metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C7×D4)⋊14D4, (C7×Q8)⋊14D4, D4⋊6(C7⋊D4), C7⋊7(D4⋊D4), Q8⋊6(C7⋊D4), C28⋊7D4⋊26C2, C28.215(C2×D4), (C2×C28).452D4, C14.78C22≀C2, (C2×D4).204D14, D4⋊Dic7⋊41C2, Q8⋊Dic7⋊41C2, (C2×Q8).173D14, C14.112(C4○D8), C28.55D4⋊18C2, (C2×C28).484C23, (C22×C4).164D14, (C22×C14).115D4, C23.32(C7⋊D4), C2.23(D4⋊D14), C14.125(C8⋊C22), (C2×D28).133C22, (D4×C14).245C22, C2.12(C24⋊D7), C4⋊Dic7.189C22, (Q8×C14).208C22, C2.30(D4.8D14), (C22×C28).210C22, (C2×C4○D4)⋊1D7, (C2×D4⋊D7)⋊24C2, (C14×C4○D4)⋊1C2, (C2×Q8⋊D7)⋊24C2, C4.62(C2×C7⋊D4), (C2×C14).567(C2×D4), (C2×C7⋊C8).178C22, (C2×C4).225(C7⋊D4), (C2×C4).568(C22×D7), C22.224(C2×C7⋊D4), SmallGroup(448,772)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C7×D4)⋊14D4
G = < a,b,c,d,e | a7=b4=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >
Subgroups: 724 in 162 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C7⋊C8, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×C14, C22×C14, D4⋊D4, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, D4⋊D7, Q8⋊D7, C2×D28, C2×C7⋊D4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C28.55D4, D4⋊Dic7, Q8⋊Dic7, C28⋊7D4, C2×D4⋊D7, C2×Q8⋊D7, C14×C4○D4, (C7×D4)⋊14D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C4○D8, C8⋊C22, C7⋊D4, C22×D7, D4⋊D4, C2×C7⋊D4, D4⋊D14, D4.8D14, C24⋊D7, (C7×D4)⋊14D4
►On 224 points
Generators in S224
(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 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 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 141)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 204)(170 205)(171 206)(172 207)(173 208)(174 209)(175 210)(176 197)(177 198)(178 199)(179 200)(180 201)(181 202)(182 203)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(1 223 36 188)(2 222 37 187)(3 221 38 186)(4 220 39 185)(5 219 40 184)(6 218 41 183)(7 224 42 189)(8 216 29 195)(9 215 30 194)(10 214 31 193)(11 213 32 192)(12 212 33 191)(13 211 34 190)(14 217 35 196)(15 202 50 181)(16 201 51 180)(17 200 52 179)(18 199 53 178)(19 198 54 177)(20 197 55 176)(21 203 56 182)(22 209 43 174)(23 208 44 173)(24 207 45 172)(25 206 46 171)(26 205 47 170)(27 204 48 169)(28 210 49 175)(57 160 92 139)(58 159 93 138)(59 158 94 137)(60 157 95 136)(61 156 96 135)(62 155 97 134)(63 161 98 140)(64 167 85 132)(65 166 86 131)(66 165 87 130)(67 164 88 129)(68 163 89 128)(69 162 90 127)(70 168 91 133)(71 153 106 118)(72 152 107 117)(73 151 108 116)(74 150 109 115)(75 149 110 114)(76 148 111 113)(77 154 112 119)(78 146 99 125)(79 145 100 124)(80 144 101 123)(81 143 102 122)(82 142 103 121)(83 141 104 120)(84 147 105 126)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 119)(8 125)(9 124)(10 123)(11 122)(12 121)(13 120)(14 126)(15 132)(16 131)(17 130)(18 129)(19 128)(20 127)(21 133)(22 139)(23 138)(24 137)(25 136)(26 135)(27 134)(28 140)(29 146)(30 145)(31 144)(32 143)(33 142)(34 141)(35 147)(36 153)(37 152)(38 151)(39 150)(40 149)(41 148)(42 154)(43 160)(44 159)(45 158)(46 157)(47 156)(48 155)(49 161)(50 167)(51 166)(52 165)(53 164)(54 163)(55 162)(56 168)(57 174)(58 173)(59 172)(60 171)(61 170)(62 169)(63 175)(64 181)(65 180)(66 179)(67 178)(68 177)(69 176)(70 182)(71 188)(72 187)(73 186)(74 185)(75 184)(76 183)(77 189)(78 195)(79 194)(80 193)(81 192)(82 191)(83 190)(84 196)(85 202)(86 201)(87 200)(88 199)(89 198)(90 197)(91 203)(92 209)(93 208)(94 207)(95 206)(96 205)(97 204)(98 210)(99 216)(100 215)(101 214)(102 213)(103 212)(104 211)(105 217)(106 223)(107 222)(108 221)(109 220)(110 219)(111 218)(112 224)
G:=sub<Sym(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), (1,22,8,15)(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), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,223,36,188)(2,222,37,187)(3,221,38,186)(4,220,39,185)(5,219,40,184)(6,218,41,183)(7,224,42,189)(8,216,29,195)(9,215,30,194)(10,214,31,193)(11,213,32,192)(12,212,33,191)(13,211,34,190)(14,217,35,196)(15,202,50,181)(16,201,51,180)(17,200,52,179)(18,199,53,178)(19,198,54,177)(20,197,55,176)(21,203,56,182)(22,209,43,174)(23,208,44,173)(24,207,45,172)(25,206,46,171)(26,205,47,170)(27,204,48,169)(28,210,49,175)(57,160,92,139)(58,159,93,138)(59,158,94,137)(60,157,95,136)(61,156,96,135)(62,155,97,134)(63,161,98,140)(64,167,85,132)(65,166,86,131)(66,165,87,130)(67,164,88,129)(68,163,89,128)(69,162,90,127)(70,168,91,133)(71,153,106,118)(72,152,107,117)(73,151,108,116)(74,150,109,115)(75,149,110,114)(76,148,111,113)(77,154,112,119)(78,146,99,125)(79,145,100,124)(80,144,101,123)(81,143,102,122)(82,142,103,121)(83,141,104,120)(84,147,105,126), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,119)(8,125)(9,124)(10,123)(11,122)(12,121)(13,120)(14,126)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,133)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,140)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,147)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,154)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,161)(50,167)(51,166)(52,165)(53,164)(54,163)(55,162)(56,168)(57,174)(58,173)(59,172)(60,171)(61,170)(62,169)(63,175)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,182)(71,188)(72,187)(73,186)(74,185)(75,184)(76,183)(77,189)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,196)(85,202)(86,201)(87,200)(88,199)(89,198)(90,197)(91,203)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,210)(99,216)(100,215)(101,214)(102,213)(103,212)(104,211)(105,217)(106,223)(107,222)(108,221)(109,220)(110,219)(111,218)(112,224)>;
G:=Group( (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), (1,22,8,15)(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), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,223,36,188)(2,222,37,187)(3,221,38,186)(4,220,39,185)(5,219,40,184)(6,218,41,183)(7,224,42,189)(8,216,29,195)(9,215,30,194)(10,214,31,193)(11,213,32,192)(12,212,33,191)(13,211,34,190)(14,217,35,196)(15,202,50,181)(16,201,51,180)(17,200,52,179)(18,199,53,178)(19,198,54,177)(20,197,55,176)(21,203,56,182)(22,209,43,174)(23,208,44,173)(24,207,45,172)(25,206,46,171)(26,205,47,170)(27,204,48,169)(28,210,49,175)(57,160,92,139)(58,159,93,138)(59,158,94,137)(60,157,95,136)(61,156,96,135)(62,155,97,134)(63,161,98,140)(64,167,85,132)(65,166,86,131)(66,165,87,130)(67,164,88,129)(68,163,89,128)(69,162,90,127)(70,168,91,133)(71,153,106,118)(72,152,107,117)(73,151,108,116)(74,150,109,115)(75,149,110,114)(76,148,111,113)(77,154,112,119)(78,146,99,125)(79,145,100,124)(80,144,101,123)(81,143,102,122)(82,142,103,121)(83,141,104,120)(84,147,105,126), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,119)(8,125)(9,124)(10,123)(11,122)(12,121)(13,120)(14,126)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,133)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,140)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,147)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,154)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,161)(50,167)(51,166)(52,165)(53,164)(54,163)(55,162)(56,168)(57,174)(58,173)(59,172)(60,171)(61,170)(62,169)(63,175)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,182)(71,188)(72,187)(73,186)(74,185)(75,184)(76,183)(77,189)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,196)(85,202)(86,201)(87,200)(88,199)(89,198)(90,197)(91,203)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,210)(99,216)(100,215)(101,214)(102,213)(103,212)(104,211)(105,217)(106,223)(107,222)(108,221)(109,220)(110,219)(111,218)(112,224) );
G=PermutationGroup([[(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)], [(1,22,8,15),(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)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(113,148),(114,149),(115,150),(116,151),(117,152),(118,153),(119,154),(120,141),(121,142),(122,143),(123,144),(124,145),(125,146),(126,147),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,204),(170,205),(171,206),(172,207),(173,208),(174,209),(175,210),(176,197),(177,198),(178,199),(179,200),(180,201),(181,202),(182,203),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(1,223,36,188),(2,222,37,187),(3,221,38,186),(4,220,39,185),(5,219,40,184),(6,218,41,183),(7,224,42,189),(8,216,29,195),(9,215,30,194),(10,214,31,193),(11,213,32,192),(12,212,33,191),(13,211,34,190),(14,217,35,196),(15,202,50,181),(16,201,51,180),(17,200,52,179),(18,199,53,178),(19,198,54,177),(20,197,55,176),(21,203,56,182),(22,209,43,174),(23,208,44,173),(24,207,45,172),(25,206,46,171),(26,205,47,170),(27,204,48,169),(28,210,49,175),(57,160,92,139),(58,159,93,138),(59,158,94,137),(60,157,95,136),(61,156,96,135),(62,155,97,134),(63,161,98,140),(64,167,85,132),(65,166,86,131),(66,165,87,130),(67,164,88,129),(68,163,89,128),(69,162,90,127),(70,168,91,133),(71,153,106,118),(72,152,107,117),(73,151,108,116),(74,150,109,115),(75,149,110,114),(76,148,111,113),(77,154,112,119),(78,146,99,125),(79,145,100,124),(80,144,101,123),(81,143,102,122),(82,142,103,121),(83,141,104,120),(84,147,105,126)], [(1,118),(2,117),(3,116),(4,115),(5,114),(6,113),(7,119),(8,125),(9,124),(10,123),(11,122),(12,121),(13,120),(14,126),(15,132),(16,131),(17,130),(18,129),(19,128),(20,127),(21,133),(22,139),(23,138),(24,137),(25,136),(26,135),(27,134),(28,140),(29,146),(30,145),(31,144),(32,143),(33,142),(34,141),(35,147),(36,153),(37,152),(38,151),(39,150),(40,149),(41,148),(42,154),(43,160),(44,159),(45,158),(46,157),(47,156),(48,155),(49,161),(50,167),(51,166),(52,165),(53,164),(54,163),(55,162),(56,168),(57,174),(58,173),(59,172),(60,171),(61,170),(62,169),(63,175),(64,181),(65,180),(66,179),(67,178),(68,177),(69,176),(70,182),(71,188),(72,187),(73,186),(74,185),(75,184),(76,183),(77,189),(78,195),(79,194),(80,193),(81,192),(82,191),(83,190),(84,196),(85,202),(86,201),(87,200),(88,199),(89,198),(90,197),(91,203),(92,209),(93,208),(94,207),(95,206),(96,205),(97,204),(98,210),(99,216),(100,215),(101,214),(102,213),(103,212),(104,211),(105,217),(106,223),(107,222),(108,221),(109,220),(110,219),(111,218),(112,224)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 56 | 2 | 2 | 2 | 2 | 4 | 4 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C7⋊D4 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4⋊D14 | D4.8D14 |
| kernel | (C7×D4)⋊14D4 | C28.55D4 | D4⋊Dic7 | Q8⋊Dic7 | C28⋊7D4 | C2×D4⋊D7 | C2×Q8⋊D7 | C14×C4○D4 | C2×C28 | C7×D4 | C7×Q8 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C14 | C2×C4 | D4 | Q8 | C23 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 6 | 12 | 12 | 6 | 1 | 6 | 6 |
Matrix representation of (C7×D4)⋊14D4 ►in GL4(𝔽113) generated by
| 16 | 0 | 0 | 0 |
| 44 | 106 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 30 |
| 0 | 0 | 15 | 1 |
| 1 | 0 | 0 | 0 |
| 107 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 98 | 112 |
| 58 | 57 | 0 | 0 |
| 50 | 55 | 0 | 0 |
| 0 | 0 | 87 | 51 |
| 0 | 0 | 82 | 26 |
| 58 | 57 | 0 | 0 |
| 54 | 55 | 0 | 0 |
| 0 | 0 | 0 | 26 |
| 0 | 0 | 100 | 0 |
G:=sub<GL(4,GF(113))| [16,44,0,0,0,106,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,112,15,0,0,30,1],[1,107,0,0,0,112,0,0,0,0,1,98,0,0,0,112],[58,50,0,0,57,55,0,0,0,0,87,82,0,0,51,26],[58,54,0,0,57,55,0,0,0,0,0,100,0,0,26,0] >;
(C7×D4)⋊14D4 in GAP, Magma, Sage, TeX
(C_7\times D_4)\rtimes_{14}D_4 % in TeX
G:=Group("(C7xD4):14D4"); // GroupNames label
G:=SmallGroup(448,772);
// by ID
G=gap.SmallGroup(448,772);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,184,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations