metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2D7⋊2C4, D4.10(C4×D7), (C4×D7).38D4, C4.157(D4×D7), D4⋊C4⋊22D7, C4⋊C4.134D14, (C2×C8).201D14, D4⋊Dic7⋊6C2, C28.106(C2×D4), C14.Q16⋊5C2, C28.7(C22×C4), C22.71(D4×D7), (C2×D4).133D14, C14.39(C4○D8), C2.2(D8⋊3D7), Dic14.1(C2×C4), (C22×D7).46D4, C28.44D4⋊19C2, (C2×C28).212C23, (C2×C56).183C22, D14.5(C22⋊C4), (C2×Dic7).201D4, (D4×C14).33C22, C7⋊1(C23.24D4), C4⋊Dic7.68C22, C2.2(SD16⋊3D7), Dic7.18(C22⋊C4), (C2×Dic14).56C22, C4.7(C2×C4×D7), (D7×C2×C8)⋊17C2, C4⋊C4⋊7D7⋊2C2, (C7×D4).4(C2×C4), (C4×D7).13(C2×C4), C2.20(D7×C22⋊C4), (C7×D4⋊C4)⋊21C2, (C2×D4⋊2D7).4C2, (C2×C14).225(C2×D4), (C7×C4⋊C4).15C22, (C2×C7⋊C8).211C22, C14.19(C2×C22⋊C4), (C2×C4×D7).223C22, (C2×C4).319(C22×D7), SmallGroup(448,306)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2D7⋊C4
G = < a,b,c,d,e | a4=b2=c7=d2=e4=1, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, ebe-1=ab, dcd=c-1, ce=ec, ede-1=a2d >
Subgroups: 756 in 158 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, D4⋊C4, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C23.24D4, C8×D7, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, D4⋊2D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, C14.Q16, C28.44D4, D4⋊Dic7, C7×D4⋊C4, C4⋊C4⋊7D7, D7×C2×C8, C2×D4⋊2D7, D4⋊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, D8⋊3D7, SD16⋊3D7, D4⋊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)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 127)(16 128)(17 129)(18 130)(19 131)(20 132)(21 133)(22 134)(23 135)(24 136)(25 137)(26 138)(27 139)(28 140)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 161)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
(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)
(1 211 43 169)(2 212 44 170)(3 213 45 171)(4 214 46 172)(5 215 47 173)(6 216 48 174)(7 217 49 175)(8 218 50 176)(9 219 51 177)(10 220 52 178)(11 221 53 179)(12 222 54 180)(13 223 55 181)(14 224 56 182)(15 204 36 183)(16 205 37 184)(17 206 38 185)(18 207 39 186)(19 208 40 187)(20 209 41 188)(21 210 42 189)(22 197 29 190)(23 198 30 191)(24 199 31 192)(25 200 32 193)(26 201 33 194)(27 202 34 195)(28 203 35 196)(57 134 99 141)(58 135 100 142)(59 136 101 143)(60 137 102 144)(61 138 103 145)(62 139 104 146)(63 140 105 147)(64 127 106 148)(65 128 107 149)(66 129 108 150)(67 130 109 151)(68 131 110 152)(69 132 111 153)(70 133 112 154)(71 113 92 155)(72 114 93 156)(73 115 94 157)(74 116 95 158)(75 117 96 159)(76 118 97 160)(77 119 98 161)(78 120 85 162)(79 121 86 163)(80 122 87 164)(81 123 88 165)(82 124 89 166)(83 125 90 167)(84 126 91 168)
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), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (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), (1,211,43,169)(2,212,44,170)(3,213,45,171)(4,214,46,172)(5,215,47,173)(6,216,48,174)(7,217,49,175)(8,218,50,176)(9,219,51,177)(10,220,52,178)(11,221,53,179)(12,222,54,180)(13,223,55,181)(14,224,56,182)(15,204,36,183)(16,205,37,184)(17,206,38,185)(18,207,39,186)(19,208,40,187)(20,209,41,188)(21,210,42,189)(22,197,29,190)(23,198,30,191)(24,199,31,192)(25,200,32,193)(26,201,33,194)(27,202,34,195)(28,203,35,196)(57,134,99,141)(58,135,100,142)(59,136,101,143)(60,137,102,144)(61,138,103,145)(62,139,104,146)(63,140,105,147)(64,127,106,148)(65,128,107,149)(66,129,108,150)(67,130,109,151)(68,131,110,152)(69,132,111,153)(70,133,112,154)(71,113,92,155)(72,114,93,156)(73,115,94,157)(74,116,95,158)(75,117,96,159)(76,118,97,160)(77,119,98,161)(78,120,85,162)(79,121,86,163)(80,122,87,164)(81,123,88,165)(82,124,89,166)(83,125,90,167)(84,126,91,168)>;
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), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (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), (1,211,43,169)(2,212,44,170)(3,213,45,171)(4,214,46,172)(5,215,47,173)(6,216,48,174)(7,217,49,175)(8,218,50,176)(9,219,51,177)(10,220,52,178)(11,221,53,179)(12,222,54,180)(13,223,55,181)(14,224,56,182)(15,204,36,183)(16,205,37,184)(17,206,38,185)(18,207,39,186)(19,208,40,187)(20,209,41,188)(21,210,42,189)(22,197,29,190)(23,198,30,191)(24,199,31,192)(25,200,32,193)(26,201,33,194)(27,202,34,195)(28,203,35,196)(57,134,99,141)(58,135,100,142)(59,136,101,143)(60,137,102,144)(61,138,103,145)(62,139,104,146)(63,140,105,147)(64,127,106,148)(65,128,107,149)(66,129,108,150)(67,130,109,151)(68,131,110,152)(69,132,111,153)(70,133,112,154)(71,113,92,155)(72,114,93,156)(73,115,94,157)(74,116,95,158)(75,117,96,159)(76,118,97,160)(77,119,98,161)(78,120,85,162)(79,121,86,163)(80,122,87,164)(81,123,88,165)(82,124,89,166)(83,125,90,167)(84,126,91,168) );
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)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,121),(10,122),(11,123),(12,124),(13,125),(14,126),(15,127),(16,128),(17,129),(18,130),(19,131),(20,132),(21,133),(22,134),(23,135),(24,136),(25,137),(26,138),(27,139),(28,140),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,161),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(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)], [(1,211,43,169),(2,212,44,170),(3,213,45,171),(4,214,46,172),(5,215,47,173),(6,216,48,174),(7,217,49,175),(8,218,50,176),(9,219,51,177),(10,220,52,178),(11,221,53,179),(12,222,54,180),(13,223,55,181),(14,224,56,182),(15,204,36,183),(16,205,37,184),(17,206,38,185),(18,207,39,186),(19,208,40,187),(20,209,41,188),(21,210,42,189),(22,197,29,190),(23,198,30,191),(24,199,31,192),(25,200,32,193),(26,201,33,194),(27,202,34,195),(28,203,35,196),(57,134,99,141),(58,135,100,142),(59,136,101,143),(60,137,102,144),(61,138,103,145),(62,139,104,146),(63,140,105,147),(64,127,106,148),(65,128,107,149),(66,129,108,150),(67,130,109,151),(68,131,110,152),(69,132,111,153),(70,133,112,154),(71,113,92,155),(72,114,93,156),(73,115,94,157),(74,116,95,158),(75,117,96,159),(76,118,97,160),(77,119,98,161),(78,120,85,162),(79,121,86,163),(80,122,87,164),(81,123,88,165),(82,124,89,166),(83,125,90,167),(84,126,91,168)]])
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 | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 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 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 7 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 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 | D8⋊3D7 | SD16⋊3D7 |
| kernel | D4⋊2D7⋊C4 | C14.Q16 | C28.44D4 | D4⋊Dic7 | C7×D4⋊C4 | C4⋊C4⋊7D7 | D7×C2×C8 | C2×D4⋊2D7 | D4⋊2D7 | C4×D7 | C2×Dic7 | C22×D7 | D4⋊C4 | C4⋊C4 | C2×C8 | C2×D4 | C14 | D4 | 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 D4⋊2D7⋊C4 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 98 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 34 | 112 | 0 | 0 |
| 2 | 103 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 103 | 0 | 0 |
| 12 | 102 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 69 |
| 0 | 0 | 18 | 0 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,15,0,0,0,0,98],[112,0,0,0,0,112,0,0,0,0,0,1,0,0,1,0],[34,2,0,0,112,103,0,0,0,0,1,0,0,0,0,1],[11,12,0,0,103,102,0,0,0,0,1,0,0,0,0,112],[15,0,0,0,0,15,0,0,0,0,0,18,0,0,69,0] >;
D4⋊2D7⋊C4 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_7\rtimes C_4
% in TeX
G:=Group("D4:2D7:C4"); // GroupNames label
G:=SmallGroup(448,306);
// by ID
G=gap.SmallGroup(448,306);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,219,58,570,136,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^7=d^2=e^4=1,b*a*b=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,e*b*e^-1=a*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations