110,135,93)(74,111,136,94)(75,112,137,95)(76,99,138,96)(77,100,139,97)(78,101,140,98)(79,102,127,85)(80,103,128,86)(81,104,129,87)(82,105,130,88)(83,106,131,89)(84,107,132,90)(141,189,173,164)(142,190,174,165)(143,191,175,166)(144,192,176,167)(145,193,177,168)(146,194,178,155)(147,195,179,156)(148,196,180,157)(149,183,181,158)(150,184,182,159)(151,185,169,160)(152,186,170,161)(153,187,171,162)(154,188,172,163), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,113)(26,114)(27,115)(28,116)(29,218)(30,219)(31,220)(32,221)(33,222)(34,223)(35,224)(36,211)(37,212)(38,213)(39,214)(40,215)(41,216)(42,217)(57,204)(58,205)(59,206)(60,207)(61,208)(62,209)(63,210)(64,197)(65,198)(66,199)(67,200)(68,201)(69,202)(70,203)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,155)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(109,180)(110,181)(111,182)(112,169)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188), (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), (1,110,8,103)(2,109,9,102)(3,108,10,101)(4,107,11,100)(5,106,12,99)(6,105,13,112)(7,104,14,111)(15,159,22,166)(16,158,23,165)(17,157,24,164)(18,156,25,163)(19,155,26,162)(20,168,27,161)(21,167,28,160)(29,140,36,133)(30,139,37,132)(31,138,38,131)(32,137,39,130)(33,136,40,129)(34,135,41,128)(35,134,42,127)(43,189,50,196)(44,188,51,195)(45,187,52,194)(46,186,53,193)(47,185,54,192)(48,184,55,191)(49,183,56,190)(57,74,64,81)(58,73,65,80)(59,72,66,79)(60,71,67,78)(61,84,68,77)(62,83,69,76)(63,82,70,75)(85,119,92,126)(86,118,93,125)(87,117,94,124)(88,116,95,123)(89,115,96,122)(90,114,97,121)(91,113,98,120)(141,224,148,217)(142,223,149,216)(143,222,150,215)(144,221,151,214)(145,220,152,213)(146,219,153,212)(147,218,154,211)(169,210,176,203)(170,209,177,202)(171,208,178,201)(172,207,179,200)(173,206,180,199)(174,205,181,198)(175,204,182,197)>;
G:=Group( (1,216)(2,217)(3,218)(4,219)(5,220)(6,221)(7,222)(8,223)(9,224)(10,211)(11,212)(12,213)(13,214)(14,215)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,180)(86,181)(87,182)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,141)(110,142)(111,143)(112,144)(113,207)(114,208)(115,209)(116,210)(117,197)(118,198)(119,199)(120,200)(121,201)(122,202)(123,203)(124,204)(125,205)(126,206)(127,196)(128,183)(129,184)(130,185)(131,186)(132,187)(133,188)(134,189)(135,190)(136,191)(137,192)(138,193)(139,194)(140,195), (1,58,118,34)(2,59,119,35)(3,60,120,36)(4,61,121,37)(5,62,122,38)(6,63,123,39)(7,64,124,40)(8,65,125,41)(9,66,126,42)(10,67,113,29)(11,68,114,30)(12,69,115,31)(13,70,116,32)(14,57,117,33)(15,204,48,222)(16,205,49,223)(17,206,50,224)(18,207,51,211)(19,208,52,212)(20,209,53,213)(21,210,54,214)(22,197,55,215)(23,198,56,216)(24,199,43,217)(25,200,44,218)(26,201,45,219)(27,202,46,220)(28,203,47,221)(71,108,133,91)(72,109,134,92)(73,110,135,93)(74,111,136,94)(75,112,137,95)(76,99,138,96)(77,100,139,97)(78,101,140,98)(79,102,127,85)(80,103,128,86)(81,104,129,87)(82,105,130,88)(83,106,131,89)(84,107,132,90)(141,189,173,164)(142,190,174,165)(143,191,175,166)(144,192,176,167)(145,193,177,168)(146,194,178,155)(147,195,179,156)(148,196,180,157)(149,183,181,158)(150,184,182,159)(151,185,169,160)(152,186,170,161)(153,187,171,162)(154,188,172,163), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,113)(26,114)(27,115)(28,116)(29,218)(30,219)(31,220)(32,221)(33,222)(34,223)(35,224)(36,211)(37,212)(38,213)(39,214)(40,215)(41,216)(42,217)(57,204)(58,205)(59,206)(60,207)(61,208)(62,209)(63,210)(64,197)(65,198)(66,199)(67,200)(68,201)(69,202)(70,203)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,155)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(109,180)(110,181)(111,182)(112,169)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188), (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), (1,110,8,103)(2,109,9,102)(3,108,10,101)(4,107,11,100)(5,106,12,99)(6,105,13,112)(7,104,14,111)(15,159,22,166)(16,158,23,165)(17,157,24,164)(18,156,25,163)(19,155,26,162)(20,168,27,161)(21,167,28,160)(29,140,36,133)(30,139,37,132)(31,138,38,131)(32,137,39,130)(33,136,40,129)(34,135,41,128)(35,134,42,127)(43,189,50,196)(44,188,51,195)(45,187,52,194)(46,186,53,193)(47,185,54,192)(48,184,55,191)(49,183,56,190)(57,74,64,81)(58,73,65,80)(59,72,66,79)(60,71,67,78)(61,84,68,77)(62,83,69,76)(63,82,70,75)(85,119,92,126)(86,118,93,125)(87,117,94,124)(88,116,95,123)(89,115,96,122)(90,114,97,121)(91,113,98,120)(141,224,148,217)(142,223,149,216)(143,222,150,215)(144,221,151,214)(145,220,152,213)(146,219,153,212)(147,218,154,211)(169,210,176,203)(170,209,177,202)(171,208,178,201)(172,207,179,200)(173,206,180,199)(174,205,181,198)(175,204,182,197) );
G=PermutationGroup([[(1,216),(2,217),(3,218),(4,219),(5,220),(6,221),(7,222),(8,223),(9,224),(10,211),(11,212),(12,213),(13,214),(14,215),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(41,49),(42,50),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,155),(78,156),(79,157),(80,158),(81,159),(82,160),(83,161),(84,162),(85,180),(86,181),(87,182),(88,169),(89,170),(90,171),(91,172),(92,173),(93,174),(94,175),(95,176),(96,177),(97,178),(98,179),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,141),(110,142),(111,143),(112,144),(113,207),(114,208),(115,209),(116,210),(117,197),(118,198),(119,199),(120,200),(121,201),(122,202),(123,203),(124,204),(125,205),(126,206),(127,196),(128,183),(129,184),(130,185),(131,186),(132,187),(133,188),(134,189),(135,190),(136,191),(137,192),(138,193),(139,194),(140,195)], [(1,58,118,34),(2,59,119,35),(3,60,120,36),(4,61,121,37),(5,62,122,38),(6,63,123,39),(7,64,124,40),(8,65,125,41),(9,66,126,42),(10,67,113,29),(11,68,114,30),(12,69,115,31),(13,70,116,32),(14,57,117,33),(15,204,48,222),(16,205,49,223),(17,206,50,224),(18,207,51,211),(19,208,52,212),(20,209,53,213),(21,210,54,214),(22,197,55,215),(23,198,56,216),(24,199,43,217),(25,200,44,218),(26,201,45,219),(27,202,46,220),(28,203,47,221),(71,108,133,91),(72,109,134,92),(73,110,135,93),(74,111,136,94),(75,112,137,95),(76,99,138,96),(77,100,139,97),(78,101,140,98),(79,102,127,85),(80,103,128,86),(81,104,129,87),(82,105,130,88),(83,106,131,89),(84,107,132,90),(141,189,173,164),(142,190,174,165),(143,191,175,166),(144,192,176,167),(145,193,177,168),(146,194,178,155),(147,195,179,156),(148,196,180,157),(149,183,181,158),(150,184,182,159),(151,185,169,160),(152,186,170,161),(153,187,171,162),(154,188,172,163)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,117),(16,118),(17,119),(18,120),(19,121),(20,122),(21,123),(22,124),(23,125),(24,126),(25,113),(26,114),(27,115),(28,116),(29,218),(30,219),(31,220),(32,221),(33,222),(34,223),(35,224),(36,211),(37,212),(38,213),(39,214),(40,215),(41,216),(42,217),(57,204),(58,205),(59,206),(60,207),(61,208),(62,209),(63,210),(64,197),(65,198),(66,199),(67,200),(68,201),(69,202),(70,203),(71,156),(72,157),(73,158),(74,159),(75,160),(76,161),(77,162),(78,163),(79,164),(80,165),(81,166),(82,167),(83,168),(84,155),(85,141),(86,142),(87,143),(88,144),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,153),(98,154),(99,170),(100,171),(101,172),(102,173),(103,174),(104,175),(105,176),(106,177),(107,178),(108,179),(109,180),(110,181),(111,182),(112,169),(127,189),(128,190),(129,191),(130,192),(131,193),(132,194),(133,195),(134,196),(135,183),(136,184),(137,185),(138,186),(139,187),(140,188)], [(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)], [(1,110,8,103),(2,109,9,102),(3,108,10,101),(4,107,11,100),(5,106,12,99),(6,105,13,112),(7,104,14,111),(15,159,22,166),(16,158,23,165),(17,157,24,164),(18,156,25,163),(19,155,26,162),(20,168,27,161),(21,167,28,160),(29,140,36,133),(30,139,37,132),(31,138,38,131),(32,137,39,130),(33,136,40,129),(34,135,41,128),(35,134,42,127),(43,189,50,196),(44,188,51,195),(45,187,52,194),(46,186,53,193),(47,185,54,192),(48,184,55,191),(49,183,56,190),(57,74,64,81),(58,73,65,80),(59,72,66,79),(60,71,67,78),(61,84,68,77),(62,83,69,76),(63,82,70,75),(85,119,92,126),(86,118,93,125),(87,117,94,124),(88,116,95,123),(89,115,96,122),(90,114,97,121),(91,113,98,120),(141,224,148,217),(142,223,149,216),(143,222,150,215),(144,221,151,214),(145,220,152,213),(146,219,153,212),(147,218,154,211),(169,210,176,203),(170,209,177,202),(171,208,178,201),(172,207,179,200),(173,206,180,199),(174,205,181,198),(175,204,182,197)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 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 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | Dic7 | D14 | C7⋊D4 | C7⋊D4 | D4⋊D7 | D4.D7 |
| kernel | C2×D4⋊Dic7 | D4⋊Dic7 | C22×C7⋊C8 | C2×C4⋊Dic7 | D4×C2×C14 | D4×C14 | C2×C28 | C22×C14 | C22×D4 | C2×C14 | C2×C14 | C22×C4 | C2×D4 | C2×D4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 4 | 4 | 3 | 12 | 6 | 18 | 6 | 6 | 6 |
Matrix representation of C2×D4⋊Dic7 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 112 | 0 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 98 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 104 | 112 | 0 | 0 |
| 0 | 0 | 0 | 13 | 100 |
| 0 | 0 | 0 | 100 | 100 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,1,0],[112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,1,0],[112,0,0,0,0,0,0,1,0,0,0,112,9,0,0,0,0,0,112,0,0,0,0,0,112],[98,0,0,0,0,0,1,104,0,0,0,0,112,0,0,0,0,0,13,100,0,0,0,100,100] >;
C2×D4⋊Dic7 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes {\rm Dic}_7 % in TeX
G:=Group("C2xD4:Dic7"); // GroupNames label
G:=SmallGroup(448,748);
// by ID
G=gap.SmallGroup(448,748);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^14=1,e^2=d^7,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