direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic7.D4, C24.28D14, C22⋊C4⋊41D14, Dic7.1(C2×D4), D14⋊C4⋊47C22, C14⋊2(C4.4D4), (C2×C14).35C24, C22.128(D4×D7), C14.38(C22×D4), (C2×C28).574C23, (C2×Dic7).119D4, (C4×Dic7)⋊74C22, (C22×Dic14)⋊6C2, (C22×C4).314D14, C23.D7⋊46C22, (C22×D7).7C23, C22.74(C23×D7), C23.81(C22×D7), (C2×Dic14)⋊49C22, C22.74(C4○D28), (C23×C14).61C22, (C23×D7).32C22, C22.68(D4⋊2D7), (C22×C28).354C22, (C22×C14).388C23, (C2×Dic7).181C23, (C22×Dic7).79C22, C2.12(C2×D4×D7), C7⋊2(C2×C4.4D4), (C2×C4×Dic7)⋊31C2, (C2×D14⋊C4)⋊18C2, (C2×C22⋊C4)⋊14D7, C14.15(C2×C4○D4), C2.17(C2×C4○D28), (C14×C22⋊C4)⋊19C2, C2.10(C2×D4⋊2D7), (C2×C14).384(C2×D4), (C2×C23.D7)⋊17C2, (C7×C22⋊C4)⋊54C22, (C2×C4).260(C22×D7), (C22×C7⋊D4).11C2, (C2×C7⋊D4).90C22, (C2×C14).103(C4○D4), SmallGroup(448,944)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic7.D4
G = < a,b,c,d,e | a2=b14=d4=1, c2=e2=b7, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b7c, ede-1=b7d-1 >
Subgroups: 1556 in 330 conjugacy classes, 119 normal (31 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C24, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, Dic14, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C4.4D4, C4×Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×Dic14, C2×Dic14, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, Dic7.D4, C2×C4×Dic7, C2×D14⋊C4, C2×C23.D7, C14×C22⋊C4, C22×Dic14, C22×C7⋊D4, C2×Dic7.D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C4.4D4, C22×D4, C2×C4○D4, C22×D7, C2×C4.4D4, C4○D28, D4×D7, D4⋊2D7, C23×D7, Dic7.D4, C2×C4○D28, C2×D4×D7, C2×D4⋊2D7, C2×Dic7.D4
►On 224 points
Generators in S224
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 71)(26 72)(27 73)(28 74)(43 67)(44 68)(45 69)(46 70)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(85 154)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 186)(100 187)(101 188)(102 189)(103 190)(104 191)(105 192)(106 193)(107 194)(108 195)(109 196)(110 183)(111 184)(112 185)(113 197)(114 198)(115 199)(116 200)(117 201)(118 202)(119 203)(120 204)(121 205)(122 206)(123 207)(124 208)(125 209)(126 210)(127 221)(128 222)(129 223)(130 224)(131 211)(132 212)(133 213)(134 214)(135 215)(136 216)(137 217)(138 218)(139 219)(140 220)(155 170)(156 171)(157 172)(158 173)(159 174)(160 175)(161 176)(162 177)(163 178)(164 179)(165 180)(166 181)(167 182)(168 169)
(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)
(1 173 8 180)(2 172 9 179)(3 171 10 178)(4 170 11 177)(5 169 12 176)(6 182 13 175)(7 181 14 174)(15 188 22 195)(16 187 23 194)(17 186 24 193)(18 185 25 192)(19 184 26 191)(20 183 27 190)(21 196 28 189)(29 166 36 159)(30 165 37 158)(31 164 38 157)(32 163 39 156)(33 162 40 155)(34 161 41 168)(35 160 42 167)(43 125 50 118)(44 124 51 117)(45 123 52 116)(46 122 53 115)(47 121 54 114)(48 120 55 113)(49 119 56 126)(57 205 64 198)(58 204 65 197)(59 203 66 210)(60 202 67 209)(61 201 68 208)(62 200 69 207)(63 199 70 206)(71 105 78 112)(72 104 79 111)(73 103 80 110)(74 102 81 109)(75 101 82 108)(76 100 83 107)(77 99 84 106)(85 215 92 222)(86 214 93 221)(87 213 94 220)(88 212 95 219)(89 211 96 218)(90 224 97 217)(91 223 98 216)(127 141 134 148)(128 154 135 147)(129 153 136 146)(130 152 137 145)(131 151 138 144)(132 150 139 143)(133 149 140 142)
(1 188 63 134)(2 189 64 135)(3 190 65 136)(4 191 66 137)(5 192 67 138)(6 193 68 139)(7 194 69 140)(8 195 70 127)(9 196 57 128)(10 183 58 129)(11 184 59 130)(12 185 60 131)(13 186 61 132)(14 187 62 133)(15 206 141 180)(16 207 142 181)(17 208 143 182)(18 209 144 169)(19 210 145 170)(20 197 146 171)(21 198 147 172)(22 199 148 173)(23 200 149 174)(24 201 150 175)(25 202 151 176)(26 203 152 177)(27 204 153 178)(28 205 154 179)(29 107 45 220)(30 108 46 221)(31 109 47 222)(32 110 48 223)(33 111 49 224)(34 112 50 211)(35 99 51 212)(36 100 52 213)(37 101 53 214)(38 102 54 215)(39 103 55 216)(40 104 56 217)(41 105 43 218)(42 106 44 219)(71 118 96 161)(72 119 97 162)(73 120 98 163)(74 121 85 164)(75 122 86 165)(76 123 87 166)(77 124 88 167)(78 125 89 168)(79 126 90 155)(80 113 91 156)(81 114 92 157)(82 115 93 158)(83 116 94 159)(84 117 95 160)
(1 195 8 188)(2 194 9 187)(3 193 10 186)(4 192 11 185)(5 191 12 184)(6 190 13 183)(7 189 14 196)(15 173 22 180)(16 172 23 179)(17 171 24 178)(18 170 25 177)(19 169 26 176)(20 182 27 175)(21 181 28 174)(29 102 36 109)(30 101 37 108)(31 100 38 107)(32 99 39 106)(33 112 40 105)(34 111 41 104)(35 110 42 103)(43 217 50 224)(44 216 51 223)(45 215 52 222)(46 214 53 221)(47 213 54 220)(48 212 55 219)(49 211 56 218)(57 133 64 140)(58 132 65 139)(59 131 66 138)(60 130 67 137)(61 129 68 136)(62 128 69 135)(63 127 70 134)(71 162 78 155)(72 161 79 168)(73 160 80 167)(74 159 81 166)(75 158 82 165)(76 157 83 164)(77 156 84 163)(85 116 92 123)(86 115 93 122)(87 114 94 121)(88 113 95 120)(89 126 96 119)(90 125 97 118)(91 124 98 117)(141 199 148 206)(142 198 149 205)(143 197 150 204)(144 210 151 203)(145 209 152 202)(146 208 153 201)(147 207 154 200)
G:=sub<Sym(224)| (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,71)(26,72)(27,73)(28,74)(43,67)(44,68)(45,69)(46,70)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(85,154)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191)(105,192)(106,193)(107,194)(108,195)(109,196)(110,183)(111,184)(112,185)(113,197)(114,198)(115,199)(116,200)(117,201)(118,202)(119,203)(120,204)(121,205)(122,206)(123,207)(124,208)(125,209)(126,210)(127,221)(128,222)(129,223)(130,224)(131,211)(132,212)(133,213)(134,214)(135,215)(136,216)(137,217)(138,218)(139,219)(140,220)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(166,181)(167,182)(168,169), (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), (1,173,8,180)(2,172,9,179)(3,171,10,178)(4,170,11,177)(5,169,12,176)(6,182,13,175)(7,181,14,174)(15,188,22,195)(16,187,23,194)(17,186,24,193)(18,185,25,192)(19,184,26,191)(20,183,27,190)(21,196,28,189)(29,166,36,159)(30,165,37,158)(31,164,38,157)(32,163,39,156)(33,162,40,155)(34,161,41,168)(35,160,42,167)(43,125,50,118)(44,124,51,117)(45,123,52,116)(46,122,53,115)(47,121,54,114)(48,120,55,113)(49,119,56,126)(57,205,64,198)(58,204,65,197)(59,203,66,210)(60,202,67,209)(61,201,68,208)(62,200,69,207)(63,199,70,206)(71,105,78,112)(72,104,79,111)(73,103,80,110)(74,102,81,109)(75,101,82,108)(76,100,83,107)(77,99,84,106)(85,215,92,222)(86,214,93,221)(87,213,94,220)(88,212,95,219)(89,211,96,218)(90,224,97,217)(91,223,98,216)(127,141,134,148)(128,154,135,147)(129,153,136,146)(130,152,137,145)(131,151,138,144)(132,150,139,143)(133,149,140,142), (1,188,63,134)(2,189,64,135)(3,190,65,136)(4,191,66,137)(5,192,67,138)(6,193,68,139)(7,194,69,140)(8,195,70,127)(9,196,57,128)(10,183,58,129)(11,184,59,130)(12,185,60,131)(13,186,61,132