metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊D7⋊2C4, Q8⋊2(C4×D7), (Q8×Dic7)⋊2C2, C56⋊C4⋊20C2, D28.4(C2×C4), C14.37(C4×D4), C4⋊C4.155D14, Q8⋊C4⋊20D7, (C2×C8).179D14, C22.80(D4×D7), D28⋊C4.2C2, C2.D56.9C2, C28.Q8⋊13C2, C7⋊3(SD16⋊C4), C2.4(D56⋊C2), C28.16(C22×C4), (C2×Q8).111D14, C28.165(C4○D4), C4.58(D4⋊2D7), C14.65(C8⋊C22), (C2×C28).257C23, (C2×C56).204C22, C2.4(Q16⋊D7), (C2×Dic7).158D4, (C2×D28).67C22, (Q8×C14).40C22, C14.65(C8.C22), C4⋊Dic7.101C22, (C4×Dic7).24C22, C2.21(Dic7⋊4D4), C7⋊C8⋊3(C2×C4), C4.16(C2×C4×D7), (C7×Q8)⋊4(C2×C4), (C2×Q8⋊D7).2C2, (C2×C7⋊C8).47C22, (C7×Q8⋊C4)⋊26C2, (C2×C14).270(C2×D4), (C7×C4⋊C4).58C22, (C2×C4).364(C22×D7), SmallGroup(448,351)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊D7⋊C4
G = < a,b,c,d,e | a4=c7=d2=e4=1, b2=a2, bab-1=dad=eae-1=a-1, ac=ca, bc=cb, dbd=a-1b, ebe-1=ab, dcd=c-1, ce=ec, de=ed >
Subgroups: 628 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, SD16⋊C4, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4⋊Dic7, C4⋊Dic7, D14⋊C4, Q8⋊D7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, Q8×C14, C28.Q8, C56⋊C4, C2.D56, C7×Q8⋊C4, D28⋊C4, C2×Q8⋊D7, Q8×Dic7, Q8⋊D7⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C8⋊C22, C8.C22, C4×D7, C22×D7, SD16⋊C4, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4, D56⋊C2, Q16⋊D7, Q8⋊D7⋊C4
►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)
(1 148 8 141)(2 149 9 142)(3 150 10 143)(4 151 11 144)(5 152 12 145)(6 153 13 146)(7 154 14 147)(15 162 22 155)(16 163 23 156)(17 164 24 157)(18 165 25 158)(19 166 26 159)(20 167 27 160)(21 168 28 161)(29 120 36 113)(30 121 37 114)(31 122 38 115)(32 123 39 116)(33 124 40 117)(34 125 41 118)(35 126 42 119)(43 134 50 127)(44 135 51 128)(45 136 52 129)(46 137 53 130)(47 138 54 131)(48 139 55 132)(49 140 56 133)(57 204 64 197)(58 205 65 198)(59 206 66 199)(60 207 67 200)(61 208 68 201)(62 209 69 202)(63 210 70 203)(71 218 78 211)(72 219 79 212)(73 220 80 213)(74 221 81 214)(75 222 82 215)(76 223 83 216)(77 224 84 217)(85 176 92 169)(86 177 93 170)(87 178 94 171)(88 179 95 172)(89 180 96 173)(90 181 97 174)(91 182 98 175)(99 190 106 183)(100 191 107 184)(101 192 108 185)(102 193 109 186)(103 194 110 187)(104 195 111 188)(105 196 112 189)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(113 161)(114 160)(115 159)(116 158)(117 157)(118 156)(119 155)(120 168)(121 167)(122 166)(123 165)(124 164)(125 163)(126 162)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 141)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)(169 224)(170 223)(171 222)(172 221)(173 220)(174 219)(175 218)(176 217)(177 216)(178 215)(179 214)(180 213)(181 212)(182 211)(183 210)(184 209)(185 208)(186 207)(187 206)(188 205)(189 204)(190 203)(191 202)(192 201)(193 200)(194 199)(195 198)(196 197)
(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)
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), (1,148,8,141)(2,149,9,142)(3,150,10,143)(4,151,11,144)(5,152,12,145)(6,153,13,146)(7,154,14,147)(15,162,22,155)(16,163,23,156)(17,164,24,157)(18,165,25,158)(19,166,26,159)(20,167,27,160)(21,168,28,161)(29,120,36,113)(30,121,37,114)(31,122,38,115)(32,123,39,116)(33,124,40,117)(34,125,41,118)(35,126,42,119)(43,134,50,127)(44,135,51,128)(45,136,52,129)(46,137,53,130)(47,138,54,131)(48,139,55,132)(49,140,56,133)(57,204,64,197)(58,205,65,198)(59,206,66,199)(60,207,67,200)(61,208,68,201)(62,209,69,202)(63,210,70,203)(71,218,78,211)(72,219,79,212)(73,220,80,213)(74,221,81,214)(75,222,82,215)(76,223,83,216)(77,224,84,217)(85,176,92,169)(86,177,93,170)(87,178,94,171)(88,179,95,172)(89,180,96,173)(90,181,97,174)(91,182,98,175)(99,190,106,183)(100,191,107,184)(101,192,108,185)(102,193,109,186)(103,194,110,187)(104,195,111,188)(105,196,112,189), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(169,224)(170,223)(171,222)(172,221)(173,220)(174,219)(175,218)(176,217)(177,216)(178,215)(179,214)(180,213)(181,212)(182,211)(183,210)(184,209)(185,208)(186,207)(187,206)(188,205)(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(195,198)(196,197), (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)>;
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), (1,148,8,141)(2,149,9,142)(3,150,10,143)(4,151,11,144)(5,152,12,145)(6,153,13,146)(7,154,14,147)(15,162,22,155)(16,163,23,156)(17,164,24,157)(18,165,25,158)(19,166,26,159)(20,167,27,160)(21,168,28,161)(29,120,36,113)(30,121,37,114)(31,122,38,115)(32,123,39,116)(33,124,40,117)(34,125,41,118)(35,126,42,119)(43,134,50,127)(44,135,51,128)(45,136,52,129)(46,137,53,130)(47,138,54,131)(48,139,55,132)(49,140,56,133)(57,204,64,197)(58,205,65,198)(59,206,66,199)(60,207,67,200)(61,208,68,201)(62,209,69,202)(63,210,70,203)(71,218,78,211)(72,219,79,212)(73,220,80,213)(74,221,81,214)(75,222,82,215)(76,223,83,216)(77,224,84,217)(85,176,92,169)(86,177,93,170)(87,178,94,171)(88,179,95,172)(89,180,96,173)(90,181,97,174)(91,182,98,175)(99,190,106,183)(100,191,107,184)(101,192,108,185)(102,193,109,186)(103,194,110,187)(104,195,111,188)(105,196,112,189), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(169,224)(170,223)(171,222)(172,221)(173,220)(174,219)(175,218)(176,217)(177,216)(178,215)(179,214)(180,213)(181,212)(182,211)(183,210)(184,209)(185,208)(186,207)(187,206)(188,205)(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(195,198)(196,197), (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) );
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)], [(1,148,8,141),(2,149,9,142),(3,150,10,143),(4,151,11,144),(5,152,12,145),(6,153,13,146),(7,154,14,147),(15,162,22,155),(16,163,23,156),(17,164,24,157),(18,165,25,158),(19,166,26,159),(20,167,27,160),(21,168,28,161),(29,120,36,113),(30,121,37,114),(31,122,38,115),(32,123,39,116),(33,124,40,117),(34,125,41,118),(35,126,42,119),(43,134,50,127),(44,135,51,128),(45,136,52,129),(46,137,53,130),(47,138,54,131),(48,139,55,132),(49,140,56,133),(57,204,64,197),(58,205,65,198),(59,206,66,199),(60,207,67,200),(61,208,68,201),(62,209,69,202),(63,210,70,203),(71,218,78,211),(72,219,79,212),(73,220,80,213),(74,221,81,214),(75,222,82,215),(76,223,83,216),(77,224,84,217),(85,176,92,169),(86,177,93,170),(87,178,94,171),(88,179,95,172),(89,180,96,173),(90,181,97,174),(91,182,98,175),(99,190,106,183),(100,191,107,184),(101,192,108,185),(102,193,109,186),(103,194,110,187),(104,195,111,188),(105,196,112,189)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(57,91),(58,90),(59,89),(60,88),(61,87),(62,86),(63,85),(64,98),(65,97),(66,96),(67,95),(68,94),(69,93),(70,92),(71,112),(72,111),(73,110),(74,109),(75,108),(76,107),(77,106),(78,105),(79,104),(80,103),(81,102),(82,101),(83,100),(84,99),(113,161),(114,160),(115,159),(116,158),(117,157),(118,156),(119,155),(120,168),(121,167),(122,166),(123,165),(124,164),(125,163),(126,162),(127,147),(128,146),(129,145),(130,144),(131,143),(132,142),(133,141),(134,154),(135,153),(136,152),(137,151),(138,150),(139,149),(140,148),(169,224),(170,223),(171,222),(172,221),(173,220),(174,219),(175,218),(176,217),(177,216),(178,215),(179,214),(180,213),(181,212),(182,211),(183,210),(184,209),(185,208),(186,207),(187,206),(188,205),(189,204),(190,203),(191,202),(192,201),(193,200),(194,199),(195,198),(196,197)], [(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)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 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 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4×D7 | C8⋊C22 | C8.C22 | D4⋊2D7 | D4×D7 | D56⋊C2 | Q16⋊D7 |
| kernel | Q8⋊D7⋊C4 | C28.Q8 | C56⋊C4 | C2.D56 | C7×Q8⋊C4 | D28⋊C4 | C2×Q8⋊D7 | Q8×Dic7 | Q8⋊D7 | C2×Dic7 | Q8⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C14 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 2 | 3 | 3 | 3 | 12 | 1 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Q8⋊D7⋊C4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 23 | 3 | 90 |
| 0 | 0 | 90 | 3 | 23 | 12 |
| 0 | 0 | 3 | 90 | 101 | 90 |
| 0 | 0 | 23 | 12 | 23 | 110 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 1 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 79 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 1 | 79 |
| 24 | 1 | 0 | 0 | 0 | 0 |
| 103 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 88 | 0 | 0 |
| 0 | 0 | 1 | 79 | 0 | 0 |
| 0 | 0 | 0 | 0 | 79 | 25 |
| 0 | 0 | 0 | 0 | 112 | 34 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 0 | 60 | 43 |
| 0 | 0 | 0 | 100 | 70 | 53 |
| 0 | 0 | 60 | 43 | 13 | 0 |
| 0 | 0 | 70 | 53 | 0 | 13 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,90,3,23,0,0,23,3,90,12,0,0,3,23,101,23,0,0,90,12,90,110],[0,1,0,0,0,0,112,24,0,0,0,0,0,0,0,1,0,0,0,0,112,79,0,0,0,0,0,0,0,1,0,0,0,0,112,79],[24,103,0,0,0,0,1,89,0,0,0,0,0,0,34,1,0,0,0,0,88,79,0,0,0,0,0,0,79,112,0,0,0,0,25,34],[15,0,0,0,0,0,0,15,0,0,0,0,0,0,100,0,60,70,0,0,0,100,43,53,0,0,60,70,13,0,0,0,43,53,0,13] >;
Q8⋊D7⋊C4 in GAP, Magma, Sage, TeX
Q_8\rtimes D_7\rtimes C_4
% in TeX
G:=Group("Q8:D7:C4"); // GroupNames label
G:=SmallGroup(448,351);
// by ID
G=gap.SmallGroup(448,351);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,232,219,58,1684,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^7=d^2=e^4=1,b^2=a^2,b*a*b^-1=d*a*d=e*a*e^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,e*b*e^-1=a*b,d*c*d=c^-1,c*e=e*c,d*e=e*d>;
// generators/relations