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G = Q8⋊C4⋊D7  order 448 = 26·7

15th semidirect product of Q8⋊C4 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C28⋊Q8⋊5C2, C4⋊C4.18D14, C56⋊C4⋊19C2, Q8⋊C4⋊15D7, Q8⋊Dic7⋊1C2, (C2×C8).173D14, (C2×Q8).11D14, C14.D8.1C2, C2.D56.7C2, C28.15(C4○D4), C4.29(C4○D28), (C2×Dic7).26D4, C22.189(D4×D7), C4.55(D4⋊2D7), C2.15(D56⋊C2), C14.60(C8⋊C22), (C2×C56).193C22, (C2×C28).235C23, C2.9(Q16⋊D7), C28.23D4.2C2, (C2×D28).57C22, C4⋊Dic7.84C22, (Q8×C14).18C22, C14.27(C4.4D4), C14.54(C8.C22), (C4×Dic7).19C22, C7⋊3(C42.28C22), C2.17(Dic7.D4), (C2×C7⋊C8).30C22, (C7×Q8⋊C4)⋊16C2, (C2×C14).248(C2×D4), (C7×C4⋊C4).36C22, (C2×C4).342(C22×D7), SmallGroup(448,329)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C28 — Q8⋊C4⋊D7
Chief series
C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — C28⋊Q8 — Q8⋊C4⋊D7
Lower central
C7 — C14 — C2×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=eae=a-1, ad=da, cbc-1=a-1b, bd=db, ebe=a2bc2, cd=dc, ece=ac-1, ede=d-1 >

Subgroups: 596 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, C2×Q8, Dic7, C28, C28, D14, C2×C14, C8⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C7⋊C8, C56, Dic14, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C42.28C22, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C2×D28, Q8×C14, C14.D8, C56⋊C4, C2.D56, Q8⋊Dic7, C7×Q8⋊C4, C28⋊Q8, C28.23D4, Q8⋊C4⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C8⋊C22, C8.C22, C22×D7, C42.28C22, C4○D28, D4×D7, D4⋊2D7, Dic7.D4, D56⋊C2, Q16⋊D7, Q8⋊C4⋊D7

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

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

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

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

58 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G7A7B7C8A8B8C8D14A···14I28A···28F28G···28R56A···56L
order122224444444777888814···1428···2828···2856···56
size11115622882828562224428282···24···48···84···4

58 irreducible representations

dim111111112222222444444
type++++++++++++++--++
imageC1C2C2C2C2C2C2C2D4D7C4○D4D14D14D14C4○D28C8⋊C22C8.C22D4⋊2D7D4×D7D56⋊C2Q16⋊D7
kernelQ8⋊C4⋊D7C14.D8C56⋊C4C2.D56Q8⋊Dic7C7×Q8⋊C4C28⋊Q8C28.23D4C2×Dic7Q8⋊C4C28C4⋊C4C2×C8C2×Q8C4C14C14C4C22C2C2
# reps1111111123433312113366

Matrix representation of Q8⋊C4⋊D7 ►in GL8(𝔽113)

1120000000
0112000000
0011200000
0001120000
00000010
00000001
0000112000
0000011200
,
84106200000
729020000
002970000
00106840000
000010610840
00001037084
00008407103
000008410106
,
87038910000
08722750000
37442600000
69760260000
000095737473
000040394018
000074731840
000040187374
,
01000000
11224000000
00010000
00112240000
00000100
00001122400
00000001
00000011224
,
01000000
10000000
0001120000
0011200000
00000100
00001000
0000000112
0000001120

G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[84,7,0,0,0,0,0,0,106,29,0,0,0,0,0,0,2,0,29,106,0,0,0,0,0,2,7,84,0,0,0,0,0,0,0,0,106,103,84,0,0,0,0,0,10,7,0,84,0,0,0,0,84,0,7,10,0,0,0,0,0,84,103,106],[87,0,37,69,0,0,0,0,0,87,44,76,0,0,0,0,38,22,26,0,0,0,0,0,91,75,0,26,0,0,0,0,0,0,0,0,95,40,74,40,0,0,0,0,73,39,73,18,0,0,0,0,74,40,18,73,0,0,0,0,73,18,40,74],[0,112,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,24],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0] >;
 

Q8⋊C4⋊D7 in GAP, Magma, Sage, TeX 

Q_8\rtimes C_4\rtimes D_7
 
% in TeX
 
G:=Group("Q8:C4:D7");
 
// GroupNames label
 
G:=SmallGroup(448,329);
 
// by ID
 
G=gap.SmallGroup(448,329);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,1094,135,184,570,297,136,18822]);
 
// 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=e*a*e=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,e*b*e=a^2*b*c^2,c*d=d*c,e*c*e=a*c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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