)(14,187,62,133)(15,206,141,180)(16,207,142,181)(17,208,143,182)(18,209,144,169)(19,210,145,170)(20,197,146,171)(21,198,147,172)(22,199,148,173)(23,200,149,174)(24,201,150,175)(25,202,151,176)(26,203,152,177)(27,204,153,178)(28,205,154,179)(29,107,45,220)(30,108,46,221)(31,109,47,222)(32,110,48,223)(33,111,49,224)(34,112,50,211)(35,99,51,212)(36,100,52,213)(37,101,53,214)(38,102,54,215)(39,103,55,216)(40,104,56,217)(41,105,43,218)(42,106,44,219)(71,118,96,161)(72,119,97,162)(73,120,98,163)(74,121,85,164)(75,122,86,165)(76,123,87,166)(77,124,88,167)(78,125,89,168)(79,126,90,155)(80,113,91,156)(81,114,92,157)(82,115,93,158)(83,116,94,159)(84,117,95,160), (1,195,8,188)(2,194,9,187)(3,193,10,186)(4,192,11,185)(5,191,12,184)(6,190,13,183)(7,189,14,196)(15,173,22,180)(16,172,23,179)(17,171,24,178)(18,170,25,177)(19,169,26,176)(20,182,27,175)(21,181,28,174)(29,102,36,109)(30,101,37,108)(31,100,38,107)(32,99,39,106)(33,112,40,105)(34,111,41,104)(35,110,42,103)(43,217,50,224)(44,216,51,223)(45,215,52,222)(46,214,53,221)(47,213,54,220)(48,212,55,219)(49,211,56,218)(57,133,64,140)(58,132,65,139)(59,131,66,138)(60,130,67,137)(61,129,68,136)(62,128,69,135)(63,127,70,134)(71,162,78,155)(72,161,79,168)(73,160,80,167)(74,159,81,166)(75,158,82,165)(76,157,83,164)(77,156,84,163)(85,116,92,123)(86,115,93,122)(87,114,94,121)(88,113,95,120)(89,126,96,119)(90,125,97,118)(91,124,98,117)(141,199,148,206)(142,198,149,205)(143,197,150,204)(144,210,151,203)(145,209,152,202)(146,208,153,201)(147,207,154,200)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,71)(26,72)(27,73)(28,74)(43,67)(44,68)(45,69)(46,70)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(85,154)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191)(105,192)(106,193)(107,194)(108,195)(109,196)(110,183)(111,184)(112,185)(113,197)(114,198)(115,199)(116,200)(117,201)(118,202)(119,203)(120,204)(121,205)(122,206)(123,207)(124,208)(125,209)(126,210)(127,221)(128,222)(129,223)(130,224)(131,211)(132,212)(133,213)(134,214)(135,215)(136,216)(137,217)(138,218)(139,219)(140,220)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(166,181)(167,182)(168,169), (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), (1,173,8,180)(2,172,9,179)(3,171,10,178)(4,170,11,177)(5,169,12,176)(6,182,13,175)(7,181,14,174)(15,188,22,195)(16,187,23,194)(17,186,24,193)(18,185,25,192)(19,184,26,191)(20,183,27,190)(21,196,28,189)(29,166,36,159)(30,165,37,158)(31,164,38,157)(32,163,39,156)(33,162,40,155)(34,161,41,168)(35,160,42,167)(43,125,50,118)(44,124,51,117)(45,123,52,116)(46,122,53,115)(47,121,54,114)(48,120,55,113)(49,119,56,126)(57,205,64,198)(58,204,65,197)(59,203,66,210)(60,202,67,209)(61,201,68,208)(62,200,69,207)(63,199,70,206)(71,105,78,112)(72,104,79,111)(73,103,80,110)(74,102,81,109)(75,101,82,108)(76,100,83,107)(77,99,84,106)(85,215,92,222)(86,214,93,221)(87,213,94,220)(88,212,95,219)(89,211,96,218)(90,224,97,217)(91,223,98,216)(127,141,134,148)(128,154,135,147)(129,153,136,146)(130,152,137,145)(131,151,138,144)(132,150,139,143)(133,149,140,142), (1,188,63,134)(2,189,64,135)(3,190,65,136)(4,191,66,137)(5,192,67,138)(6,193,68,139)(7,194,69,140)(8,195,70,127)(9,196,57,128)(10,183,58,129)(11,184,59,130)(12,185,60,131)(13,186,61,132)(14,187,62,133)(15,206,141,180)(16,207,142,181)(17,208,143,182)(18,209,144,169)(19,210,145,170)(20,197,146,171)(21,198,147,172)(22,199,148,173)(23,200,149,174)(24,201,150,175)(25,202,151,176)(26,203,152,177)(27,204,153,178)(28,205,154,179)(29,107,45,220)(30,108,46,221)(31,109,47,222)(32,110,48,223)(33,111,49,224)(34,112,50,211)(35,99,51,212)(36,100,52,213)(37,101,53,214)(38,102,54,215)(39,103,55,216)(40,104,56,217)(41,105,43,218)(42,106,44,219)(71,118,96,161)(72,119,97,162)(73,120,98,163)(74,121,85,164)(75,122,86,165)(76,123,87,166)(77,124,88,167)(78,125,89,168)(79,126,90,155)(80,113,91,156)(81,114,92,157)(82,115,93,158)(83,116,94,159)(84,117,95,160), (1,195,8,188)(2,194,9,187)(3,193,10,186)(4,192,11,185)(5,191,12,184)(6,190,13,183)(7,189,14,196)(15,173,22,180)(16,172,23,179)(17,171,24,178)(18,170,25,177)(19,169,26,176)(20,182,27,175)(21,181,28,174)(29,102,36,109)(30,101,37,108)(31,100,38,107)(32,99,39,106)(33,112,40,105)(34,111,41,104)(35,110,42,103)(43,217,50,224)(44,216,51,223)(45,215,52,222)(46,214,53,221)(47,213,54,220)(48,212,55,219)(49,211,56,218)(57,133,64,140)(58,132,65,139)(59,131,66,138)(60,130,67,137)(61,129,68,136)(62,128,69,135)(63,127,70,134)(71,162,78,155)(72,161,79,168)(73,160,80,167)(74,159,81,166)(75,158,82,165)(76,157,83,164)(77,156,84,163)(85,116,92,123)(86,115,93,122)(87,114,94,121)(88,113,95,120)(89,126,96,119)(90,125,97,118)(91,124,98,117)(141,199,148,206)(142,198,149,205)(143,197,150,204)(144,210,151,203)(145,209,152,202)(146,208,153,201)(147,207,154,200) );
G=PermutationGroup([[(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,71),(26,72),(27,73),(28,74),(43,67),(44,68),(45,69),(46,70),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(85,154),(86,141),(87,142),(88,143),(89,144),(90,145),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,186),(100,187),(101,188),(102,189),(103,190),(104,191),(105,192),(106,193),(107,194),(108,195),(109,196),(110,183),(111,184),(112,185),(113,197),(114,198),(115,199),(116,200),(117,201),(118,202),(119,203),(120,204),(121,205),(122,206),(123,207),(124,208),(125,209),(126,210),(127,221),(128,222),(129,223),(130,224),(131,211),(132,212),(133,213),(134,214),(135,215),(136,216),(137,217),(138,218),(139,219),(140,220),(155,170),(156,171),(157,172),(158,173),(159,174),(160,175),(161,176),(162,177),(163,178),(164,179),(165,180),(166,181),(167,182),(168,169)], [(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)], [(1,173,8,180),(2,172,9,179),(3,171,10,178),(4,170,11,177),(5,169,12,176),(6,182,13,175),(7,181,14,174),(15,188,22,195),(16,187,23,194),(17,186,24,193),(18,185,25,192),(19,184,26,191),(20,183,27,190),(21,196,28,189),(29,166,36,159),(30,165,37,158),(31,164,38,157),(32,163,39,156),(33,162,40,155),(34,161,41,168),(35,160,42,167),(43,125,50,118),(44,124,51,117),(45,123,52,116),(46,122,53,115),(47,121,54,114),(48,120,55,113),(49,119,56,126),(57,205,64,198),(58,204,65,197),(59,203,66,210),(60,202,67,209),(61,201,68,208),(62,200,69,207),(63,199,70,206),(71,105,78,112),(72,104,79,111),(73,103,80,110),(74,102,81,109),(75,101,82,108),(76,100,83,107),(77,99,84,106),(85,215,92,222),(86,214,93,221),(87,213,94,220),(88,212,95,219),(89,211,96,218),(90,224,97,217),(91,223,98,216),(127,141,134,148),(128,154,135,147),(129,153,136,146),(130,152,137,145),(131,151,138,144),(132,150,139,143),(133,149,140,142)], [(1,188,63,134),(2,189,64,135),(3,190,65,136),(4,191,66,137),(5,192,67,138),(6,193,68,139),(7,194,69,140),(8,195,70,127),(9,196,57,128),(10,183,58,129),(11,184,59,130),(12,185,60,131),(13,186,61,132),(14,187,62,133),(15,206,141,180),(16,207,142,181),(17,208,143,182),(18,209,144,169),(19,210,145,170),(20,197,146,171),(21,198,147,172),(22,199,148,173),(23,200,149,174),(24,201,150,175),(25,202,151,176),(26,203,152,177),(27,204,153,178),(28,205,154,179),(29,107,45,220),(30,108,46,221),(31,109,47,222),(32,110,48,223),(33,111,49,224),(34,112,50,211),(35,99,51,212),(36,100,52,213),(37,101,53,214),(38,102,54,215),(39,103,55,216),(40,104,56,217),(41,105,43,218),(42,106,44,219),(71,118,96,161),(72,119,97,162),(73,120,98,163),(74,121,85,164),(75,122,86,165),(76,123,87,166),(77,124,88,167),(78,125,89,168),(79,126,90,155),(80,113,91,156),(81,114,92,157),(82,115,93,158),(83,116,94,159),(84,117,95,160)], [(1,195,8,188),(2,194,9,187),(3,193,10,186),(4,192,11,185),(5,191,12,184),(6,190,13,183),(7,189,14,196),(15,173,22,180),(16,172,23,179),(17,171,24,178),(18,170,25,177),(19,169,26,176),(20,182,27,175),(21,181,28,174),(29,102,36,109),(30,101,37,108),(31,100,38,107),(32,99,39,106),(33,112,40,105),(34,111,41,104),(35,110,42,103),(43,217,50,224),(44,216,51,223),(45,215,52,222),(46,214,53,221),(47,213,54,220),(48,212,55,219),(49,211,56,218),(57,133,64,140),(58,132,65,139),(59,131,66,138),(60,130,67,137),(61,129,68,136),(62,128,69,135),(63,127,70,134),(71,162,78,155),(72,161,79,168),(73,160,80,167),(74,159,81,166),(75,158,82,165),(76,157,83,164),(77,156,84,163),(85,116,92,123),(86,115,93,122),(87,114,94,121),(88,113,95,120),(89,126,96,119),(90,125,97,118),(91,124,98,117),(141,199,148,206),(142,198,149,205),(143,197,150,204),(144,210,151,203),(145,209,152,202),(146,208,153,201),(147,207,154,200)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 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 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | D4×D7 | D4⋊2D7 |
| kernel | C2×Dic7.D4 | Dic7.D4 | C2×C4×Dic7 | C2×D14⋊C4 | C2×C23.D7 | C14×C22⋊C4 | C22×Dic14 | C22×C7⋊D4 | C2×Dic7 | C2×C22⋊C4 | C2×C14 | C22⋊C4 | C22×C4 | C24 | C22 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 8 | 12 | 6 | 3 | 24 | 6 | 6 |
Matrix representation of C2×Dic7.D4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 25 | 28 | 0 | 0 | 0 | 0 |
| 2 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 13 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 15 | 18 | 0 | 0 | 0 | 0 |
| 23 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 10 | 0 | 0 |
| 0 | 0 | 14 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 15 | 18 | 0 | 0 | 0 | 0 |
| 23 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 2 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,2,0,0,0,0,28,22,0,0,0,0,0,0,4,13,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,23,0,0,0,0,18,14,0,0,0,0,0,0,2,14,0,0,0,0,10,27,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,2,0,0,0,0,14,0],[15,23,0,0,0,0,18,14,0,0,0,0,0,0,0,10,0,0,0,0,26,0,0,0,0,0,0,0,0,2,0,0,0,0,15,0] >;
C2×Dic7.D4 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_7.D_4 % in TeX
G:=Group("C2xDic7.D4"); // GroupNames label
G:=SmallGroup(448,944);
// by ID
G=gap.SmallGroup(448,944);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,1571,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^14=d^4=1,c^2=e^2=b^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^7*c,e*d*e^-1=b^7*d^-1>;
// generators/relations