direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×C7⋊D4, C14.432- 1+4, C7⋊5(D4×Q8), (C7×Q8)⋊16D4, D14⋊8(C2×Q8), C22⋊3(Q8×D7), Dic7⋊5(C2×Q8), (C22×Q8)⋊7D7, (Q8×Dic7)⋊26C2, C28.260(C2×D4), D14⋊3Q8⋊40C2, (C2×Q8).187D14, Dic7⋊Q8⋊29C2, C28.48D4⋊38C2, C14.54(C22×Q8), (C2×C14).307C24, (C2×C28).647C23, C14.155(C22×D4), (C22×C4).277D14, D14⋊C4.158C22, Dic7⋊C4.89C22, C4⋊Dic7.257C22, (Q8×C14).234C22, C23.238(C22×D7), C22.318(C23×D7), (C22×C28).286C22, (C22×C14).425C23, (C2×Dic7).158C23, (C4×Dic7).170C22, (C22×D7).242C23, C23.D7.131C22, C2.43(Q8.10D14), (C2×Dic14).202C22, (C2×Q8×D7)⋊18C2, (Q8×C2×C14)⋊6C2, C2.36(C2×Q8×D7), (C2×C14)⋊9(C2×Q8), C4.69(C2×C7⋊D4), (C4×C7⋊D4).12C2, (C2×C4×D7).164C22, C2.28(C22×C7⋊D4), (C2×C4).633(C22×D7), (C2×C7⋊D4).149C22, SmallGroup(448,1268)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C7⋊D4
G = < a,b,c,d,e | a4=c7=d4=e2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 1044 in 280 conjugacy classes, 123 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C22×Q8, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×C14, D4×Q8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C2×Dic14, C2×C4×D7, Q8×D7, C2×C7⋊D4, C22×C28, Q8×C14, Q8×C14, Q8×C14, C28.48D4, C4×C7⋊D4, Dic7⋊Q8, Q8×Dic7, D14⋊3Q8, C2×Q8×D7, Q8×C2×C14, Q8×C7⋊D4
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C24, D14, C22×D4, C22×Q8, 2- 1+4, C7⋊D4, C22×D7, D4×Q8, Q8×D7, C2×C7⋊D4, C23×D7, C2×Q8×D7, Q8.10D14, C22×C7⋊D4, Q8×C7⋊D4
►On 224 points
Generators in S224
(1 78 8 71)(2 79 9 72)(3 80 10 73)(4 81 11 74)(5 82 12 75)(6 83 13 76)(7 84 14 77)(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)(113 183 120 190)(114 184 121 191)(115 185 122 192)(116 186 123 193)(117 187 124 194)(118 188 125 195)(119 189 126 196)(127 169 134 176)(128 170 135 177)(129 171 136 178)(130 172 137 179)(131 173 138 180)(132 174 139 181)(133 175 140 182)(141 211 148 218)(142 212 149 219)(143 213 150 220)(144 214 151 221)(145 215 152 222)(146 216 153 223)(147 217 154 224)(155 197 162 204)(156 198 163 205)(157 199 164 206)(158 200 165 207)(159 201 166 208)(160 202 167 209)(161 203 168 210)
(1 134 8 127)(2 135 9 128)(3 136 10 129)(4 137 11 130)(5 138 12 131)(6 139 13 132)(7 140 14 133)(15 120 22 113)(16 121 23 114)(17 122 24 115)(18 123 25 116)(19 124 26 117)(20 125 27 118)(21 126 28 119)(29 162 36 155)(30 163 37 156)(31 164 38 157)(32 165 39 158)(33 166 40 159)(34 167 41 160)(35 168 42 161)(43 148 50 141)(44 149 51 142)(45 150 52 143)(46 151 53 144)(47 152 54 145)(48 153 55 146)(49 154 56 147)(57 190 64 183)(58 191 65 184)(59 192 66 185)(60 193 67 186)(61 194 68 187)(62 195 69 188)(63 196 70 189)(71 176 78 169)(72 177 79 170)(73 178 80 171)(74 179 81 172)(75 180 82 173)(76 181 83 174)(77 182 84 175)(85 218 92 211)(86 219 93 212)(87 220 94 213)(88 221 95 214)(89 222 96 215)(90 223 97 216)(91 224 98 217)(99 204 106 197)(100 205 107 198)(101 206 108 199)(102 207 109 200)(103 208 110 201)(104 209 111 202)(105 210 112 203)
