metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D7⋊2C4, D28.3(C2×C4), (C4×D7).40D4, C4.165(D4×D7), Q8.10(C4×D7), C4⋊C4.150D14, Q8⋊C4⋊22D7, C28.119(C2×D4), Q8⋊Dic7⋊6C2, (C2×C8).209D14, C14.D8⋊10C2, C2.D56⋊23C2, C22.79(D4×D7), C14.47(C4○D8), C28.15(C22×C4), (C2×Q8).106D14, (C22×D7).49D4, C2.2(Q8.D14), (C2×C28).244C23, (C2×C56).199C22, D14.7(C22⋊C4), (C2×Dic7).204D4, (C2×D28).61C22, C7⋊2(C23.24D4), C4⋊Dic7.92C22, (Q8×C14).27C22, C2.4(SD16⋊3D7), Dic7.19(C22⋊C4), (D7×C2×C8)⋊20C2, C4.15(C2×C4×D7), C4⋊C4⋊7D7⋊5C2, (C7×Q8).4(C2×C4), (C4×D7).15(C2×C4), C2.24(D7×C22⋊C4), (C7×Q8⋊C4)⋊21C2, (C2×C14).257(C2×D4), (C7×C4⋊C4).45C22, C14.23(C2×C22⋊C4), (C2×C7⋊C8).219C22, (C2×Q8⋊2D7).3C2, (C2×C4×D7).227C22, (C2×C4).351(C22×D7), SmallGroup(448,338)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D7⋊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, bd=db, ebe-1=a-1b, dcd=c-1, ce=ec, ede-1=a-1d >
Subgroups: 852 in 158 conjugacy classes, 55 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, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, D4⋊C4, Q8⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C7⋊C8, C56, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×D7, C23.24D4, C8×D7, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, Q8⋊2D7, Q8⋊2D7, Q8×C14, C14.D8, C2.D56, Q8⋊Dic7, C7×Q8⋊C4, C4⋊C4⋊7D7, D7×C2×C8, C2×Q8⋊2D7, Q8⋊2D7⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4○D8, C4×D7, C22×D7, C23.24D4, C2×C4×D7, D4×D7, D7×C22⋊C4, SD16⋊3D7, Q8.D14, Q8⋊2D7⋊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 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)(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)(169 183 176 190)(170 184 177 191)(171 185 178 192)(172 186 179 193)(173 187 180 194)(174 188 181 195)(175 189 182 196)(197 211 204 218)(198 212 205 219)(199 213 206 220)(200 214 207 221)(201 215 208 222)(202 216 209 223)(203 217 210 224)
(1 127 8 134)(2 128 9 135)(3 129 10 136)(4 130 11 137)(5 131 12 138)(6 132 13 139)(7 133 14 140)(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)(29 155 36 162)(30 156 37 163)(31 157 38 164)(32 158 39 165)(33 159 40 166)(34 160 41 167)(35 161 42 168)(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)(57 183 64 190)(58 184 65 191)(59 185 66 192)(60 186 67 193)(61 187 68 194)(62 188 69 195)(63 189 70 196)(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)(85 211 92 218)(86 212 93 219)(87 213 94 220)(88 214 95 221)(89 215 96 222)(90 216 97 223)(91 217 98 224)(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)
(1 161)(2 160)(3 159)(4 158)(5 157)(6 156)(7 155)(8 168)(9 167)(10 166)(11 165)(12 164)(13 163)(14 162)(15 154)(16 153)(17 152)(18 151)(19 150)(20 149)(21 148)(22 147)(23 146)(24 145)(25 144)(26 143)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 136)(34 135)(35 134)(36 133)(37 132)(38 131)(39 130)(40 129)(41 128)(42 127)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 113)(50 126)(51 125)(52 124)(53 123)(54 122)(55 121)(56 120)(57 217)(58 216)(59 215)(60 214)(61 213)(62 212)(63 211)(64 224)(65 223)(66 222)(67 221)(68 220)(69 219)(70 218)(71 210)(72 209)(73 208)(74 207)(75 206)(76 205)(77 204)(78 203)(79 202)(80 201)(81 200)(82 199)(83 198)(84 197)(85 196)(86 195)(87 194)(88 193)(89 192)(90 191)(91 190)(92 189)(93 188)(94 187)(95 186)(96 185)(97 184)(98 183)(99 175)(100 174)(101 173)(102 172)(103 171)(104 170)(105 169)(106 182)(107 181)(108 180)(109 179)(110 178)(111 177)(112 176)
(1 218 50 169)(2 219 51 170)(3 220 52 171)(4 221 53 172)(5 222 54 173)(6 223 55 174)(7 224 56 175)(8 211 43 176)(9 212 44 177)(10 213 45 178)(11 214 46 179)(12 215 47 180)(13 216 48 181)(14 217 49 182)(15 197 29 183)(16 198 30 184)(17 199 31 185)(18 200 32 186)(19 201 33 187)(20 202 34 188)(21 203 35 189)(22 204 36 190)(23 205 37 191)(24 206 38 192)(25 207 39 193)(26 208 40 194)(27 209 41 195)(28 210 42 196)(57 127 106 141)(58 128 107 142)(59 129 108 143)(60 130 109 144)(61 131 110 145)(62 132 111 146)(63 133 112 147)(64 134 99 148)(65 135 100 149)(66 136 101 150)(67 137 102 151)(68 138 103 152)(69 139 104 153)(70 140 105 154)(71 120 85 155)(72 121 86 156)(73 122 87 157)(74 123 88 158)(75 124 89 159)(76 125 90 160)(77 126 91 161)(78 113 92 162)(79 114 93 163)(80 115 94 164)(81 116 95 165)(82 117 96 166)(83 118 97 167)(84 119 98 168)
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)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(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)(169,183,176,190)(170,184,177,191)(171,185,178,192)(172,186,179,193)(173,187,180,194)(174,188,181,195)(175,189,182,196)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,127,8,134)(2,128,9,135)(3,129,10,136)(4,130,11,137)(5,131,12,138)(6,132,13,139)(7,133,14,140)(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)(29,155,36,162)(30,156,37,163)(31,157,38,164)(32,158,39,165)(33,159,40,166)(34,160,41,167)(35,161,42,168)(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)(57,183,64,190)(58,184,65,191)(59,185,66,192)(60,186,67,193)(61,187,68,194)(62,188,69,195)(63,189,70,196)(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)(85,211,92,218)(86,212,93,219)(87,213,94,220)(88,214,95,221)(89,215,96,222)(90,216,97,223)(91,217,98,224)(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), (1,161)(2,160)(3,159)(4,158)(5,157)(6,156)(7,155)(8,168)(9,167)(10,166)(11,165)(12,164)(13,163)(14,162)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,120)(57,217)(58,216)(59,215)(60,214)(61,213)(62,212)(63,211)(64,224)(65,223)(66,222)(67,221)(68,220)(69,219)(70,218)(71,210)(72,209)(73,208)(74,207)(75,206)(76,205)(77,204)(78,203)(79,202)(80,201)(81,200)(82,199)(83,198)(84,197)(85,196)(86,195)(87,194)(88,193)(89,192)(90,191)(91,190)(92,189)(93,188)(94,187)(95,186)(96,185)(97,184)(98,183)(99,175)(100,174)(101,173)(102,172)(103,171)(104,170)(105,169)(106,182)(107,181)(108,180)(109,179)(110,178)(111,177)(112,176), (1,218,50,169)(2,219,51,170)(3,220,52,171)(4,221,53,172)(5,222,54,173)(6,223,55,174)(7,224,56,175)(8,211,43,176)(9,212,44,177)(10,213,45,178)(11,214,46,179)(12,215,47,180)(13,216,48,181)(14,217,49,182)(15,197,29,183)(16,198,30,184)(17,199,31,185)(18,200,32,186)(19,201,33,187)(20,202,34,188)(21,203,35,189)(22,204,36,190)(23,205,37,191)(24,206,38,192)(25,207,39,193)(26,208,40,194)(27,209,41,195)(28,210,42,196)(57,127,106,141)(58,128,107,142)(59,129,108,143)(60,130,109,144)(61,131,110,145)(62,132,111,146)(63,133,112,147)(64,134,99,148)(65,135,100,149)(66,136,101,150)(67,137,102,151)(68,138,103,152)(69,139,104,153)(70,140,105,154)(71,120,85,155)(72,121,86,156)(73,122,87,157)(74,123,88,158)(75,124,89,159)(76,125,90,160)(77,126,91,161)(78,113,92,162)(79,114,93,163)(80,115,94,164)(81,116,95,165)(82,117,96,166)(83,118,97,167)(84,119,98,168)>;
