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G = Q8.(C2×Dic7)  order 448 = 26·7

2nd non-split extension by Q8 of C2×Dic7 acting via C2×Dic7/C2×C14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D42Dic7, (C2×C28).501D4, C28.213(C2×D4), Q8.6(C2×Dic7), D4.6(C2×Dic7), (C2×D4).202D14, D4⋊Dic744C2, Q8⋊Dic744C2, C28.84(C22×C4), (C2×Q8).171D14, C14.111(C4○D8), C28.98(C22⋊C4), (C2×C28).480C23, (C22×C14).112D4, (C22×C4).355D14, C23.44(C7⋊D4), C75(C23.24D4), C4.32(C23.D7), C4.14(C22×Dic7), C2.7(D4.8D14), (D4×C14).243C22, C4⋊Dic7.355C22, (Q8×C14).206C22, C22.3(C23.D7), C23.21D1419C2, (C22×C28).206C22, (C7×C4○D4)⋊2C4, (C22×C7⋊C8)⋊8C2, (C2×C4○D4).2D7, C4.95(C2×C7⋊D4), (C14×C4○D4).2C2, (C7×D4).23(C2×C4), (C7×Q8).24(C2×C4), (C2×C28).123(C2×C4), (C2×C14).566(C2×D4), (C2×C7⋊C8).282C22, C14.82(C2×C22⋊C4), (C2×C4).52(C2×Dic7), C22.97(C2×C7⋊D4), C2.18(C2×C23.D7), (C2×C4).281(C7⋊D4), (C2×C4).565(C22×D7), (C2×C14).25(C22⋊C4), SmallGroup(448,767)

Series: Derived Chief Lower central Upper central

C1C28 — Q8.(C2×Dic7)
C1C7C14C28C2×C28C4⋊Dic7C23.21D14 — Q8.(C2×Dic7)
C7C14C28 — Q8.(C2×Dic7)
C1C2×C4C22×C4C2×C4○D4

Generators and relations for Q8.(C2×Dic7)
 G = < a,b,c,d,e | a4=d14=1, b2=c2=a2, e2=d7, bab-1=eae-1=a-1, ac=ca, ad=da, bc=cb, bd=db, ebe-1=a-1b, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 468 in 158 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C7, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C14, C14 [×2], C14 [×4], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C22×C4, C22×C4, 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], C42⋊C2, C22×C8, C2×C4○D4, C7⋊C8 [×2], C2×Dic7 [×2], C2×C28 [×2], C2×C28 [×4], C2×C28 [×5], C7×D4 [×2], C7×D4 [×5], C7×Q8 [×2], C7×Q8, C22×C14, C22×C14, C23.24D4, C2×C7⋊C8 [×2], C2×C7⋊C8 [×2], C4×Dic7, C4⋊Dic7 [×2], C23.D7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4 [×4], C7×C4○D4 [×2], D4⋊Dic7 [×2], Q8⋊Dic7 [×2], C22×C7⋊C8, C23.21D14, C14×C4○D4, Q8.(C2×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, C4○D8 [×2], C2×Dic7 [×6], C7⋊D4 [×4], C22×D7, C23.24D4, C23.D7 [×4], C22×Dic7, C2×C7⋊D4 [×2], D4.8D14 [×2], C2×C23.D7, Q8.(C2×Dic7)

Smallest permutation representation of Q8.(C2×Dic7)
On 224 points
Generators in S224
(1 128 149 209)(2 129 150 210)(3 130 151 197)(4 131 152 198)(5 132 153 199)(6 133 154 200)(7 134 141 201)(8 135 142 202)(9 136 143 203)(10 137 144 204)(11 138 145 205)(12 139 146 206)(13 140 147 207)(14 127 148 208)(15 213 98 61)(16 214 85 62)(17 215 86 63)(18 216 87 64)(19 217 88 65)(20 218 89 66)(21 219 90 67)(22 220 91 68)(23 221 92 69)(24 222 93 70)(25 223 94 57)(26 224 95 58)(27 211 96 59)(28 212 97 60)(29 124 108 195)(30 125 109 196)(31 126 110 183)(32 113 111 184)(33 114 112 185)(34 115 99 186)(35 116 100 187)(36 117 101 188)(37 118 102 189)(38 119 103 190)(39 120 104 191)(40 121 105 192)(41 122 106 193)(42 123 107 194)(43 71 157 175)(44 72 158 176)(45 73 159 177)(46 74 160 178)(47 75 161 179)(48 76 162 180)(49 77 163 181)(50 78 164 182)(51 79 165 169)(52 80 166 170)(53 81 167 171)(54 82 168 172)(55 83 155 173)(56 84 156 174)
(1 181 149 77)(2 182 150 78)(3 169 151 79)(4 170 152 80)(5 171 153 81)(6 172 154 82)(7 173 141 83)(8 174 142 84)(9 175 143 71)(10 176 144 72)(11 177 145 73)(12 178 146 74)(13 179 147 75)(14 180 148 76)(15 194 98 123)(16 195 85 124)(17 196 86 125)(18 183 87 126)(19 184 88 113)(20 185 89 114)(21 186 90 115)(22 187 91 116)(23 188 92 117)(24 189 93 118)(25 190 94 119)(26 191 95 120)(27 192 96 121)(28 193 97 122)(29 214 108 62)(30 215 109 63)(31 216 110 64)(32 217 111 65)(33 218 112 66)(34 219 99 67)(35 220 100 68)(36 221 101 69)(37 222 102 70)(38 223 103 57)(39 224 104 58)(40 211 105 59)(41 212 106 60)(42 213 107 61)(43 136 157 203)(44 137 158 204)(45 138 159 205)(46 139 160 206)(47 140 161 207)(48 127 162 208)(49 128 163 209)(50 129 164 210)(51 130 165 197)(52 131 166 198)(53 132 167 199)(54 133 168 200)(55 134 155 201)(56 135 156 202)
(1 202 149 135)(2 203 150 136)(3 204 151 137)(4 205 152 138)(5 206 153 139)(6 207 154 140)(7 208 141 127)(8 209 142 128)(9 210 143 129)(10 197 144 130)(11 198 145 131)(12 199 146 132)(13 200 147 133)(14 201 148 134)(15 68 98 220)(16 69 85 221)(17 70 86 222)(18 57 87 223)(19 58 88 224)(20 59 89 211)(21 60 90 212)(22 61 91 213)(23 62 92 214)(24 63 93 215)(25 64 94 216)(26 65 95 217)(27 66 96 218)(28 67 97 219)(29 117 108 188)(30 118 109 189)(31 119 110 190)(32 120 111 191)(33 121 112 192)(34 122 99 193)(35 123 100 194)(36 124 101 195)(37 125 102 196)(38 126 103 183)(39 113 104 184)(40 114 105 185)(41 115 106 186)(42 116 107 187)(43 78 157 182)(44 79 158 169)(45 80 159 170)(46 81 160 171)(47 82 161 172)(48 83 162 173)(49 84 163 174)(50 71 164 175)(51 72 165 176)(52 73 166 177)(53 74 167 178)(54 75 168 179)(55 76 155 180)(56 77 156 181)
(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 29 8 36)(2 42 9 35)(3 41 10 34)(4 40 11 33)(5 39 12 32)(6 38 13 31)(7 37 14 30)(15 71 22 78)(16 84 23 77)(17 83 24 76)(18 82 25 75)(19 81 26 74)(20 80 27 73)(21 79 28 72)(43 220 50 213)(44 219 51 212)(45 218 52 211)(46 217 53 224)(47 216 54 223)(48 215 55 222)(49 214 56 221)(57 161 64 168)(58 160 65 167)(59 159 66 166)(60 158 67 165)(61 157 68 164)(62 156 69 163)(63 155 70 162)(85 174 92 181)(86 173 93 180)(87 172 94 179)(88 171 95 178)(89 170 96 177)(90 169 97 176)(91 182 98 175)(99 151 106 144)(100 150 107 143)(101 149 108 142)(102 148 109 141)(103 147 110 154)(104 146 111 153)(105 145 112 152)(113 199 120 206)(114 198 121 205)(115 197 122 204)(116 210 123 203)(117 209 124 202)(118 208 125 201)(119 207 126 200)(127 196 134 189)(128 195 135 188)(129 194 136 187)(130 193 137 186)(131 192 138 185)(132 191 139 184)(133 190 140 183)

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

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

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

88 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L7A7B7C8A···8H14A···14I14J···14AA28A···28L28M···28AD
order122222224444444444447778···814···1414···1428···2828···28
size11112244111122442828282822214···142···24···42···24···4

88 irreducible representations

dim111111122222222224
type++++++++++++-
imageC1C2C2C2C2C2C4D4D4D7D14D14D14Dic7C4○D8C7⋊D4C7⋊D4D4.8D14
kernelQ8.(C2×Dic7)D4⋊Dic7Q8⋊Dic7C22×C7⋊C8C23.21D14C14×C4○D4C7×C4○D4C2×C28C22×C14C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C14C2×C4C23C2
# reps122111831333312818612

Matrix representation of Q8.(C2×Dic7) in GL4(𝔽113) generated by

1000
0100
00980
006515
,
112000
011200
006530
005548
,
112000
011200
00980
00098
,
0100
1128900
001120
000112
,
10510400
70800
007325
005840
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,98,65,0,0,0,15],[112,0,0,0,0,112,0,0,0,0,65,55,0,0,30,48],[112,0,0,0,0,112,0,0,0,0,98,0,0,0,0,98],[0,112,0,0,1,89,0,0,0,0,112,0,0,0,0,112],[105,70,0,0,104,8,0,0,0,0,73,58,0,0,25,40] >;

Q8.(C2×Dic7) in GAP, Magma, Sage, TeX

Q_8.(C_2\times {\rm Dic}_7)
% in TeX

G:=Group("Q8.(C2xDic7)");
// GroupNames label

G:=SmallGroup(448,767);
// by ID

G=gap.SmallGroup(448,767);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,254,1123,297,136,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=d^14=1,b^2=c^2=a^2,e^2=d^7,b*a*b^-1=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^-1*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