(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 15 43)(2 35 16 49)(3 34 17 48)(4 33 18 47)(5 32 19 46)(6 31 20 45)(7 30 21 44)(8 36 22 50)(9 42 23 56)(10 41 24 55)(11 40 25 54)(12 39 26 53)(13 38 27 52)(14 37 28 51)(57 85 71 99)(58 91 72 105)(59 90 73 104)(60 89 74 103)(61 88 75 102)(62 87 76 101)(63 86 77 100)(64 92 78 106)(65 98 79 112)(66 97 80 111)(67 96 81 110)(68 95 82 109)(69 94 83 108)(70 93 84 107)(113 141 127 155)(114 147 128 161)(115 146 129 160)(116 145 130 159)(117 144 131 158)(118 143 132 157)(119 142 133 156)(120 148 134 162)(121 154 135 168)(122 153 136 167)(123 152 137 166)(124 151 138 165)(125 150 139 164)(126 149 140 163)(169 197 183 211)(170 203 184 217)(171 202 185 216)(172 201 186 215)(173 200 187 214)(174 199 188 213)(175 198 189 212)(176 204 190 218)(177 210 191 224)(178 209 192 223)(179 208 193 222)(180 207 194 221)(181 206 195 220)(182 205 196 219)
(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 43)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 50)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(58 63)(59 62)(60 61)(65 70)(66 69)(67 68)(72 77)(73 76)(74 75)(79 84)(80 83)(81 82)(85 99)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 106)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(114 119)(115 118)(116 117)(121 126)(122 125)(123 124)(128 133)(129 132)(130 131)(135 140)(136 139)(137 138)(141 155)(142 161)(143 160)(144 159)(145 158)(146 157)(147 156)(148 162)(149 168)(150 167)(151 166)(152 165)(153 164)(154 163)(170 175)(171 174)(172 173)(177 182)(178 181)(179 180)(184 189)(185 188)(186 187)(191 196)(192 195)(193 194)(197 211)(198 217)(199 216)(200 215)(201 214)(202 213)(203 212)(204 218)(205 224)(206 223)(207 222)(208 221)(209 220)(210 219)
G:=sub<Sym(224)| (1,78,8,71)(2,79,9,72)(3,80,10,73)(4,81,11,74)(5,82,12,75)(6,83,13,76)(7,84,14,77)(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)(113,183,120,190)(114,184,121,191)(115,185,122,192)(116,186,123,193)(117,187,124,194)(118,188,125,195)(119,189,126,196)(127,169,134,176)(128,170,135,177)(129,171,136,178)(130,172,137,179)(131,173,138,180)(132,174,139,181)(133,175,140,182)(141,211,148,218)(142,212,149,219)(143,213,150,220)(144,214,151,221)(145,215,152,222)(146,216,153,223)(147,217,154,224)(155,197,162,204)(156,198,163,205)(157,199,164,206)(158,200,165,207)(159,201,166,208)(160,202,167,209)(161,203,168,210), (1,134,8,127)(2,135,9,128)(3,136,10,129)(4,137,11,130)(5,138,12,131)(6,139,13,132)(7,140,14,133)(15,120,22,113)(16,121,23,114)(17,122,24,115)(18,123,25,116)(19,124,26,117)(20,125,27,118)(21,126,28,119)(29,162,36,155)(30,163,37,156)(31,164,38,157)(32,165,39,158)(33,166,40,159)(34,167,41,160)(35,168,42,161)(43,148,50,141)(44,149,51,142)(45,150,52,143)(46,151,53,144)(47,152,54,145)(48,153,55,146)(49,154,56,147)(57,190,64,183)(58,191,65,184)(59,192,66,185)(60,193,67,186)(61,194,68,187)(62,195,69,188)(63,196,70,189)(71,176,78,169)(72,177,79,170)(73,178,80,171)(74,179,81,172)(75,180,82,173)(76,181,83,174)(77,182,84,175)(85,218,92,211)(86,219,93,212)(87,220,94,213)(88,221,95,214)(89,222,96,215)(90,223,97,216)(91,224,98,217)(99,204,106,197)(100,205,107,198)(101,206,108,199)(102,207,109,200)(103,208,110,201)(104,209,111,202)(105,210,112,203), (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), (1,29,15,43)(2,35,16,49)(3,34,17,48)(4,33,18,47)(5,32,19,46)(6,31,20,45)(7,30,21,44)(8,36,22,50)(9,42,23,56)(10,41,24,55)(11,40,25,54)(12,39,26,53)(13,38,27,52)(14,37,28,51)(57,85,71,99)(58,91,72,105)(59,90,73,104)(60,89,74,103)(61,88,75,102)(62,87,76,101)(63,86,77,100)(64,92,78,106)(65,98,79,112)(66,97,80,111)(67,96,81,110)(68,95,82,109)(69,94,83,108)(70,93,84,107)(113,141,127,155)(114,147,128,161)(115,146,129,160)(116,145,130,159)(117,144,131,158)(118,143,132,157)(119,142,133,156)(120,148,134,162)(121,154,135,168)(122,153,136,167)(123,152,137,166)(124,151,138,165)(125,150,139,164)(126,149,140,163)(169,197,183,211)(170,203,184,217)(171,202,185,216)(172,201,186,215)(173,200,187,214)(174,199,188,213)(175,198,189,212)(176,204,190,218)(177,210,191,224)(178,209,192,223)(179,208,193,222)(180,207,194,221)(181,206,195,220)(182,205,196,219), (2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,43)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,50)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,99)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,106)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)(129,132)(130,131)(135,140)(136,139)(137,138)(141,155)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,162)(149,168)(150,167)(151,166)(152,165)(153,164)(154,163)(170,175)(171,174)(172,173)(177,182)(178,181)(179,180)(184,189)(185,188)(186,187)(191,196)(192,195)(193,194)(197,211)(198,217)(199,216)(200,215)(201,214)(202,213)(203,212)(204,218)(205,224)(206,223)(207,222)(208,221)(209,220)(210,219)>;
G:=Group( (1,78,8,71)(2,79,9,72)(3,80,10,73)(4,81,11,74)(5,82,12,75)(6,83,13,76)(7,84,14,77)(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)(113,183,120,190)(114,184,121,191)(115,185,122,192)(116,186,123,193)(117,187,124,194)(118,188,125,195)(119,189,126,196)(127,169,134,176)(128,170,135,177)(129,171,136,178)(130,172,137,179)(131,173,138,180)(132,174,139,181)(133,175,140,182)(141,211,148,218)(142,212,149,219)(143,213,150,220)(144,214,151,221)(145,215,152,222)(146,216,153,223)(147,217,154,224)(155,197,162,204)(156,198,163,205)(157,199,164,206)(158,200,165,207)(159,201,166,208)(160,202,167,209)(161,203,168,210), (1,134,8,127)(2,135,9,128)(3,136,10,129)(4,137,11,130)(5,138,12,131)(6,139,13,132)(7,140,14,133)(15,120,22,113)(16,121,23,114)(17,122,24,115)(18,123,25,116)(19,124,26,117)(20,125,27,118)(21,126,28,119)(29,162,36,155)(30,163,37,156)(31,164,38,157)(32,165,39,158)(33,166,40,159)(34,167,41,160)(35,168,42,161)(43,148,50,141)(44,149,51,142)(45,150,52,143)(46,151,53,144)(47,152,54,145)(48,153,55,146)(49,154,56,147)(57,190,64,183)(58,191,65,184)(59,192,66,185)(60,193,67,186)(61,194,68,187)(62,195,69,188)(63,196,70,189)(71,176,78,169)(72,177,79,170)(73,178,80,171)(74,179,81,172)(75,180,82,173)(76,181,83,174)(77,182,84,175)(85,218,92,211)(86,219,93,212)(87,220,94,213)(88,221,95,214)(89,222,96,215)(90,223,97,216)(91,224,98,217)(99,204,106,197)(100,205,107,198)(101,206,108,199)(102,207,109,200)(103,208,110,201)(104,209,111,202)(105,210,112,203), (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), (1,29,15,43)(2,35,16,49)(3,34,17,48)(4,33,18,47)(5,32,19,46)(6,31,20,45)(7,30,21,44)(8,36,22,50)(9,42,23,56)(10,41,24,55)(11,40,25,54)(12,39,26,53)(13,38,27,52)(14,37,28,51)(57,85,71,99)(58,91,72,105)(59,90,73,104)(60,89,74,103)(61,88,75,102)(62,87,76,101)(63,86,77,100)(64,92,78,106)(65,98,79,112)(66,97,80,111)(67,96,81,110)(68,95,82,109)(69,94,83,108)(70,93,84,107)(113,141,127,155)(114,147,128,161)(115,146,129,160)(116,145,130,159)(117,144,131,158)(118,143,132,157)(119,142,133,156)(120,148,134,162)(121,154,135,168)(122,153,136,167)(123,152,137,166)(124,151,138,165)(125,150,139,164)(126,149,140,163)(169,197,183,211)(170,203,184,217)(171,202,185,216)(172,201,186,215)(173,200,187,214)(174,199,188,213)(175,198,189,212)(176,204,190,218)(177,210,191,224)(178,209,192,223)(179,208,193,222)(180,207,194,221)(181,206,195,220)(182,205,196,219), (2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,43)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,50)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,99)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,106)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)(129,132)(130,131)(135,140)(136,139)(137,138)(141,155)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,162)(149,168)(150,167)(151,166)(152,165)(153,164)(154,163)(170,175)(171,174)(172,173)(177,182)(178,181)(179,180)(184,189)(185,188)(186,187)(191,196)(192,195)(193,194)(197,211)(198,217)(199,216)(200,215)(201,214)(202,213)(203,212)(204,218)(205,224)(206,223)(207,222)(208,221)(209,220)(210,219) );
G=PermutationGroup([[(1,78,8,71),(2,79,9,72),(3,80,10,73),(4,81,11,74),(5,82,12,75),(6,83,13,76),(7,84,14,77),(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),(113,183,120,190),(114,184,121,191),(115,185,122,192),(116,186,123,193),(117,187,124,194),(118,188,125,195),(119,189,126,196),(127,169,134,176),(128,170,135,177),(129,171,136,178),(130,172,137,179),(131,173,138,180),(132,174,139,181),(133,175,140,182),(141,211,148,218),(142,212,149,219),(143,213,150,220),(144,214,151,221),(145,215,152,222),(146,216,153,223),(147,217,154,224),(155,197,162,204),(156,198,163,205),(157,199,164,206),(158,200,165,207),(159,201,166,208),(160,202,167,209),(161,203,168,210)], [(1,134,8,127),(2,135,9,128),(3,136,10,129),(4,137,11,130),(5,138,12,131),(6,139,13,132),(7,140,14,133),(15,120,22,113),(16,121,23,114),(17,122,24,115),(18,123,25,116),(19,124,26,117),(20,125,27,118),(21,126,28,119),(29,162,36,155),(30,163,37,156),(31,164,38,157),(32,165,39,158),(33,166,40,159),(34,167,41,160),(35,168,42,161),(43,148,50,141),(44,149,51,142),(45,150,52,143),(46,151,53,144),(47,152,54,145),(48,153,55,146),(49,154,56,147),(57,190,64,183),(58,191,65,184),(59,192,66,185),(60,193,67,186),(61,194,68,187),(62,195,69,188),(63,196,70,189),(71,176,78,169),(72,177,79,170),(73,178,80,171),(74,179,81,172),(75,180,82,173),(76,181,83,174),(77,182,84,175),(85,218,92,211),(86,219,93,212),(87,220,94,213),(88,221,95,214),(89,222,96,215),(90,223,97,216),(91,224,98,217),(99,204,106,197),(100,205,107,198),(101,206,108,199),(102,207,109,200),(103,208,110,201),(104,209,111,202),(105,210,112,203)], [(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)], [(1,29,15,43),(2,35,16,49),(3,34,17,48),(4,33,18,47),(5,32,19,46),(6,31,20,45),(7,30,21,44),(8,36,22,50),(9,42,23,56),(10,41,24,55),(11,40,25,54),(12,39,26,53),(13,38,27,52),(14,37,28,51),(57,85,71,99),(58,91,72,105),(59,90,73,104),(60,89,74,103),(61,88,75,102),(62,87,76,101),(63,86,77,100),(64,92,78,106),(65,98,79,112),(66,97,80,111),(67,96,81,110),(68,95,82,109),(69,94,83,108),(70,93,84,107),(113,141,127,155),(114,147,128,161),(115,146,129,160),(116,145,130,159),(117,144,131,158),(118,143,132,157),(119,142,133,156),(120,148,134,162),(121,154,135,168),(122,153,136,167),(123,152,137,166),(124,151,138,165),(125,150,139,164),(126,149,140,163),(169,197,183,211),(170,203,184,217),(171,202,185,216),(172,201,186,215),(173,200,187,214),(174,199,188,213),(175,198,189,212),(176,204,190,218),(177,210,191,224),(178,209,192,223),(179,208,193,222),(180,207,194,221),(181,206,195,220),(182,205,196,219)], [(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(29,43),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,50),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(58,63),(59,62),(60,61),(65,70),(66,69),(67,68),(72,77),(73,76),(74,75),(79,84),(80,83),(81,82),(85,99),(86,105),(87,104),(88,103),(89,102),(90,101),(91,100),(92,106),(93,112),(94,111),(95,110),(96,109),(97,108),(98,107),(114,119),(115,118),(116,117),(121,126),(122,125),(123,124),(128,133),(129,132),(130,131),(135,140),(136,139),(137,138),(141,155),(142,161),(143,160),(144,159),(145,158),(146,157),(147,156),(148,162),(149,168),(150,167),(151,166),(152,165),(153,164),(154,163),(170,175),(171,174),(172,173),(177,182),(178,181),(179,180),(184,189),(185,188),(186,187),(191,196),(192,195),(193,194),(197,211),(198,217),(199,216),(200,215),(201,214),(202,213),(203,212),(204,218),(205,224),(206,223),(207,222),(208,221),(209,220),(210,219)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | D14 | D14 | C7⋊D4 | 2- 1+4 | Q8×D7 | Q8.10D14 |
| kernel | Q8×C7⋊D4 | C28.48D4 | C4×C7⋊D4 | Dic7⋊Q8 | Q8×Dic7 | D14⋊3Q8 | C2×Q8×D7 | Q8×C2×C14 | C7⋊D4 | C7×Q8 | C22×Q8 | C22×C4 | C2×Q8 | Q8 | C14 | C22 | C2 |
| # reps | 1 | 3 | 3 | 3 | 1 | 3 | 1 | 1 | 4 | 4 | 3 | 9 | 12 | 24 | 1 | 6 | 6 |
Matrix representation of Q8×C7⋊D4 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 11 |
| 0 | 0 | 13 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 17 |
| 0 | 0 | 27 | 2 |
| 26 | 28 | 0 | 0 |
| 2 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 8 | 0 | 0 |
| 8 | 15 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 19 | 0 | 0 |
| 18 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,13,0,0,11,1],[1,0,0,0,0,1,0,0,0,0,27,27,0,0,17,2],[26,2,0,0,28,10,0,0,0,0,1,0,0,0,0,1],[14,8,0,0,8,15,0,0,0,0,1,0,0,0,0,1],[6,18,0,0,19,23,0,0,0,0,1,0,0,0,0,1] >;
Q8×C7⋊D4 in GAP, Magma, Sage, TeX
Q_8\times C_7\rtimes D_4
% in TeX
G:=Group("Q8xC7:D4"); // GroupNames label
G:=SmallGroup(448,1268);
// by ID
G=gap.SmallGroup(448,1268);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,675,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^7=d^4=e^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations