Copied to
clipboard

G = C4○D4⋊Dic7order 448 = 26·7

1st semidirect product of C4○D4 and Dic7 acting via Dic7/C14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D41Dic7, Q84(C2×Dic7), D44(C2×Dic7), C28.212(C2×D4), (C2×C28).476D4, (C2×D4).201D14, Q8⋊Dic740C2, D4⋊Dic740C2, C28.83(C22×C4), (C2×Q8).170D14, C2.5(D4⋊D14), C28.90(C22⋊C4), (C2×C28).479C23, (C22×C4).161D14, (C22×C14).111D4, C75(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

C1C28 — C4○D4⋊Dic7
C1C7C14C2×C14C2×C28C4⋊Dic7C2×C4⋊Dic7 — C4○D4⋊Dic7
C7C14C28 — C4○D4⋊Dic7
C1C22C22×C4C2×C4○D4

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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C7, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C14 [×3], C14 [×4], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], Dic7 [×2], C28 [×2], C28 [×2], C28 [×2], C2×C14, C2×C14 [×2], C2×C14 [×6], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C7⋊C8 [×2], C2×Dic7 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×5], C7×D4 [×2], C7×D4 [×5], C7×Q8 [×2], C7×Q8, C22×C14, C22×C14, C23.36D4, C2×C7⋊C8 [×2], C4.Dic7 [×2], C4⋊Dic7 [×2], C4⋊Dic7, C22×Dic7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4 [×4], C7×C4○D4 [×2], D4⋊Dic7 [×2], Q8⋊Dic7 [×2], C2×C4.Dic7, C2×C4⋊Dic7, C14×C4○D4, C4○D4⋊Dic7
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D7, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic7 [×4], D14 [×3], C2×C22⋊C4, C8⋊C22, C8.C22, C2×Dic7 [×6], C7⋊D4 [×4], C22×D7, C23.36D4, C23.D7 [×4], C22×Dic7, C2×C7⋊D4 [×2], D4⋊D14, D4.9D14, C2×C23.D7, C4○D4⋊Dic7

Smallest permutation representation of C4○D4⋊Dic7
On 224 points
Generators in S224
(1 78 126 167)(2 79 113 168)(3 80 114 155)(4 81 115 156)(5 82 116 157)(6 83 117 158)(7 84 118 159)(8 71 119 160)(9 72 120 161)(10 73 121 162)(11 74 122 163)(12 75 123 164)(13 76 124 165)(14 77 125 166)(15 88 47 141)(16 89 48 142)(17 90 49 143)(18 91 50 144)(19 92 51 145)(20 93 52 146)(21 94 53 147)(22 95 54 148)(23 96 55 149)(24 97 56 150)(25 98 43 151)(26 85 44 152)(27 86 45 153)(28 87 46 154)(29 169 206 65)(30 170 207 66)(31 171 208 67)(32 172 209 68)(33 173 210 69)(34 174 197 70)(35 175 198 57)(36 176 199 58)(37 177 200 59)(38 178 201 60)(39 179 202 61)(40 180 203 62)(41 181 204 63)(42 182 205 64)(99 183 217 136)(100 184 218 137)(101 185 219 138)(102 186 220 139)(103 187 221 140)(104 188 222 127)(105 189 223 128)(106 190 224 129)(107 191 211 130)(108 192 212 131)(109 193 213 132)(110 194 214 133)(111 195 215 134)(112 196 216 135)
(1 28 126 46)(2 15 113 47)(3 16 114 48)(4 17 115 49)(5 18 116 50)(6 19 117 51)(7 20 118 52)(8 21 119 53)(9 22 120 54)(10 23 121 55)(11 24 122 56)(12 25 123 43)(13 26 124 44)(14 27 125 45)(29 102 206 220)(30 103 207 221)(31 104 208 222)(32 105 209 223)(33 106 210 224)(34 107 197 211)(35 108 198 212)(36 109 199 213)(37 110 200 214)(38 111 201 215)(39 112 202 216)(40 99 203 217)(41 100 204 218)(42 101 205 219)(57 131 175 192)(58 132 176 193)(59 133 177 194)(60 134 178 195)(61 135 179 196)(62 136 180 183)(63 137 181 184)(64 138 182 185)(65 139 169 186)(66 140 170 187)(67 127 171 188)(68 128 172 189)(69 129 173 190)(70 130 174 191)(71 94 160 147)(72 95 161 148)(73 96 162 149)(74 97 163 150)(75 98 164 151)(76 85 165 152)(77 86 166 153)(78 87 167 154)(79 88 168 141)(80 89 155 142)(81 90 156 143)(82 91 157 144)(83 92 158 145)(84 93 159 146)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 54)(16 55)(17 56)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 199)(30 200)(31 201)(32 202)(33 203)(34 204)(35 205)(36 206)(37 207)(38 208)(39 209)(40 210)(41 197)(42 198)(57 182)(58 169)(59 170)(60 171)(61 172)(62 173)(63 174)(64 175)(65 176)(66 177)(67 178)(68 179)(69 180)(70 181)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 141)(96 142)(97 143)(98 144)(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)(127 134)(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)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(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)
(1 212 8 219)(2 211 9 218)(3 224 10 217)(4 223 11 216)(5 222 12 215)(6 221 13 214)(7 220 14 213)(15 63 22 70)(16 62 23 69)(17 61 24 68)(18 60 25 67)(19 59 26 66)(20 58 27 65)(21 57 28 64)(29 146 36 153)(30 145 37 152)(31 144 38 151)(32 143 39 150)(33 142 40 149)(34 141 41 148)(35 154 42 147)(43 171 50 178)(44 170 51 177)(45 169 52 176)(46 182 53 175)(47 181 54 174)(48 180 55 173)(49 179 56 172)(71 185 78 192)(72 184 79 191)(73 183 80 190)(74 196 81 189)(75 195 82 188)(76 194 83 187)(77 193 84 186)(85 207 92 200)(86 206 93 199)(87 205 94 198)(88 204 95 197)(89 203 96 210)(90 202 97 209)(91 201 98 208)(99 114 106 121)(100 113 107 120)(101 126 108 119)(102 125 109 118)(103 124 110 117)(104 123 111 116)(105 122 112 115)(127 164 134 157)(128 163 135 156)(129 162 136 155)(130 161 137 168)(131 160 138 167)(132 159 139 166)(133 158 140 165)

G:=sub<Sym(224)| (1,78,126,167)(2,79,113,168)(3,80,114,155)(4,81,115,156)(5,82,116,157)(6,83,117,158)(7,84,118,159)(8,71,119,160)(9,72,120,161)(10,73,121,162)(11,74,122,163)(12,75,123,164)(13,76,124,165)(14,77,125,166)(15,88,47,141)(16,89,48,142)(17,90,49,143)(18,91,50,144)(19,92,51,145)(20,93,52,146)(21,94,53,147)(22,95,54,148)(23,96,55,149)(24,97,56,150)(25,98,43,151)(26,85,44,152)(27,86,45,153)(28,87,46,154)(29,169,206,65)(30,170,207,66)(31,171,208,67)(32,172,209,68)(33,173,210,69)(34,174,197,70)(35,175,198,57)(36,176,199,58)(37,177,200,59)(38,178,201,60)(39,179,202,61)(40,180,203,62)(41,181,204,63)(42,182,205,64)(99,183,217,136)(100,184,218,137)(101,185,219,138)(102,186,220,139)(103,187,221,140)(104,188,222,127)(105,189,223,128)(106,190,224,129)(107,191,211,130)(108,192,212,131)(109,193,213,132)(110,194,214,133)(111,195,215,134)(112,196,216,135), (1,28,126,46)(2,15,113,47)(3,16,114,48)(4,17,115,49)(5,18,116,50)(6,19,117,51)(7,20,118,52)(8,21,119,53)(9,22,120,54)(10,23,121,55)(11,24,122,56)(12,25,123,43)(13,26,124,44)(14,27,125,45)(29,102,206,220)(30,103,207,221)(31,104,208,222)(32,105,209,223)(33,106,210,224)(34,107,197,211)(35,108,198,212)(36,109,199,213)(37,110,200,214)(38,111,201,215)(39,112,202,216)(40,99,203,217)(41,100,204,218)(42,101,205,219)(57,131,175,192)(58,132,176,193)(59,133,177,194)(60,134,178,195)(61,135,179,196)(62,136,180,183)(63,137,181,184)(64,138,182,185)(65,139,169,186)(66,140,170,187)(67,127,171,188)(68,128,172,189)(69,129,173,190)(70,130,174,191)(71,94,160,147)(72,95,161,148)(73,96,162,149)(74,97,163,150)(75,98,164,151)(76,85,165,152)(77,86,166,153)(78,87,167,154)(79,88,168,141)(80,89,155,142)(81,90,156,143)(82,91,157,144)(83,92,158,145)(84,93,159,146), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,199)(30,200)(31,201)(32,202)(33,203)(34,204)(35,205)(36,206)(37,207)(38,208)(39,209)(40,210)(41,197)(42,198)(57,182)(58,169)(59,170)(60,171)(61,172)(62,173)(63,174)(64,175)(65,176)(66,177)(67,178)(68,179)(69,180)(70,181)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,141)(96,142)(97,143)(98,144)(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)(127,134)(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)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(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), (1,212,8,219)(2,211,9,218)(3,224,10,217)(4,223,11,216)(5,222,12,215)(6,221,13,214)(7,220,14,213)(15,63,22,70)(16,62,23,69)(17,61,24,68)(18,60,25,67)(19,59,26,66)(20,58,27,65)(21,57,28,64)(29,146,36,153)(30,145,37,152)(31,144,38,151)(32,143,39,150)(33,142,40,149)(34,141,41,148)(35,154,42,147)(43,171,50,178)(44,170,51,177)(45,169,52,176)(46,182,53,175)(47,181,54,174)(48,180,55,173)(49,179,56,172)(71,185,78,192)(72,184,79,191)(73,183,80,190)(74,196,81,189)(75,195,82,188)(76,194,83,187)(77,193,84,186)(85,207,92,200)(86,206,93,199)(87,205,94,198)(88,204,95,197)(89,203,96,210)(90,202,97,209)(91,201,98,208)(99,114,106,121)(100,113,107,120)(101,126,108,119)(102,125,109,118)(103,124,110,117)(104,123,111,116)(105,122,112,115)(127,164,134,157)(128,163,135,156)(129,162,136,155)(130,161,137,168)(131,160,138,167)(132,159,139,166)(133,158,140,165)>;

G:=Group( (1,78,126,167)(2,79,113,168)(3,80,114,155)(4,81,115,156)(5,82,116,157)(6,83,117,158)(7,84,118,159)(8,71,119,160)(9,72,120,161)(10,73,121,162)(11,74,122,163)(12,75,123,164)(13,76,124,165)(14,77,125,166)(15,88,47,141)(16,89,48,142)(17,90,49,143)(18,91,50,144)(19,92,51,145)(20,93,52,146)(21,94,53,147)(22,95,54,148)(23,96,55,149)(24,97,56,150)(25,98,43,151)(26,85,44,152)(27,86,45,153)(28,87,46,154)(29,169,206,65)(30,170,207,66)(31,171,208,67)(32,172,209,68)(33,173,210,69)(34,174,197,70)(35,175,198,57)(36,176,199,58)(37,177,200,59)(38,178,201,60)(39,179,202,61)(40,180,203,62)(41,181,204,63)(42,182,205,64)(99,183,217,136)(100,184,218,137)(101,185,219,138)(102,186,220,139)(103,187,221,140)(104,188,222,127)(105,189,223,128)(106,190,224,129)(107,191,211,130)(108,192,212,131)(109,193,213,132)(110,194,214,133)(111,195,215,134)(112,196,216,135), (1,28,126,46)(2,15,113,47)(3,16,114,48)(4,17,115,49)(5,18,116,50)(6,19,117,51)(7,20,118,52)(8,21,119,53)(9,22,120,54)(10,23,121,55)(11,24,122,56)(12,25,123,43)(13,26,124,44)(14,27,125,45)(29,102,206,220)(30,103,207,221)(31,104,208,222)(32,105,209,223)(33,106,210,224)(34,107,197,211)(35,108,198,212)(36,109,199,213)(37,110,200,214)(38,111,201,215)(39,112,202,216)(40,99,203,217)(41,100,204,218)(42,101,205,219)(57,131,175,192)(58,132,176,193)(59,133,177,194)(60,134,178,195)(61,135,179,196)(62,136,180,183)(63,137,181,184)(64,138,182,185)(65,139,169,186)(66,140,170,187)(67,127,171,188)(68,128,172,189)(69,129,173,190)(70,130,174,191)(71,94,160,147)(72,95,161,148)(73,96,162,149)(74,97,163,150)(75,98,164,151)(76,85,165,152)(77,86,166,153)(78,87,167,154)(79,88,168,141)(80,89,155,142)(81,90,156,143)(82,91,157,144)(83,92,158,145)(84,93,159,146), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,199)(30,200)(31,201)(32,202)(33,203)(34,204)(35,205)(36,206)(37,207)(38,208)(39,209)(40,210)(41,197)(42,198)(57,182)(58,169)(59,170)(60,171)(61,172)(62,173)(63,174)(64,175)(65,176)(66,177)(67,178)(68,179)(69,180)(70,181)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,141)(96,142)(97,143)(98,144)(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)(127,134)(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)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(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), (1,212,8,219)(2,211,9,218)(3,224,10,217)(4,223,11,216)(5,222,12,215)(6,221,13,214)(7,220,14,213)(15,63,22,70)(16,62,23,69)(17,61,24,68)(18,60,25,67)(19,59,26,66)(20,58,27,65)(21,57,28,64)(29,146,36,153)(30,145,37,152)(31,144,38,151)(32,143,39,150)(33,142,40,149)(34,141,41,148)(35,154,42,147)(43,171,50,178)(44,170,51,177)(45,169,52,176)(46,182,53,175)(47,181,54,174)(48,180,55,173)(49,179,56,172)(71,185,78,192)(72,184,79,191)(73,183,80,190)(74,196,81,189)(75,195,82,188)(76,194,83,187)(77,193,84,186)(85,207,92,200)(86,206,93,199)(87,205,94,198)(88,204,95,197)(89,203,96,210)(90,202,97,209)(91,201,98,208)(99,114,106,121)(100,113,107,120)(101,126,108,119)(102,125,109,118)(103,124,110,117)(104,123,111,116)(105,122,112,115)(127,164,134,157)(128,163,135,156)(129,162,136,155)(130,161,137,168)(131,160,138,167)(132,159,139,166)(133,158,140,165) );

G=PermutationGroup([(1,78,126,167),(2,79,113,168),(3,80,114,155),(4,81,115,156),(5,82,116,157),(6,83,117,158),(7,84,118,159),(8,71,119,160),(9,72,120,161),(10,73,121,162),(11,74,122,163),(12,75,123,164),(13,76,124,165),(14,77,125,166),(15,88,47,141),(16,89,48,142),(17,90,49,143),(18,91,50,144),(19,92,51,145),(20,93,52,146),(21,94,53,147),(22,95,54,148),(23,96,55,149),(24,97,56,150),(25,98,43,151),(26,85,44,152),(27,86,45,153),(28,87,46,154),(29,169,206,65),(30,170,207,66),(31,171,208,67),(32,172,209,68),(33,173,210,69),(34,174,197,70),(35,175,198,57),(36,176,199,58),(37,177,200,59),(38,178,201,60),(39,179,202,61),(40,180,203,62),(41,181,204,63),(42,182,205,64),(99,183,217,136),(100,184,218,137),(101,185,219,138),(102,186,220,139),(103,187,221,140),(104,188,222,127),(105,189,223,128),(106,190,224,129),(107,191,211,130),(108,192,212,131),(109,193,213,132),(110,194,214,133),(111,195,215,134),(112,196,216,135)], [(1,28,126,46),(2,15,113,47),(3,16,114,48),(4,17,115,49),(5,18,116,50),(6,19,117,51),(7,20,118,52),(8,21,119,53),(9,22,120,54),(10,23,121,55),(11,24,122,56),(12,25,123,43),(13,26,124,44),(14,27,125,45),(29,102,206,220),(30,103,207,221),(31,104,208,222),(32,105,209,223),(33,106,210,224),(34,107,197,211),(35,108,198,212),(36,109,199,213),(37,110,200,214),(38,111,201,215),(39,112,202,216),(40,99,203,217),(41,100,204,218),(42,101,205,219),(57,131,175,192),(58,132,176,193),(59,133,177,194),(60,134,178,195),(61,135,179,196),(62,136,180,183),(63,137,181,184),(64,138,182,185),(65,139,169,186),(66,140,170,187),(67,127,171,188),(68,128,172,189),(69,129,173,190),(70,130,174,191),(71,94,160,147),(72,95,161,148),(73,96,162,149),(74,97,163,150),(75,98,164,151),(76,85,165,152),(77,86,166,153),(78,87,167,154),(79,88,168,141),(80,89,155,142),(81,90,156,143),(82,91,157,144),(83,92,158,145),(84,93,159,146)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,54),(16,55),(17,56),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,199),(30,200),(31,201),(32,202),(33,203),(34,204),(35,205),(36,206),(37,207),(38,208),(39,209),(40,210),(41,197),(42,198),(57,182),(58,169),(59,170),(60,171),(61,172),(62,173),(63,174),(64,175),(65,176),(66,177),(67,178),(68,179),(69,180),(70,181),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150),(91,151),(92,152),(93,153),(94,154),(95,141),(96,142),(97,143),(98,144),(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),(127,134),(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),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(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)], [(1,212,8,219),(2,211,9,218),(3,224,10,217),(4,223,11,216),(5,222,12,215),(6,221,13,214),(7,220,14,213),(15,63,22,70),(16,62,23,69),(17,61,24,68),(18,60,25,67),(19,59,26,66),(20,58,27,65),(21,57,28,64),(29,146,36,153),(30,145,37,152),(31,144,38,151),(32,143,39,150),(33,142,40,149),(34,141,41,148),(35,154,42,147),(43,171,50,178),(44,170,51,177),(45,169,52,176),(46,182,53,175),(47,181,54,174),(48,180,55,173),(49,179,56,172),(71,185,78,192),(72,184,79,191),(73,183,80,190),(74,196,81,189),(75,195,82,188),(76,194,83,187),(77,193,84,186),(85,207,92,200),(86,206,93,199),(87,205,94,198),(88,204,95,197),(89,203,96,210),(90,202,97,209),(91,201,98,208),(99,114,106,121),(100,113,107,120),(101,126,108,119),(102,125,109,118),(103,124,110,117),(104,123,111,116),(105,122,112,115),(127,164,134,157),(128,163,135,156),(129,162,136,155),(130,161,137,168),(131,160,138,167),(132,159,139,166),(133,158,140,165)])

82 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J7A7B7C8A8B8C8D14A···14I14J···14AA28A···28L28M···28AD
order122222224444444444777888814···1414···1428···2828···28
size1111224422224428282828222282828282···24···42···24···4

82 irreducible representations

dim11111112222222224444
type++++++++++++-+-+-
imageC1C2C2C2C2C2C4D4D4D7D14D14D14Dic7C7⋊D4C7⋊D4C8⋊C22C8.C22D4⋊D14D4.9D14
kernelC4○D4⋊Dic7D4⋊Dic7Q8⋊Dic7C2×C4.Dic7C2×C4⋊Dic7C14×C4○D4C7×C4○D4C2×C28C22×C14C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C2×C4C23C14C14C2C2
# reps1221118313333121861166

Matrix representation of C4○D4⋊Dic7 in GL6(𝔽113)

100000
010000
009610500
0081700
0000104105
0000959
,
100000
010000
0013563341
005713141
0074754357
001122810056
,
100000
010000
00112000
00011200
00695510
00148801
,
11120000
26880000
00011200
0018900
00001112
00002688
,
33570000
80800000
001068400
0029700
001834150
0032165198

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

׿
×
𝔽