metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.65D14, C23.28D28, (C2×C28)⋊35D4, (C23×C4)⋊1D7, (C23×C28)⋊1C2, C14.85(C4×D4), C23.38(C4×D7), C14.70C22≀C2, C22⋊2(D14⋊C4), C2.5(C28⋊7D4), C22.61(C2×D28), C14.80(C4⋊D4), (C22×C14).193D4, (C22×C4).408D14, C2.2(C24⋊D7), C7⋊4(C23.23D4), C23.83(C7⋊D4), C14.C42⋊25C2, C22.64(C4○D28), (C23×D7).24C22, C23.304(C22×D7), (C23×C14).100C22, (C22×C14).364C23, (C22×C28).485C22, C14.69(C22.D4), C2.5(C23.23D14), (C22×Dic7).67C22, (C2×C7⋊D4)⋊7C4, C2.29(C4×C7⋊D4), (C2×D14⋊C4)⋊11C2, (C2×C4)⋊15(C7⋊D4), C2.36(C2×D14⋊C4), (C2×Dic7)⋊9(C2×C4), (C22×D7)⋊5(C2×C4), (C2×C23.D7)⋊7C2, C22.150(C2×C4×D7), (C2×C14)⋊5(C22⋊C4), (C2×C14).550(C2×D4), C14.65(C2×C22⋊C4), (C22×C7⋊D4).7C2, C22.88(C2×C7⋊D4), (C2×C14).92(C4○D4), (C22×C14).100(C2×C4), (C2×C14).143(C22×C4), SmallGroup(448,747)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.65D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=b, f2=db=bd, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de13 >
Subgroups: 1284 in 286 conjugacy classes, 87 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×8], C22 [×3], C22 [×8], C22 [×22], C7, C2×C4 [×4], C2×C4 [×22], D4 [×8], C23, C23 [×6], C23 [×12], D7 [×2], C14 [×3], C14 [×4], C14 [×4], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×9], C2×D4 [×8], C24, C24, Dic7 [×4], C28 [×4], D14 [×10], C2×C14 [×3], C2×C14 [×8], C2×C14 [×12], C2.C42 [×2], C2×C22⋊C4 [×3], C23×C4, C22×D4, C2×Dic7 [×2], C2×Dic7 [×8], C7⋊D4 [×8], C2×C28 [×4], C2×C28 [×12], C22×D7 [×2], C22×D7 [×6], C22×C14, C22×C14 [×6], C22×C14 [×4], C23.23D4, D14⋊C4 [×4], C23.D7 [×2], C22×Dic7, C22×Dic7 [×2], C2×C7⋊D4 [×4], C2×C7⋊D4 [×4], C22×C28 [×2], C22×C28 [×6], C23×D7, C23×C14, C14.C42 [×2], C2×D14⋊C4 [×2], C2×C23.D7, C22×C7⋊D4, C23×C28, C24.65D14
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D7, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D14 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C4×D7 [×2], D28 [×2], C7⋊D4 [×6], C22×D7, C23.23D4, D14⋊C4 [×4], C2×C4×D7, C2×D28, C4○D28 [×2], C2×C7⋊D4 [×3], C2×D14⋊C4, C4×C7⋊D4 [×2], C23.23D14, C28⋊7D4 [×2], C24⋊D7, C24.65D14
►On 224 points
Generators in S224
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)(57 220)(58 221)(59 222)(60 223)(61 224)(62 197)(63 198)(64 199)(65 200)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 208)(74 209)(75 210)(76 211)(77 212)(78 213)(79 214)(80 215)(81 216)(82 217)(83 218)(84 219)(85 136)(86 137)(87 138)(88 139)(89 140)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(141 182)(142 183)(143 184)(144 185)(145 186)(146 187)(147 188)(148 189)(149 190)(150 191)(151 192)(152 193)(153 194)(154 195)(155 196)(156 169)(157 170)(158 171)(159 172)(160 173)(161 174)(162 175)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 220)(30 221)(31 222)(32 223)(33 224)(34 197)(35 198)(36 199)(37 200)(38 201)(39 202)(40 203)(41 204)(42 205)(43 206)(44 207)(45 208)(46 209)(47 210)(48 211)(49 212)(50 213)(51 214)(52 215)(53 216)(54 217)(55 218)(56 219)(85 148)(86 149)(87 150)(88 151)(89 152)(90 153)(91 154)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 161)(99 162)(100 163)(101 164)(102 165)(103 166)(104 167)(105 168)(106 141)(107 142)(108 143)(109 144)(110 145)(111 146)(112 147)(113 194)(114 195)(115 196)(116 169)(117 170)(118 171)(119 172)(120 173)(121 174)(122 175)(123 176)(124 177)(125 178)(126 179)(127 180)(128 181)(129 182)(130 183)(131 184)(132 185)(133 186)(134 187)(135 188)(136 189)(137 190)(138 191)(139 192)(140 193)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 206)(58 207)(59 208)(60 209)(61 210)(62 211)(63 212)(64 213)(65 214)(66 215)(67 216)(68 217)(69 218)(70 219)(71 220)(72 221)(73 222)(74 223)(75 224)(76 197)(77 198)(78 199)(79 200)(80 201)(81 202)(82 203)(83 204)(84 205)(85 175)(86 176)(87 177)(88 178)(89 179)(90 180)(91 181)(92 182)(93 183)(94 184)(95 185)(96 186)(97 187)(98 188)(99 189)(100 190)(101 191)(102 192)(103 193)(104 194)(105 195)(106 196)(107 169)(108 170)(109 171)(110 172)(111 173)(112 174)(113 167)(114 168)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 147)(122 148)(123 149)(124 150)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)(131 157)(132 158)(133 159)(134 160)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)
(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)
(1 135 36 147)(2 146 37 134)(3 133 38 145)(4 144 39 132)(5 131 40 143)(6 142 41 130)(7 129 42 141)(8 168 43 128)(9 127 44 167)(10 166 45 126)(11 125 46 165)(12 164 47 124)(13 123 48 163)(14 162 49 122)(15 121 50 161)(16 160 51 120)(17 119 52 159)(18 158 53 118)(19 117 54 157)(20 156 55 116)(21 115 56 155)(22 154 29 114)(23 113 30 153)(24 152 31 140)(25 139 32 151)(26 150 33 138)(27 137 34 149)(28 148 35 136)(57 91 220 195)(58 194 221 90)(59 89 222 193)(60 192 223 88)(61 87 224 191)(62 190 197 86)(63 85 198 189)(64 188 199 112)(65 111 200 187)(66 186 201 110)(67 109 202 185)(68 184 203 108)(69 107 204 183)(70 182 205 106)(71 105 206 181)(72 180 207 104)(73 103 208 179)(74 178 209 102)(75 101 210 177)(76 176 211 100)(77 99 212 175)(78 174 213 98)(79 97 214 173)(80 172 215 96)(81 95 216 171)(82 170 217 94)(83 93 218 169)(84 196 219 92)
G:=sub<Sym(224)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(57,220)(58,221)(59,222)(60,223)(61,224)(62,197)(63,198)(64,199)(65,200)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(85,136)(86,137)(87,138)(88,139)(89,140)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(141,182)(142,183)(143,184)(144,185)(145,186)(146,187)(147,188)(148,189)(149,190)(150,191)(151,192)(152,193)(153,194)(154,195)(155,196)(156,169)(157,170)(158,171)(159,172)(160,173)(161,174)(162,175)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,220)(30,221)(31,222)(32,223)(33,224)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,209)(47,210)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)(112,147)(113,194)(114,195)(115,196)(116,169)(117,170)(118,171)(119,172)(120,173)(121,174)(122,175)(123,176)(124,177)(125,178)(126,179)(127,180)(128,181)(129,182)(130,183)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,206)(58,207)(59,208)(60,209)(61,210)(62,211)(63,212)(64,213)(65,214)(66,215)(67,216)(68,217)(69,218)(70,219)(71,220)(72,221)(73,222)(74,223)(75,224)(76,197)(77,198)(78,199)(79,200)(80,201)(81,202)(82,203)(83,204)(84,205)(85,175)(86,176)(87,177)(88,178)(89,179)(90,180)(91,181)(92,182)(93,183)(94,184)(95,185)(96,186)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,193)(104,194)(105,195)(106,196)(107,169)(108,170)(109,171)(110,172)(111,173)(112,174)(113,167)(114,168)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166), (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), (1,135,36,147)(2,146,37,134)(3,133,38,145)(4,144,39,132)(5,131,40,143)(6,142,41,130)(7,129,42,141)(8,168,43,128)(9,127,44,167)(10,166,45,126)(11,125,46,165)(12,164,47,124)(13,123,48,163)(14,162,49,122)(15,121,50,161)(16,160,51,120)(17,119,52,159)(18,158,53,118)(19,117,54,157)(20,156,55,116)(21,115,56,155)(22,154,29,114)(23,113,30,153)(24,152,31,140)(25,139,32,151)(26,150,33,138)(27,137,34,149)(28,148,35,136)(57,91,220,195)(58,194,221,90)(59,89,222,193)(60,192,223,88)(61,87,224,191)(62,190,197,86)(63,85,198,189)(64,188,199,112)(65,111,200,187)(66,186,201,110)(67,109,202,185)(68,184,203,108)(69,107,204,183)(70,182,205,106)(71,105,206,181)(72,180,207,104)(73,103,208,179)(74,178,209,102)(75,101,210,177)(76,176,211,100)(77,99,212,175)(78,174,213,98)(79,97,214,173)(80,172,215,96)(81,95,216,171)(82,170,217,94)(83,93,218,169)(84,196,219,92)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(57,220)(58,221)(59,222)(60,223)(61,224)(62,197)(63,198)(64,199)(65,200)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(85,136)(86,137)(87,138)(88,139)(89,140)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(141,182)(142,183)(143,184)(144,185)(145,186)(146,187)(147,188)(148,189)(149,190)(150,191)(151,192)(152,193)(153,194)(154,195)(155,196)(156,169)(157,170)(158,171)(159,172)(160,173)(161,174)(162,175)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,220)(30,221)(31,222)(32,223)(33,224)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,209)(47,210)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)(112,147)(113,194)(114,195)(115,196)(116,169)(117,170)(118,171)(119,172)(120,173)(121,174)(122,175)(123,176)(124,177)(125,178)(126,179)(127,180)(128,181)(129,182)(130,183)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,206)(58,207)(59,208)(60,209)(61,210)(62,211)(63,212)(64,213)(65,214)(66,215)(67,216)(68,217)(69,218)(70,219)(71,220)(72,221)(73,222)(74,223)(75,224)(76,197)(77,198)(78,199)(79,200)(80,201)(81,202)(82,203)(83,204)(84,205)(85,175)(86,176)(87,177)(88,178)(89,179)(90,180)(91,181)(92,182)(93,183)(94,184)(95,185)(96,186)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,193)(104,194)(105,195)(106,196)(107,169)(108,170)(109,171)(110,172)(111,173)(112,174)(113,167)(114,168)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166), (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), (1,135,36,147)(2,146,37,134)(3,133,38,145)(4,144,39,132)(5,131,40,143)(6,142,41,130)(7,129,42,141)(8,168,43,128)(9,127,44,167)(10,166,45,126)(11,125,46,165)(12,164,47,124)(13,123,48,163)(14,162,49,122)(15,121,50,161)(16,160,51,120)(17,119,52,159)(18,158,53,118)(19,117,54,157)(20,156,55,116)(21,115,56,155)(22,154,29,114)(23,113,30,153)(24,152,31,140)(25,139,32,151)(26,150,33,138)(27,137,34,149)(28,148,35,136)(57,91,220,195)(58,194,221,90)(59,89,222,193)(60,192,223,88)(61,87,224,191)(62,190,197,86)(63,85,198,189)(64,188,199,112)(65,111,200,187)(66,186,201,110)(67,109,202,185)(68,184,203,108)(69,107,204,183)(70,182,205,106)(71,105,206,181)(72,180,207,104)(73,103,208,179)(74,178,209,102)(75,101,210,177)(76,176,211,100)(77,99,212,175)(78,174,213,98)(79,97,214,173)(80,172,215,96)(81,95,216,171)(82,170,217,94)(83,93,218,169)(84,196,219,92) );
G=PermutationGroup([(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(28,35),(57,220),(58,221),(59,222),(60,223),(61,224),(62,197),(63,198),(64,199),(65,200),(66,201),(67,202),(68,203),(69,204),(70,205),(71,206),(72,207),(73,208),(74,209),(75,210),(76,211),(77,212),(78,213),(79,214),(80,215),(81,216),(82,217),(83,218),(84,219),(85,136),(86,137),(87,138),(88,139),(89,140),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131),(109,132),(110,133),(111,134),(112,135),(141,182),(142,183),(143,184),(144,185),(145,186),(146,187),(147,188),(148,189),(149,190),(150,191),(151,192),(152,193),(153,194),(154,195),(155,196),(156,169),(157,170),(158,171),(159,172),(160,173),(161,174),(162,175),(163,176),(164,177),(165,178),(166,179),(167,180),(168,181)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,220),(30,221),(31,222),(32,223),(33,224),(34,197),(35,198),(36,199),(37,200),(38,201),(39,202),(40,203),(41,204),(42,205),(43,206),(44,207),(45,208),(46,209),(47,210),(48,211),(49,212),(50,213),(51,214),(52,215),(53,216),(54,217),(55,218),(56,219),(85,148),(86,149),(87,150),(88,151),(89,152),(90,153),(91,154),(92,155),(93,156),(94,157),(95,158),(96,159),(97,160),(98,161),(99,162),(100,163),(101,164),(102,165),(103,166),(104,167),(105,168),(106,141),(107,142),(108,143),(109,144),(110,145),(111,146),(112,147),(113,194),(114,195),(115,196),(116,169),(117,170),(118,171),(119,172),(120,173),(121,174),(122,175),(123,176),(124,177),(125,178),(126,179),(127,180),(128,181),(129,182),(130,183),(131,184),(132,185),(133,186),(134,187),(135,188),(136,189),(137,190),(138,191),(139,192),(140,193)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,206),(58,207),(59,208),(60,209),(61,210),(62,211),(63,212),(64,213),(65,214),(66,215),(67,216),(68,217),(69,218),(70,219),(71,220),(72,221),(73,222),(74,223),(75,224),(76,197),(77,198),(78,199),(79,200),(80,201),(81,202),(82,203),(83,204),(84,205),(85,175),(86,176),(87,177),(88,178),(89,179),(90,180),(91,181),(92,182),(93,183),(94,184),(95,185),(96,186),(97,187),(98,188),(99,189),(100,190),(101,191),(102,192),(103,193),(104,194),(105,195),(106,196),(107,169),(108,170),(109,171),(110,172),(111,173),(112,174),(113,167),(114,168),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146),(121,147),(122,148),(123,149),(124,150),(125,151),(126,152),(127,153),(128,154),(129,155),(130,156),(131,157),(132,158),(133,159),(134,160),(135,161),(136,162),(137,163),(138,164),(139,165),(140,166)], [(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)], [(1,135,36,147),(2,146,37,134),(3,133,38,145),(4,144,39,132),(5,131,40,143),(6,142,41,130),(7,129,42,141),(8,168,43,128),(9,127,44,167),(10,166,45,126),(11,125,46,165),(12,164,47,124),(13,123,48,163),(14,162,49,122),(15,121,50,161),(16,160,51,120),(17,119,52,159),(18,158,53,118),(19,117,54,157),(20,156,55,116),(21,115,56,155),(22,154,29,114),(23,113,30,153),(24,152,31,140),(25,139,32,151),(26,150,33,138),(27,137,34,149),(28,148,35,136),(57,91,220,195),(58,194,221,90),(59,89,222,193),(60,192,223,88),(61,87,224,191),(62,190,197,86),(63,85,198,189),(64,188,199,112),(65,111,200,187),(66,186,201,110),(67,109,202,185),(68,184,203,108),(69,107,204,183),(70,182,205,106),(71,105,206,181),(72,180,207,104),(73,103,208,179),(74,178,209,102),(75,101,210,177),(76,176,211,100),(77,99,212,175),(78,174,213,98),(79,97,214,173),(80,172,215,96),(81,95,216,171),(82,170,217,94),(83,93,218,169),(84,196,219,92)])
124 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14AS | 28A | ··· | 28AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C4×D7 | D28 | C7⋊D4 | C4○D28 |
| kernel | C24.65D14 | C14.C42 | C2×D14⋊C4 | C2×C23.D7 | C22×C7⋊D4 | C23×C28 | C2×C7⋊D4 | C2×C28 | C22×C14 | C23×C4 | C2×C14 | C22×C4 | C24 | C2×C4 | C23 | C23 | C23 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 3 | 4 | 6 | 3 | 24 | 12 | 12 | 12 | 24 |
Matrix representation of C24.65D14 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 12 |
| 0 | 0 | 0 | 0 | 7 | 9 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 16 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 28 |
| 0 | 0 | 0 | 0 | 22 | 20 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,25,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[4,0,0,0,0,0,0,22,0,0,0,0,0,0,19,3,0,0,0,0,0,3,0,0,0,0,0,0,3,7,0,0,0,0,12,9],[0,4,0,0,0,0,22,0,0,0,0,0,0,0,26,3,0,0,0,0,16,3,0,0,0,0,0,0,9,22,0,0,0,0,28,20] >;
C24.65D14 in GAP, Magma, Sage, TeX
C_2^4._{65}D_{14} % in TeX
G:=Group("C2^4.65D14"); // GroupNames label
G:=SmallGroup(448,747);
// by ID
G=gap.SmallGroup(448,747);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,758,58,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=b,f^2=d*b=b*d,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^13>;
// generators/relations