metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4⋊1Dic7, Q8⋊4(C2×Dic7), D4⋊4(C2×Dic7), C28.212(C2×D4), (C2×C28).476D4, (C2×D4).201D14, Q8⋊Dic7⋊40C2, D4⋊Dic7⋊40C2, C28.83(C22×C4), (C2×Q8).170D14, C2.5(D4⋊D14), C28.90(C22⋊C4), (C2×C28).479C23, (C22×C4).161D14, (C22×C14).111D4, C7⋊5(C23.36D4), C23.65(C7⋊D4), C4.22(C23.D7), C4.13(C22×Dic7), C14.124(C8⋊C22), C2.5(D4.9D14), (D4×C14).242C22, C4⋊Dic7.354C22, (Q8×C14).205C22, C22.2(C23.D7), C14.124(C8.C22), (C22×C28).205C22, (C7×C4○D4)⋊1C4, (C7×D4)⋊17(C2×C4), (C7×Q8)⋊16(C2×C4), (C2×C4○D4).1D7, C4.94(C2×C7⋊D4), (C2×C4⋊Dic7)⋊36C2, (C14×C4○D4).1C2, (C2×C28).122(C2×C4), (C2×C14).565(C2×D4), C14.81(C2×C22⋊C4), (C2×C7⋊C8).177C22, (C2×C4.Dic7)⋊19C2, (C2×C4).28(C2×Dic7), C22.96(C2×C7⋊D4), C2.17(C2×C23.D7), (C2×C4).261(C7⋊D4), (C2×C4).564(C22×D7), (C2×C14).24(C22⋊C4), SmallGroup(448,766)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4⋊Dic7
G = < a,b,c,d,e | a4=c2=d14=1, b2=a2, e2=d7, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, ebe-1=abc, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 532 in 162 conjugacy classes, 71 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, C28, C2×C14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C23.36D4, C2×C7⋊C8, C4.Dic7, C4⋊Dic7, C4⋊Dic7, C22×Dic7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, D4⋊Dic7, Q8⋊Dic7, C2×C4.Dic7, C2×C4⋊Dic7, C14×C4○D4, C4○D4⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C8⋊C22, C8.C22, C2×Dic7, C7⋊D4, C22×D7, C23.36D4, C23.D7, C22×Dic7, C2×C7⋊D4, D4⋊D14, D4.9D14, C2×C23.D7, C4○D4⋊Dic7
►On 224 points
Generators in S224
(1 220 147 88)(2 221 148 89)(3 222 149 90)(4 223 150 91)(5 224 151 92)(6 211 152 93)(7 212 153 94)(8 213 154 95)(9 214 141 96)(10 215 142 97)(11 216 143 98)(12 217 144 85)(13 218 145 86)(14 219 146 87)(15 194 77 46)(16 195 78 47)(17 196 79 48)(18 183 80 49)(19 184 81 50)(20 185 82 51)(21 186 83 52)(22 187 84 53)(23 188 71 54)(24 189 72 55)(25 190 73 56)(26 191 74 43)(27 192 75 44)(28 193 76 45)(29 138 156 178)(30 139 157 179)(31 140 158 180)(32 127 159 181)(33 128 160 182)(34 129 161 169)(35 130 162 170)(36 131 163 171)(37 132 164 172)(38 133 165 173)(39 134 166 174)(40 135 167 175)(41 136 168 176)(42 137 155 177)(57 203 103 113)(58 204 104 114)(59 205 105 115)(60 206 106 116)(61 207 107 117)(62 208 108 118)(63 209 109 119)(64 210 110 120)(65 197 111 121)(66 198 112 122)(67 199 99 123)(68 200 100 124)(69 201 101 125)(70 202 102 126)
(1 77 147 15)(2 78 148 16)(3 79 149 17)(4 80 150 18)(5 81 151 19)(6 82 152 20)(7 83 153 21)(8 84 154 22)(9 71 141 23)(10 72 142 24)(11 73 143 25)(12 74 144 26)(13 75 145 27)(14 76 146 28)(29 102 156 70)(30 103 157 57)(31 104 158 58)(32 105 159 59)(33 106 160 60)(34 107 161 61)(35 108 162 62)(36 109 163 63)(37 110 164 64)(38 111 165 65)(39 112 166 66)(40 99 167 67)(41 100 168 68)(42 101 155 69)(43 85 191 217)(44 86 192 218)(45 87 193 219)(46 88 194 220)(47 89 195 221)(48 90 196 222)(49 91 183 223)(50 92 184 224)(51 93 185 211)(52 94 186 212)(53 95 187 213)(54 96 188 214)(55 97 189 215)(56 98 190 216)(113 179 203 139)(114 180 204 140)(115 181 205 127)(116 182 206 128)(117 169 207 129)(118 170 208 130)(119 171 209 131)(120 172 210 132)(121 173 197 133)(122 174 198 134)(123 175 199 135)(124 176 200 136)(125 177 201 137)(126 178 202 138)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 84)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 163)(30 164)(31 165)(32 166)(33 167)(34 168)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 184)(44 185)(45 186)(46 187)(47 188)(48 189)(49 190)(50 191)(51 192)(52 193)(53 194)(54 195)(55 196)(56 183)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 174)(128 175)(129 176)(130 177)(131 178)(132 179)(133 180)(134 181)(135 182)(136 169)(137 170)(138 171)(139 172)(140 173)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(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)
(1 202 8 209)(2 201 9 208)(3 200 10 207)(4 199 11 206)(5 198 12 205)(6 197 13 204)(7 210 14 203)(15 163 22 156)(16 162 23 155)(17 161 24 168)(18 160 25 167)(19 159 26 166)(20 158 27 165)(21 157 28 164)(29 77 36 84)(30 76 37 83)(31 75 38 82)(32 74 39 81)(33 73 40 80)(34 72 41 79)(35 71 42 78)(43 174 50 181)(44 173 51 180)(45 172 52 179)(46 171 53 178)(47 170 54 177)(48 169 55 176)(49 182 56 175)(57 212 64 219)(58 211 65 218)(59 224 66 217)(60 223 67 216)(61 222 68 215)(62 221 69 214)(63 220 70 213)(85 105 92 112)(86 104 93 111)(87 103 94 110)(88 102 95 109)(89 101 96 108)(90 100 97 107)(91 99 98 106)(113 153 120 146)(114 152 121 145)(115 151 122 144)(116 150 123 143)(117 149 124 142)(118 148 125 141)(119 147 126 154)(127 191 134 184)(128 190 135 183)(129 189 136 196)(130 188 137 195)(131 187 138 194)(132 186 139 193)(133 185 140 192)
G:=sub<Sym(224)| (1,220,147,88)(2,221,148,89)(3,222,149,90)(4,223,150,91)(5,224,151,92)(6,211,152,93)(7,212,153,94)(8,213,154,95)(9,214,141,96)(10,215,142,97)(11,216,143,98)(12,217,144,85)(13,218,145,86)(14,219,146,87)(15,194,77,46)(16,195,78,47)(17,196,79,48)(18,183,80,49)(19,184,81,50)(20,185,82,51)(21,186,83,52)(22,187,84,53)(23,188,71,54)(24,189,72,55)(25,190,73,56)(26,191,74,43)(27,192,75,44)(28,193,76,45)(29,138,156,178)(30,139,157,179)(31,140,158,180)(32,127,159,181)(33,128,160,182)(34,129,161,169)(35,130,162,170)(36,131,163,171)(37,132,164,172)(38,133,165,173)(39,134,166,174)(40,135,167,175)(41,136,168,176)(42,137,155,177)(57,203,103,113)(58,204,104,114)(59,205,105,115)(60,206,106,116)(61,207,107,117)(62,208,108,118)(63,209,109,119)(64,210,110,120)(65,197,111,121)(66,198,112,122)(67,199,99,123)(68,200,100,124)(69,201,101,125)(70,202,102,126), (1,77,147,15)(2,78,148,16)(3,79,149,17)(4,80,150,18)(5,81,151,19)(6,82,152,20)(7,83,153,21)(8,84,154,22)(9,71,141,23)(10,72,142,24)(11,73,143,25)(12,74,144,26)(13,75,145,27)(14,76,146,28)(29,102,156,70)(30,103,157,57)(31,104,158,58)(32,105,159,59)(33,106,160,60)(34,107,161,61)(35,108,162,62)(36,109,163,63)(37,110,164,64)(38,111,165,65)(39,112,166,66)(40,99,167,67)(41,100,168,68)(42,101,155,69)(43,85,191,217)(44,86,192,218)(45,87,193,219)(46,88,194,220)(47,89,195,221)(48,90,196,222)(49,91,183,223)(50,92,184,224)(51,93,185,211)(52,94,186,212)(53,95,187,213)(54,96,188,214)(55,97,189,215)(56,98,190,216)(113,179,203,139)(114,180,204,140)(115,181,205,127)(116,182,206,128)(117,169,207,129)(118,170,208,130)(119,171,209,131)(120,172,210,132)(121,173,197,133)(122,174,198,134)(123,175,199,135)(124,176,200,136)(125,177,201,137)(126,178,202,138), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,84)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,163)(30,164)(31,165)(32,166)(33,167)(34,168)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,184)(44,185)(45,186)(46,187)(47,188)(48,189)(49,190)(50,191)(51,192)(52,193)(53,194)(54,195)(55,196)(56,183)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,181)(135,182)(136,169)(137,170)(138,171)(139,172)(140,173)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (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), (1,202,8,209)(2,201,9,208)(3,200,10,207)(4,199,11,206)(5,198,12,205)(6,197,13,204)(7,210,14,203)(15,163,22,156)(16,162,23,155)(17,161,24,168)(18,160,25,167)(19,159,26,166)(20,158,27,165)(21,157,28,164)(29,77,36,84)(30,76,37,83)(31,75,38,82)(32,74,39,81)(33,73,40,80)(34,72,41,79)(35,71,42,78)(43,174,50,181)(44,173,51,180)(45,172,52,179)(46,171,53,178)(47,170,54,177)(48,169,55,176)(49,182,56,175)(57,212,64,219)(58,211,65,218)(59,224,66,217)(60,223,67,216)(61,222,68,215)(62,221,69,214)(63,220,70,213)(85,105,92,112)(86,104,93,111)(87,103,94,110)(88,102,95,109)(89,101,96,108)(90,100,97,107)(91,99,98,106)(113,153,120,146)(114,152,121,145)(115,151,122,144)(116,150,123,143)(117,149,124,142)(118,148,125,141)(119,147,126,154)(127,191,134,184)(128,190,135,183)(129,189,136,196)(130,188,137,195)(131,187,138,194)(132,186,139,193)(133,185,140,192)>;
G:=Group( (1,220,147,88)(2,221,148,89)(3,222,149,90)(4,223,150,91)(5,224,151,92)(6,211,152,93)(7,212,153,94)(8,213,154,95)(9,214,141,96)(10,215,142,97)(11,216,143,98)(12,217,144,85)(13,218,145,86)(14,219,146,87)(15,194,77,46)(16,195,78,47)(17,196,79,48)(18,183,80,49)(19,184,81,50)(20,185,82,51)(21,186,83,52)(22,187,84,53)(23,188,71,54)(24,189,72,55)(25,190,73,56)(26,191,74,43)(27,192,75,44)(28,193,76,45)(29,138,156,178)(30,139,157,179)(31,140,158,180)(32,127,159,181)(33,128,160,182)(34,129,161,169)(35,130,162,170)(36,131,163,171)(37,132,164,172)(38,133,165,173)(39,134,166,174)(40,135,167,175)(41,136,168,176)(42,137,155,177)(57,203,103,113)(58,204,104,114)(59,205,105,115)(60,206,106,116)(61,207,107,117)(62,208,108,118)(63,209,109,119)(64,210,110,120)(65,197,111,121)(66,198,112,122)(67,199,99,123)(68,200,100,124)(69,201,101,125)(70,202,102,126), (1,77,147,15)(2,78,148,16)(3,79,149,17)(4,80,150,18)(5,81,151,19)(6,82,152,20)(7,83,153,21)(8,84,154,22)(9,71,141,23)(10,72,142,24)(11,73,143,25)(12,74,144,26)(13,75,145,27)(14,76,146,28)(29,102,156,70)(30,103,157,57)(31,104,158,58)(32,105,159,59)(33,106,160,60)(34,107,161,61)(35,108,162,62)(36,109,163,63)(37,110,164,64)(38,111,165,65)(39,112,166,66)(40,99,167,67)(41,100,168,68)(42,101,155,69)(43,85,191,217)(44,86,192,218)(45,87,193,219)(46,88,194,220)(47,89,195,221)(48,90,196,222)(49,91,183,223)(50,92,184,224)(51,93,185,211)(52,94,186,212)(53,95,187,213)(54,96,188,214)(55,97,189,215)(56,98,190,216)(113,179,203,139)(114,180,204,140)(115,181,205,127)(116,182,206,128)(117,169,207,129)(118,170,208,130)(119,171,209,131)(120,172,210,132)(121,173,197,133)(122,174,198,134)(123,175,199,135)(124,176,200,136)(125,177,201,137)(126,178,202,138), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,84)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,163)(30,164)(31,165)(32,166)(33,167)(34,168)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,184)(44,185)(45,186)(46,187)(47,188)(48,189)(49,190)(50,191)(51,192)(52,193)(53,194)(54,195)(55,196)(56,183)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,181)(135,182)(136,169)(137,170)(138,171)(139,172)(140,173)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (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), (1,202,8,209)(2,201,9,208)(3,200,10,207)(4,199,11,206)(5,198,12,205)(6,197,13,204)(7,210,14,203)(15,163,22,156)(16,162,23,155)(17,161,24,168)(18,160,25,167)(19,159,26,166)(20,158,27,165)(21,157,28,164)(29,77,36,84)(30,76,37,83)(31,75,38,82)(32,74,39,81)(33,73,40,80)(34,72,41,79)(35,71,42,78)(43,174,50,181)(44,173,51,180)(45,172,52,179)(46,171,53,178)(47,170,54,177)(48,169,55,176)(49,182,56,175)(57,212,64,219)(58,211,65,218)(59,224,66,217)(60,223,67,216)(61,222,68,215)(62,221,69,214)(63,220,70,213)(85,105,92,112)(86,104,93,111)(87,103,94,110)(88,102,95,109)(89,101,96,108)(90,100,97,107)(91,99,98,106)(113,153,120,146)(114,152,121,145)(115,151,122,144)(116,150,123,143)(117,149,124,142)(118,148,125,141)(119,147,126,154)(127,191,134,184)(128,190,135,183)(129,189,136,196)(130,188,137,195)(131,187,138,194)(132,186,139,193)(133,185,140,192) );
G=PermutationGroup([[(1,220,147,88),(2,221,148,89),(3,222,149,90),(4,223,150,91),(5,224,151,92),(6,211,152,93),(7,212,153,94),(8,213,154,95),(9,214,141,96),(10,215,142,97),(11,216,143,98),(12,217,144,85),(13,218,145,86),(14,219,146,87),(15,194,77,46),(16,195,78,47),(17,196,79,48),(18,183,80,49),(19,184,81,50),(20,185,82,51),(21,186,83,52),(22,187,84,53),(23,188,71,54),(24,189,72,55),(25,190,73,56),(26,191,74,43),(27,192,75,44),(28,193,76,45),(29,138,156,178),(30,139,157,179),(31,140,158,180),(32,127,159,181),(33,128,160,182),(34,129,161,169),(35,130,162,170),(36,131,163,171),(37,132,164,172),(38,133,165,173),(39,134,166,174),(40,135,167,175),(41,136,168,176),(42,137,155,177),(57,203,103,113),(58,204,104,114),(59,205,105,115),(60,206,106,116),(61,207,107,117),(62,208,108,118),(63,209,109,119),(64,210,110,120),(65,197,111,121),(66,198,112,122),(67,199,99,123),(68,200,100,124),(69,201,101,125),(70,202,102,126)], [(1,77,147,15),(2,78,148,16),(3,79,149,17),(4,80,150,18),(5,81,151,19),(6,82,152,20),(7,83,153,21),(8,84,154,22),(9,71,141,23),(10,72,142,24),(11,73,143,25),(12,74,144,26),(13,75,145,27),(14,76,146,28),(29,102,156,70),(30,103,157,57),(31,104,158,58),(32,105,159,59),(33,106,160,60),(34,107,161,61),(35,108,162,62),(36,109,163,63),(37,110,164,64),(38,111,165,65),(39,112,166,66),(40,99,167,67),(41,100,168,68),(42,101,155,69),(43,85,191,217),(44,86,192,218),(45,87,193,219),(46,88,194,220),(47,89,195,221),(48,90,196,222),(49,91,183,223),(50,92,184,224),(51,93,185,211),(52,94,186,212),(53,95,187,213),(54,96,188,214),(55,97,189,215),(56,98,190,216),(113,179,203,139),(114,180,204,140),(115,181,205,127),(116,182,206,128),(117,169,207,129),(118,170,208,130),(119,171,209,131),(120,172,210,132),(121,173,197,133),(122,174,198,134),(123,175,199,135),(124,176,200,136),(125,177,201,137),(126,178,202,138)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,84),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,163),(30,164),(31,165),(32,166),(33,167),(34,168),(35,155),(36,156),(37,157),(38,158),(39,159),(40,160),(41,161),(42,162),(43,184),(44,185),(45,186),(46,187),(47,188),(48,189),(49,190),(50,191),(51,192),(52,193),(53,194),(54,195),(55,196),(56,183),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(127,174),(128,175),(129,176),(130,177),(131,178),(132,179),(133,180),(134,181),(135,182),(136,169),(137,170),(138,171),(139,172),(140,173),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)], [(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)], [(1,202,8,209),(2,201,9,208),(3,200,10,207),(4,199,11,206),(5,198,12,205),(6,197,13,204),(7,210,14,203),(15,163,22,156),(16,162,23,155),(17,161,24,168),(18,160,25,167),(19,159,26,166),(20,158,27,165),(21,157,28,164),(29,77,36,84),(30,76,37,83),(31,75,38,82),(32,74,39,81),(33,73,40,80),(34,72,41,79),(35,71,42,78),(43,174,50,181),(44,173,51,180),(45,172,52,179),(46,171,53,178),(47,170,54,177),(48,169,55,176),(49,182,56,175),(57,212,64,219),(58,211,65,218),(59,224,66,217),(60,223,67,216),(61,222,68,215),(62,221,69,214),(63,220,70,213),(85,105,92,112),(86,104,93,111),(87,103,94,110),(88,102,95,109),(89,101,96,108),(90,100,97,107),(91,99,98,106),(113,153,120,146),(114,152,121,145),(115,151,122,144),(116,150,123,143),(117,149,124,142),(118,148,125,141),(119,147,126,154),(127,191,134,184),(128,190,135,183),(129,189,136,196),(130,188,137,195),(131,187,138,194),(132,186,139,193),(133,185,140,192)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | D14 | D14 | Dic7 | C7⋊D4 | C7⋊D4 | C8⋊C22 | C8.C22 | D4⋊D14 | D4.9D14 |
| kernel | C4○D4⋊Dic7 | D4⋊Dic7 | Q8⋊Dic7 | C2×C4.Dic7 | C2×C4⋊Dic7 | C14×C4○D4 | C7×C4○D4 | C2×C28 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 3 | 3 | 3 | 12 | 18 | 6 | 1 | 1 | 6 | 6 |
Matrix representation of C4○D4⋊Dic7 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 105 | 0 | 0 |
| 0 | 0 | 8 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 104 | 105 |
| 0 | 0 | 0 | 0 | 95 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 56 | 33 | 41 |
| 0 | 0 | 57 | 1 | 31 | 41 |
| 0 | 0 | 74 | 75 | 43 | 57 |
| 0 | 0 | 112 | 28 | 100 | 56 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 69 | 55 | 1 | 0 |
| 0 | 0 | 14 | 88 | 0 | 1 |
| 1 | 112 | 0 | 0 | 0 | 0 |
| 26 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 89 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 112 |
| 0 | 0 | 0 | 0 | 26 | 88 |
| 33 | 57 | 0 | 0 | 0 | 0 |
| 80 | 80 | 0 | 0 | 0 | 0 |
| 0 | 0 | 106 | 84 | 0 | 0 |
| 0 | 0 | 29 | 7 | 0 | 0 |
| 0 | 0 | 18 | 34 | 15 | 0 |
| 0 | 0 | 32 | 16 | 51 | 98 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,8,0,0,0,0,105,17,0,0,0,0,0,0,104,95,0,0,0,0,105,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,57,74,112,0,0,56,1,75,28,0,0,33,31,43,100,0,0,41,41,57,56],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,69,14,0,0,0,112,55,88,0,0,0,0,1,0,0,0,0,0,0,1],[1,26,0,0,0,0,112,88,0,0,0,0,0,0,0,1,0,0,0,0,112,89,0,0,0,0,0,0,1,26,0,0,0,0,112,88],[33,80,0,0,0,0,57,80,0,0,0,0,0,0,106,29,18,32,0,0,84,7,34,16,0,0,0,0,15,51,0,0,0,0,0,98] >;
C4○D4⋊Dic7 in GAP, Magma, Sage, TeX
C_4\circ D_4\rtimes {\rm Dic}_7 % in TeX
G:=Group("C4oD4:Dic7"); // GroupNames label
G:=SmallGroup(448,766);
// by ID
G=gap.SmallGroup(448,766);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,387,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=d^14=1,b^2=a^2,e^2=d^7,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c=a^2*b,b*d=d*b,e*b*e^-1=a*b*c,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations