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## G = Q8⋊(C4×D7)  order 448 = 26·7

### 2nd semidirect product of Q8 and C4×D7 acting via C4×D7/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Q8⋊(C4×D7)
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×C4×D7 — C2×Q8⋊2D7 — Q8⋊(C4×D7)
 Lower central C7 — C14 — C28 — Q8⋊(C4×D7)
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Q8⋊(C4×D7)
G = < a,b,c,d,e | a4=c4=d7=e2=1, b2=a2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=ab, bd=db, ebe=a2b, cd=dc, ce=ec, ede=d-1 >

Subgroups: 916 in 162 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×8], C7, C8 [×2], C2×C4, C2×C4 [×14], D4 [×7], Q8 [×2], Q8, C23 [×2], D7 [×4], C14 [×3], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], C22×C4 [×3], C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic7 [×2], Dic7, C28 [×2], C28 [×3], D14 [×2], D14 [×6], C2×C14, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C7⋊C8, C56, C4×D7 [×4], C4×D7 [×6], D28 [×2], D28 [×5], C2×Dic7, C2×Dic7, C2×C28, C2×C28 [×2], C7×Q8 [×2], C7×Q8, C22×D7, C22×D7, C23.36D4, C8⋊D7 [×2], C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×C4×D7 [×2], C2×D28, C2×D28, Q82D7 [×4], Q82D7 [×2], Q8×C14, C14.D8, C2.D56, Q8⋊Dic7, C7×Q8⋊C4, D7×C4⋊C4, C2×C8⋊D7, C2×Q82D7, Q8⋊(C4×D7)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D7, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D14 [×3], C2×C22⋊C4, C8⋊C22, C8.C22, C4×D7 [×2], C22×D7, C23.36D4, C2×C4×D7, D4×D7 [×2], D7×C22⋊C4, D56⋊C2, Q16⋊D7, Q8⋊(C4×D7)

Smallest permutation representation of Q8⋊(C4×D7)
On 224 points
Generators in S224
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 71 64 78)(58 72 65 79)(59 73 66 80)(60 74 67 81)(61 75 68 82)(62 76 69 83)(63 77 70 84)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)(113 127 120 134)(114 128 121 135)(115 129 122 136)(116 130 123 137)(117 131 124 138)(118 132 125 139)(119 133 126 140)(141 155 148 162)(142 156 149 163)(143 157 150 164)(144 158 151 165)(145 159 152 166)(146 160 153 167)(147 161 154 168)(169 190 176 183)(170 191 177 184)(171 192 178 185)(172 193 179 186)(173 194 180 187)(174 195 181 188)(175 196 182 189)(197 218 204 211)(198 219 205 212)(199 220 206 213)(200 221 207 214)(201 222 208 215)(202 223 209 216)(203 224 210 217)
(1 120 8 113)(2 121 9 114)(3 122 10 115)(4 123 11 116)(5 124 12 117)(6 125 13 118)(7 126 14 119)(15 134 22 127)(16 135 23 128)(17 136 24 129)(18 137 25 130)(19 138 26 131)(20 139 27 132)(21 140 28 133)(29 148 36 141)(30 149 37 142)(31 150 38 143)(32 151 39 144)(33 152 40 145)(34 153 41 146)(35 154 42 147)(43 162 50 155)(44 163 51 156)(45 164 52 157)(46 165 53 158)(47 166 54 159)(48 167 55 160)(49 168 56 161)(57 176 64 169)(58 177 65 170)(59 178 66 171)(60 179 67 172)(61 180 68 173)(62 181 69 174)(63 182 70 175)(71 190 78 183)(72 191 79 184)(73 192 80 185)(74 193 81 186)(75 194 82 187)(76 195 83 188)(77 196 84 189)(85 204 92 197)(86 205 93 198)(87 206 94 199)(88 207 95 200)(89 208 96 201)(90 209 97 202)(91 210 98 203)(99 218 106 211)(100 219 107 212)(101 220 108 213)(102 221 109 214)(103 222 110 215)(104 223 111 216)(105 224 112 217)
(1 85 29 57)(2 86 30 58)(3 87 31 59)(4 88 32 60)(5 89 33 61)(6 90 34 62)(7 91 35 63)(8 92 36 64)(9 93 37 65)(10 94 38 66)(11 95 39 67)(12 96 40 68)(13 97 41 69)(14 98 42 70)(15 99 43 71)(16 100 44 72)(17 101 45 73)(18 102 46 74)(19 103 47 75)(20 104 48 76)(21 105 49 77)(22 106 50 78)(23 107 51 79)(24 108 52 80)(25 109 53 81)(26 110 54 82)(27 111 55 83)(28 112 56 84)(113 211 141 183)(114 212 142 184)(115 213 143 185)(116 214 144 186)(117 215 145 187)(118 216 146 188)(119 217 147 189)(120 218 148 190)(121 219 149 191)(122 220 150 192)(123 221 151 193)(124 222 152 194)(125 223 153 195)(126 224 154 196)(127 204 155 176)(128 205 156 177)(129 206 157 178)(130 207 158 179)(131 208 159 180)(132 209 160 181)(133 210 161 182)(134 197 162 169)(135 198 163 170)(136 199 164 171)(137 200 165 172)(138 201 166 173)(139 202 167 174)(140 203 168 175)
(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 7)(2 6)(3 5)(8 14)(9 13)(10 12)(15 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 35)(30 34)(31 33)(36 42)(37 41)(38 40)(43 49)(44 48)(45 47)(50 56)(51 55)(52 54)(57 63)(58 62)(59 61)(64 70)(65 69)(66 68)(71 77)(72 76)(73 75)(78 84)(79 83)(80 82)(85 91)(86 90)(87 89)(92 98)(93 97)(94 96)(99 105)(100 104)(101 103)(106 112)(107 111)(108 110)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 168)(156 167)(157 166)(158 165)(159 164)(160 163)(161 162)(169 182)(170 181)(171 180)(172 179)(173 178)(174 177)(175 176)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(197 210)(198 209)(199 208)(200 207)(201 206)(202 205)(203 204)(211 224)(212 223)(213 222)(214 221)(215 220)(216 219)(217 218)

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

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

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

64 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 28A ··· 28F 28G ··· 28R 56A ··· 56L 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 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 14 14 28 28 2 2 4 4 4 4 14 14 28 28 2 2 2 4 4 28 28 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D7 D14 D14 D14 C4×D7 C8⋊C22 C8.C22 D4×D7 D4×D7 D56⋊C2 Q16⋊D7 kernel Q8⋊(C4×D7) C14.D8 C2.D56 Q8⋊Dic7 C7×Q8⋊C4 D7×C4⋊C4 C2×C8⋊D7 C2×Q8⋊2D7 Q8⋊2D7 C4×D7 C2×Dic7 C22×D7 Q8⋊C4 C4⋊C4 C2×C8 C2×Q8 Q8 C14 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 1 1 3 3 3 3 12 1 1 3 3 6 6

Matrix representation of Q8⋊(C4×D7) in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 57 0 98 0 0 0 0 57 0 98
,
 25 55 0 0 0 0 77 88 0 0 0 0 0 0 84 3 45 46 0 0 107 29 21 68 0 0 53 14 29 110 0 0 85 60 6 84
,
 98 111 0 0 0 0 0 15 0 0 0 0 0 0 49 0 111 0 0 0 0 49 0 111 0 0 70 0 64 0 0 0 0 70 0 64
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 33 112 0 0 0 0 2 89 0 0 0 0 0 0 33 112 0 0 0 0 2 89
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 33 0 0 0 0 82 81 0 0 0 0 0 0 32 33 0 0 0 0 82 81

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,15,0,57,0,0,0,0,15,0,57,0,0,0,0,98,0,0,0,0,0,0,98],[25,77,0,0,0,0,55,88,0,0,0,0,0,0,84,107,53,85,0,0,3,29,14,60,0,0,45,21,29,6,0,0,46,68,110,84],[98,0,0,0,0,0,111,15,0,0,0,0,0,0,49,0,70,0,0,0,0,49,0,70,0,0,111,0,64,0,0,0,0,111,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,2,0,0,0,0,112,89,0,0,0,0,0,0,33,2,0,0,0,0,112,89],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,82,0,0,0,0,33,81,0,0,0,0,0,0,32,82,0,0,0,0,33,81] >;

Q8⋊(C4×D7) in GAP, Magma, Sage, TeX

Q_8\rtimes (C_4\times D_7)
% in TeX

G:=Group("Q8:(C4xD7)");
// GroupNames label

G:=SmallGroup(448,337);
// by ID

G=gap.SmallGroup(448,337);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,232,219,58,851,438,102,18822]);
// Polycyclic

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

׿
×
𝔽