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)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(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)(169,183,176,190)(170,184,177,191)(171,185,178,192)(172,186,179,193)(173,187,180,194)(174,188,181,195)(175,189,182,196)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,127,8,134)(2,128,9,135)(3,129,10,136)(4,130,11,137)(5,131,12,138)(6,132,13,139)(7,133,14,140)(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)(29,155,36,162)(30,156,37,163)(31,157,38,164)(32,158,39,165)(33,159,40,166)(34,160,41,167)(35,161,42,168)(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)(57,183,64,190)(58,184,65,191)(59,185,66,192)(60,186,67,193)(61,187,68,194)(62,188,69,195)(63,189,70,196)(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)(85,211,92,218)(86,212,93,219)(87,213,94,220)(88,214,95,221)(89,215,96,222)(90,216,97,223)(91,217,98,224)(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), (1,161)(2,160)(3,159)(4,158)(5,157)(6,156)(7,155)(8,168)(9,167)(10,166)(11,165)(12,164)(13,163)(14,162)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,120)(57,217)(58,216)(59,215)(60,214)(61,213)(62,212)(63,211)(64,224)(65,223)(66,222)(67,221)(68,220)(69,219)(70,218)(71,210)(72,209)(73,208)(74,207)(75,206)(76,205)(77,204)(78,203)(79,202)(80,201)(81,200)(82,199)(83,198)(84,197)(85,196)(86,195)(87,194)(88,193)(89,192)(90,191)(91,190)(92,189)(93,188)(94,187)(95,186)(96,185)(97,184)(98,183)(99,175)(100,174)(101,173)(102,172)(103,171)(104,170)(105,169)(106,182)(107,181)(108,180)(109,179)(110,178)(111,177)(112,176), (1,218,50,169)(2,219,51,170)(3,220,52,171)(4,221,53,172)(5,222,54,173)(6,223,55,174)(7,224,56,175)(8,211,43,176)(9,212,44,177)(10,213,45,178)(11,214,46,179)(12,215,47,180)(13,216,48,181)(14,217,49,182)(15,197,29,183)(16,198,30,184)(17,199,31,185)(18,200,32,186)(19,201,33,187)(20,202,34,188)(21,203,35,189)(22,204,36,190)(23,205,37,191)(24,206,38,192)(25,207,39,193)(26,208,40,194)(27,209,41,195)(28,210,42,196)(57,127,106,141)(58,128,107,142)(59,129,108,143)(60,130,109,144)(61,131,110,145)(62,132,111,146)(63,133,112,147)(64,134,99,148)(65,135,100,149)(66,136,101,150)(67,137,102,151)(68,138,103,152)(69,139,104,153)(70,140,105,154)(71,120,85,155)(72,121,86,156)(73,122,87,157)(74,123,88,158)(75,124,89,159)(76,125,90,160)(77,126,91,161)(78,113,92,162)(79,114,93,163)(80,115,94,164)(81,116,95,165)(82,117,96,166)(83,118,97,167)(84,119,98,168) );
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),(57,78,64,71),(58,79,65,72),(59,80,66,73),(60,81,67,74),(61,82,68,75),(62,83,69,76),(63,84,70,77),(85,106,92,99),(86,107,93,100),(87,108,94,101),(88,109,95,102),(89,110,96,103),(90,111,97,104),(91,112,98,105),(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),(169,183,176,190),(170,184,177,191),(171,185,178,192),(172,186,179,193),(173,187,180,194),(174,188,181,195),(175,189,182,196),(197,211,204,218),(198,212,205,219),(199,213,206,220),(200,214,207,221),(201,215,208,222),(202,216,209,223),(203,217,210,224)], [(1,127,8,134),(2,128,9,135),(3,129,10,136),(4,130,11,137),(5,131,12,138),(6,132,13,139),(7,133,14,140),(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),(29,155,36,162),(30,156,37,163),(31,157,38,164),(32,158,39,165),(33,159,40,166),(34,160,41,167),(35,161,42,168),(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),(57,183,64,190),(58,184,65,191),(59,185,66,192),(60,186,67,193),(61,187,68,194),(62,188,69,195),(63,189,70,196),(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),(85,211,92,218),(86,212,93,219),(87,213,94,220),(88,214,95,221),(89,215,96,222),(90,216,97,223),(91,217,98,224),(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)], [(1,161),(2,160),(3,159),(4,158),(5,157),(6,156),(7,155),(8,168),(9,167),(10,166),(11,165),(12,164),(13,163),(14,162),(15,154),(16,153),(17,152),(18,151),(19,150),(20,149),(21,148),(22,147),(23,146),(24,145),(25,144),(26,143),(27,142),(28,141),(29,140),(30,139),(31,138),(32,137),(33,136),(34,135),(35,134),(36,133),(37,132),(38,131),(39,130),(40,129),(41,128),(42,127),(43,119),(44,118),(45,117),(46,116),(47,115),(48,114),(49,113),(50,126),(51,125),(52,124),(53,123),(54,122),(55,121),(56,120),(57,217),(58,216),(59,215),(60,214),(61,213),(62,212),(63,211),(64,224),(65,223),(66,222),(67,221),(68,220),(69,219),(70,218),(71,210),(72,209),(73,208),(74,207),(75,206),(76,205),(77,204),(78,203),(79,202),(80,201),(81,200),(82,199),(83,198),(84,197),(85,196),(86,195),(87,194),(88,193),(89,192),(90,191),(91,190),(92,189),(93,188),(94,187),(95,186),(96,185),(97,184),(98,183),(99,175),(100,174),(101,173),(102,172),(103,171),(104,170),(105,169),(106,182),(107,181),(108,180),(109,179),(110,178),(111,177),(112,176)], [(1,218,50,169),(2,219,51,170),(3,220,52,171),(4,221,53,172),(5,222,54,173),(6,223,55,174),(7,224,56,175),(8,211,43,176),(9,212,44,177),(10,213,45,178),(11,214,46,179),(12,215,47,180),(13,216,48,181),(14,217,49,182),(15,197,29,183),(16,198,30,184),(17,199,31,185),(18,200,32,186),(19,201,33,187),(20,202,34,188),(21,203,35,189),(22,204,36,190),(23,205,37,191),(24,206,38,192),(25,207,39,193),(26,208,40,194),(27,209,41,195),(28,210,42,196),(57,127,106,141),(58,128,107,142),(59,129,108,143),(60,130,109,144),(61,131,110,145),(62,132,111,146),(63,133,112,147),(64,134,99,148),(65,135,100,149),(66,136,101,150),(67,137,102,151),(68,138,103,152),(69,139,104,153),(70,140,105,154),(71,120,85,155),(72,121,86,156),(73,122,87,157),(74,123,88,158),(75,124,89,159),(76,125,90,160),(77,126,91,161),(78,113,92,162),(79,114,93,163),(80,115,94,164),(81,116,95,165),(82,117,96,166),(83,118,97,167),(84,119,98,168)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 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 | 7 | 7 | 7 | 7 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | C4×D7 | D4×D7 | D4×D7 | SD16⋊3D7 | Q8.D14 |
| kernel | Q8⋊2D7⋊C4 | C14.D8 | C2.D56 | Q8⋊Dic7 | C7×Q8⋊C4 | C4⋊C4⋊7D7 | D7×C2×C8 | C2×Q8⋊2D7 | Q8⋊2D7 | C4×D7 | C2×Dic7 | C22×D7 | Q8⋊C4 | C4⋊C4 | C2×C8 | C2×Q8 | C14 | Q8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 3 | 3 | 3 | 3 | 8 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of Q8⋊2D7⋊C4 ►in GL4(𝔽113) generated by
| 15 | 0 | 0 | 0 |
| 56 | 98 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 85 | 15 | 0 | 0 |
| 23 | 28 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 79 |
| 81 | 1 | 0 | 0 |
| 107 | 32 | 0 | 0 |
| 0 | 0 | 34 | 112 |
| 0 | 0 | 25 | 79 |
| 61 | 44 | 0 | 0 |
| 49 | 52 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(113))| [15,56,0,0,0,98,0,0,0,0,1,0,0,0,0,1],[85,23,0,0,15,28,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,112,79],[81,107,0,0,1,32,0,0,0,0,34,25,0,0,112,79],[61,49,0,0,44,52,0,0,0,0,15,0,0,0,0,15] >;
Q8⋊2D7⋊C4 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_7\rtimes C_4
% in TeX
G:=Group("Q8:2D7:C4"); // GroupNames label
G:=SmallGroup(448,338);
// by ID
G=gap.SmallGroup(448,338);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,232,219,58,570,136,851,102,18822]);
// 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,b*d=d*b,e*b*e^-1=a^-1*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=a^-1*d>;
// generators/relations