metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C6×D5)⋊6D4, C3⋊4(C20⋊2D4), (C5×Dic3)⋊3D4, C6.157(D4×D5), D10⋊3(C3⋊D4), C15⋊17(C4⋊D4), D6⋊Dic5⋊28C2, C10.156(S3×D4), C30.229(C2×D4), C23.18(S3×D5), Dic3⋊2(C5⋊D4), C5⋊5(C23.14D6), C6.Dic10⋊33C2, (C2×Dic5).60D6, (C22×D5).62D6, (C22×C10).46D6, (C22×C6).30D10, C30.144(C4○D4), C30.38D4⋊22C2, C6.82(D4⋊2D5), (C2×C30).191C23, (C22×S3).27D10, C10.81(D4⋊2S3), (C2×Dic3).120D10, (C22×C30).53C22, C2.27(C30.C23), (C6×Dic5).110C22, (C10×Dic3).110C22, (C2×Dic15).131C22, (C6×C5⋊D4)⋊2C2, (C2×C3⋊D4)⋊1D5, (C2×C5⋊D4)⋊2S3, (C10×C3⋊D4)⋊1C2, (C2×D5×Dic3)⋊15C2, C6.59(C2×C5⋊D4), C2.39(D5×C3⋊D4), C2.37(S3×C5⋊D4), (C2×C15⋊D4)⋊11C2, C10.61(C2×C3⋊D4), (D5×C2×C6).49C22, C22.228(C2×S3×D5), (S3×C2×C10).47C22, (C2×C6).203(C22×D5), (C2×C10).203(C22×S3), SmallGroup(480,625)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C6×D5)⋊D4
G = < a,b,c,d,e | a6=b5=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3c, ede=d-1 >
Subgroups: 924 in 188 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, C30, C30, C4⋊D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, C6×D5, S3×C10, C2×C30, C2×C30, C4⋊Dic5, C23.D5, C2×C4×D5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23.14D6, D5×Dic3, C15⋊D4, C6×Dic5, C3×C5⋊D4, C10×Dic3, C5×C3⋊D4, C2×Dic15, D5×C2×C6, S3×C2×C10, C22×C30, C20⋊2D4, D6⋊Dic5, C6.Dic10, C30.38D4, C2×D5×Dic3, C2×C15⋊D4, C6×C5⋊D4, C10×C3⋊D4, (C6×D5)⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4⋊D4, C5⋊D4, C22×D5, S3×D4, D4⋊2S3, C2×C3⋊D4, S3×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C23.14D6, C2×S3×D5, C20⋊2D4, C30.C23, D5×C3⋊D4, S3×C5⋊D4, (C6×D5)⋊D4
►On 240 points
Generators in S240
(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)(223 224 225 226 227 228)(229 230 231 232 233 234)(235 236 237 238 239 240)
(1 59 51 43 19)(2 60 52 44 20)(3 55 53 45 21)(4 56 54 46 22)(5 57 49 47 23)(6 58 50 48 24)(7 35 41 208 219)(8 36 42 209 220)(9 31 37 210 221)(10 32 38 205 222)(11 33 39 206 217)(12 34 40 207 218)(13 66 94 67 29)(14 61 95 68 30)(15 62 96 69 25)(16 63 91 70 26)(17 64 92 71 27)(18 65 93 72 28)(73 107 83 90 117)(74 108 84 85 118)(75 103 79 86 119)(76 104 80 87 120)(77 105 81 88 115)(78 106 82 89 116)(97 154 132 126 110)(98 155 127 121 111)(99 156 128 122 112)(100 151 129 123 113)(101 152 130 124 114)(102 153 131 125 109)(133 165 169 143 179)(134 166 170 144 180)(135 167 171 139 175)(136 168 172 140 176)(137 163 173 141 177)(138 164 174 142 178)(145 162 211 187 185)(146 157 212 188 186)(147 158 213 189 181)(148 159 214 190 182)(149 160 215 191 183)(150 161 216 192 184)(193 225 229 203 239)(194 226 230 204 240)(195 227 231 199 235)(196 228 232 200 236)(197 223 233 201 237)(198 224 234 202 238)
(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(25 72)(26 67)(27 68)(28 69)(29 70)(30 71)(31 221)(32 222)(33 217)(34 218)(35 219)(36 220)(37 210)(38 205)(39 206)(40 207)(41 208)(42 209)(43 56)(44 57)(45 58)(46 59)(47 60)(48 55)(49 52)(50 53)(51 54)(61 64)(62 65)(63 66)(79 86)(80 87)(81 88)(82 89)(83 90)(84 85)(97 132)(98 127)(99 128)(100 129)(101 130)(102 131)(103 119)(104 120)(105 115)(106 116)(107 117)(108 118)(109 125)(110 126)(111 121)(112 122)(113 123)(114 124)(133 136)(134 137)(135 138)(139 174)(140 169)(141 170)(142 171)(143 172)(144 173)(145 182)(146 183)(147 184)(148 185)(149 186)(150 181)(157 191)(158 192)(159 187)(160 188)(161 189)(162 190)(163 180)(164 175)(165 176)(166 177)(167 178)(168 179)(199 231)(200 232)(201 233)(202 234)(203 229)(204 230)(211 214)(212 215)(213 216)(223 237)(224 238)(225 239)(226 240)(227 235)(228 236)
(1 149 28 139)(2 148 29 144)(3 147 30 143)(4 146 25 142)(5 145 26 141)(6 150 27 140)(7 156 194 76)(8 155 195 75)(9 154 196 74)(10 153 197 73)(11 152 198 78)(12 151 193 77)(13 180 60 159)(14 179 55 158)(15 178 56 157)(16 177 57 162)(17 176 58 161)(18 175 59 160)(19 183 72 171)(20 182 67 170)(21 181 68 169)(22 186 69 174)(23 185 70 173)(24 184 71 172)(31 132 228 108)(32 131 223 107)(33 130 224 106)(34 129 225 105)(35 128 226 104)(36 127 227 103)(37 126 232 84)(38 125 233 83)(39 124 234 82)(40 123 229 81)(41 122 230 80)(42 121 231 79)(43 191 93 167)(44 190 94 166)(45 189 95 165)(46 188 96 164)(47 187 91 163)(48 192 92 168)(49 211 63 137)(50 216 64 136)(51 215 65 135)(52 214 66 134)(53 213 61 133)(54 212 62 138)(85 210 110 200)(86 209 111 199)(87 208 112 204)(88 207 113 203)(89 206 114 202)(90 205 109 201)(97 236 118 221)(98 235 119 220)(99 240 120 219)(100 239 115 218)(101 238 116 217)(102 237 117 222)
(1 199)(2 204)(3 203)(4 202)(5 201)(6 200)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 219)(14 218)(15 217)(16 222)(17 221)(18 220)(19 231)(20 230)(21 229)(22 234)(23 233)(24 232)(25 206)(26 205)(27 210)(28 209)(29 208)(30 207)(31 92)(32 91)(33 96)(34 95)(35 94)(36 93)(37 71)(38 70)(39 69)(40 68)(41 67)(42 72)(43 227)(44 226)(45 225)(46 224)(47 223)(48 228)(49 197)(50 196)(51 195)(52 194)(53 193)(54 198)(55 239)(56 238)(57 237)(58 236)(59 235)(60 240)(73 137)(74 136)(75 135)(76 134)(77 133)(78 138)(79 171)(80 170)(81 169)(82 174)(83 173)(84 172)(85 140)(86 139)(87 144)(88 143)(89 142)(90 141)(97 161)(98 160)(99 159)(100 158)(101 157)(102 162)(103 167)(104 166)(105 165)(106 164)(107 163)(108 168)(109 145)(110 150)(111 149)(112 148)(113 147)(114 146)(115 179)(116 178)(117 177)(118 176)(119 175)(120 180)(121 183)(122 182)(123 181)(124 186)(125 185)(126 184)(127 191)(128 190)(129 189)(130 188)(131 187)(132 192)(151 213)(152 212)(153 211)(154 216)(155 215)(156 214)
G:=sub<Sym(240)| (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)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,59,51,43,19)(2,60,52,44,20)(3,55,53,45,21)(4,56,54,46,22)(5,57,49,47,23)(6,58,50,48,24)(7,35,41,208,219)(8,36,42,209,220)(9,31,37,210,221)(10,32,38,205,222)(11,33,39,206,217)(12,34,40,207,218)(13,66,94,67,29)(14,61,95,68,30)(15,62,96,69,25)(16,63,91,70,26)(17,64,92,71,27)(18,65,93,72,28)(73,107,83,90,117)(74,108,84,85,118)(75,103,79,86,119)(76,104,80,87,120)(77,105,81,88,115)(78,106,82,89,116)(97,154,132,126,110)(98,155,127,121,111)(99,156,128,122,112)(100,151,129,123,113)(101,152,130,124,114)(102,153,131,125,109)(133,165,169,143,179)(134,166,170,144,180)(135,167,171,139,175)(136,168,172,140,176)(137,163,173,141,177)(138,164,174,142,178)(145,162,211,187,185)(146,157,212,188,186)(147,158,213,189,181)(148,159,214,190,182)(149,160,215,191,183)(150,161,216,192,184)(193,225,229,203,239)(194,226,230,204,240)(195,227,231,199,235)(196,228,232,200,236)(197,223,233,201,237)(198,224,234,202,238), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(25,72)(26,67)(27,68)(28,69)(29,70)(30,71)(31,221)(32,222)(33,217)(34,218)(35,219)(36,220)(37,210)(38,205)(39,206)(40,207)(41,208)(42,209)(43,56)(44,57)(45,58)(46,59)(47,60)(48,55)(49,52)(50,53)(51,54)(61,64)(62,65)(63,66)(79,86)(80,87)(81,88)(82,89)(83,90)(84,85)(97,132)(98,127)(99,128)(100,129)(101,130)(102,131)(103,119)(104,120)(105,115)(106,116)(107,117)(108,118)(109,125)(110,126)(111,121)(112,122)(113,123)(114,124)(133,136)(134,137)(135,138)(139,174)(140,169)(141,170)(142,171)(143,172)(144,173)(145,182)(146,183)(147,184)(148,185)(149,186)(150,181)(157,191)(158,192)(159,187)(160,188)(161,189)(162,190)(163,180)(164,175)(165,176)(166,177)(167,178)(168,179)(199,231)(200,232)(201,233)(202,234)(203,229)(204,230)(211,214)(212,215)(213,216)(223,237)(224,238)(225,239)(226,240)(227,235)(228,236), (1,149,28,139)(2,148,29,144)(3,147,30,143)(4,146,25,142)(5,145,26,141)(6,150,27,140)(7,156,194,76)(8,155,195,75)(9,154,196,74)(10,153,197,73)(11,152,198,78)(12,151,193,77)(13,180,60,159)(14,179,55,158)(15,178,56,157)(16,177,57,162)(17,176,58,161)(18,175,59,160)(19,183,72,171)(20,182,67,170)(21,181,68,169)(22,186,69,174)(23,185,70,173)(24,184,71,172)(31,132,228,108)(32,131,223,107)(33,130,224,106)(34,129,225,105)(35,128,226,104)(36,127,227,103)(37,126,232,84)(38,125,233,83)(39,124,234,82)(40,123,229,81)(41,122,230,80)(42,121,231,79)(43,191,93,167)(44,190,94,166)(45,189,95,165)(46,188,96,164)(47,187,91,163)(48,192,92,168)(49,211,63,137)(50,216,64,136)(51,215,65,135)(52,214,66,134)(53,213,61,133)(54,212,62,138)(85,210,110,200)(86,209,111,199)(87,208,112,204)(88,207,113,203)(89,206,114,202)(90,205,109,201)(97,236,118,221)(98,235,119,220)(99,240,120,219)(100,239,115,218)(101,238,116,217)(102,237,117,222), (1,199)(2,204)(3,203)(4,202)(5,201)(6,200)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,219)(14,218)(15,217)(16,222)(17,221)(18,220)(19,231)(20,230)(21,229)(22,234)(23,233)(24,232)(25,206)(26,205)(27,210)(28,209)(29,208)(30,207)(31,92)(32,91)(33,96)(34,95)(35,94)(36,93)(37,71)(38,70)(39,69)(40,68)(41,67)(42,72)(43,227)(44,226)(45,225)(46,224)(47,223)(48,228)(49,197)(50,196)(51,195)(52,194)(53,193)(54,198)(55,239)(56,238)(57,237)(58,236)(59,235)(60,240)(73,137)(74,136)(75,135)(76,134)(77,133)(78,138)(79,171)(80,170)(81,169)(82,174)(83,173)(84,172)(85,140)(86,139)(87,144)(88,143)(89,142)(90,141)(97,161)(98,160)(99,159)(100,158)(101,157)(102,162)(103,167)(104,166)(105,165)(106,164)(107,163)(108,168)(109,145)(110,150)(111,149)(112,148)(113,147)(114,146)(115,179)(116,178)(117,177)(118,176)(119,175)(120,180)(121,183)(122,182)(123,181)(124,186)(125,185)(126,184)(127,191)(128,190)(129,189)(130,188)(131,187)(132,192)(151,213)(152,212)(153,211)(154,216)(155,215)(156,214)>;
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)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,59,51,43,19)(2,60,52,44,20)(3,55,53,45,21)(4,56,54,46,22)(5,57,49,47,23)(6,58,50,48,24)(7,35,41,208,219)(8,36,42,209,220)(9,31,37,210,221)(10,32,38,205,222)(11,33,39,206,217)(12,34,40,207,218)(13,66,94,67,29)(14,61,95,68,30)(15,62,96,69,25)(16,63,91,70,26)(17,64,92,71,27)(18,65,93,72,28)(73,107,83,90,117)(74,108,84,85,118)(75,103,79,86,119)(76,104,80,87,120)(77,105,81,88,115)(78,106,82,89,116)(97,154,132,126,110)(98,155,127,121,111)(99,156,128,122,112)(100,151,129,123,113)(101,152,130,124,114)(102,153,131,125,109)(133,165,169,143,179)(134,166,170,144,180)(135,167,171,139,175)(136,168,172,140,176)(137,163,173,141,177)(138,164,174,142,178)(145,162,211,187,185)(146,157,212,188,186)(147,158,213,189,181)(148,159,214,190,182)(149,160,215,191,183)(150,161,216,192,184)(193,225,229,203,239)(194,226,230,204,240)(195,227,231,199,235)(196,228,232,200,236)(197,223,233,201,237)(198,224,234,202,238), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(25,72)(26,67)(27,68)(28,69)(29,70)(30,71)(31,221)(32,222)(33,217)(34,218)(35,219)(36,220)(37,210)(38,205)(39,206)(40,207)(41,208)(42,209)(43,56)(44,57)(45,58)(46,59)(47,60)(48,55)(49,52)(50,53)(51,54)(61,64)(62,65)(63,66)(79,86)(80,87)(81,88)(82,89)(83,90)(84,85)(97,132)(98,127)(99,128)(100,129)(101,130)(102,131)(103,119)(104,120)(105,115)(106,116)(107,117)(108,118)(109,125)(110,126)(111,121)(112,122)(113,123)(114,124)(133,136)(134,137)(135,138)(139,174)(140,169)(141,170)(142,171)(143,172)(144,173)(145,182)(146,183)(147,184)(148,185)(149,186)(150,181)(157,191)(158,192)(159,187)(160,188)(161,189)(162,190)(163,180)(164,175)(165,176)(166,177)(167,178)(168,179)(199,231)(200,232)(201,233)(202,234)(203,229)(204,230)(211,214)(212,215)(213,216)(223,237)(224,238)(225,239)(226,240)(227,235)(228,236), (1,149,28,139)(2,148,29,144)(3,147,30,143)(4,146,25,142)(5,145,26,141)(6,150,27,140)(7,156,194,76)(8,155,195,75)(9,154,196,74)(10,153,197,73)(11,152,198,78)(12,151,193,77)(13,180,60,159)(14,179,55,158)(15,178,56,157)(16,177,57,162)(17,176,58,161)(18,175,59,160)(19,183,72,171)(20,182,67,170)(21,181,68,169)(22,186,69,174)(23,185,70,173)(24,184,71,172)(31,132,228,108)(32,131,223,107)(33,130,224,106)(34,129,225,105)(35,128,226,104)(36,127,227,103)(37,126,232,84)(38,125,233,83)(39,124,234,82)(40,123,229,81)(41,122,230,80)(42,121,231,79)(43,191,93,167)(44,190,94,166)(45,189,95,165)(46,188,96,164)(47,187,91,163)(48,192,92,168)(49,211,63,137)(50,216,64,136)(51,215,65,135)(52,214,66,134)(53,213,61,133)(54,212,62,138)(85,210,110,200)(86,209,111,199)(87,208,112,204)(88,207,113,203)(89,206,114,202)(90,205,109,201)(97,236,118,221)(98,235,119,220)(99,240,120,219)(100,239,115,218)(101,238,116,217)(102,237,117,222), (1,199)(2,204)(3,203)(4,202)(5,201)(6,200)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,219)(14,218)(15,217)(16,222)(17,221)(18,220)(19,231)(20,230)(21,229)(22,234)(23,233)(24,232)(25,206)(26,205)(27,210)(28,209)(29,208)(30,207)(31,92)(32,91)(33,96)(34,95)(35,94)(36,93)(37,71)(38,70)(39,69)(40,68)(41,67)(42,72)(43,227)(44,226)(45,225)(46,224)(47,223)(48,228)(49,197)(50,196)(51,195)(52,194)(53,193)(54,198)(55,239)(56,238)(57,237)(58,236)(59,235)(60,240)(73,137)(74,136)(75,135)(76,134)(77,133)(78,138)(79,171)(80,170)(81,169)(82,174)(83,173)(84,172)(85,140)(86,139)(87,144)(88,143)(89,142)(90,141)(97,161)(98,160)(99,159)(100,158)(101,157)(102,162)(103,167)(104,166)(105,165)(106,164)(107,163)(108,168)(109,145)(110,150)(111,149)(112,148)(113,147)(114,146)(115,179)(116,178)(117,177)(118,176)(119,175)(120,180)(121,183)(122,182)(123,181)(124,186)(125,185)(126,184)(127,191)(128,190)(129,189)(130,188)(131,187)(132,192)(151,213)(152,212)(153,211)(154,216)(155,215)(156,214) );
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),(223,224,225,226,227,228),(229,230,231,232,233,234),(235,236,237,238,239,240)], [(1,59,51,43,19),(2,60,52,44,20),(3,55,53,45,21),(4,56,54,46,22),(5,57,49,47,23),(6,58,50,48,24),(7,35,41,208,219),(8,36,42,209,220),(9,31,37,210,221),(10,32,38,205,222),(11,33,39,206,217),(12,34,40,207,218),(13,66,94,67,29),(14,61,95,68,30),(15,62,96,69,25),(16,63,91,70,26),(17,64,92,71,27),(18,65,93,72,28),(73,107,83,90,117),(74,108,84,85,118),(75,103,79,86,119),(76,104,80,87,120),(77,105,81,88,115),(78,106,82,89,116),(97,154,132,126,110),(98,155,127,121,111),(99,156,128,122,112),(100,151,129,123,113),(101,152,130,124,114),(102,153,131,125,109),(133,165,169,143,179),(134,166,170,144,180),(135,167,171,139,175),(136,168,172,140,176),(137,163,173,141,177),(138,164,174,142,178),(145,162,211,187,185),(146,157,212,188,186),(147,158,213,189,181),(148,159,214,190,182),(149,160,215,191,183),(150,161,216,192,184),(193,225,229,203,239),(194,226,230,204,240),(195,227,231,199,235),(196,228,232,200,236),(197,223,233,201,237),(198,224,234,202,238)], [(1,22),(2,23),(3,24),(4,19),(5,20),(6,21),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(25,72),(26,67),(27,68),(28,69),(29,70),(30,71),(31,221),(32,222),(33,217),(34,218),(35,219),(36,220),(37,210),(38,205),(39,206),(40,207),(41,208),(42,209),(43,56),(44,57),(45,58),(46,59),(47,60),(48,55),(49,52),(50,53),(51,54),(61,64),(62,65),(63,66),(79,86),(80,87),(81,88),(82,89),(83,90),(84,85),(97,132),(98,127),(99,128),(100,129),(101,130),(102,131),(103,119),(104,120),(105,115),(106,116),(107,117),(108,118),(109,125),(110,126),(111,121),(112,122),(113,123),(114,124),(133,136),(134,137),(135,138),(139,174),(140,169),(141,170),(142,171),(143,172),(144,173),(145,182),(146,183),(147,184),(148,185),(149,186),(150,181),(157,191),(158,192),(159,187),(160,188),(161,189),(162,190),(163,180),(164,175),(165,176),(166,177),(167,178),(168,179),(199,231),(200,232),(201,233),(202,234),(203,229),(204,230),(211,214),(212,215),(213,216),(223,237),(224,238),(225,239),(226,240),(227,235),(228,236)], [(1,149,28,139),(2,148,29,144),(3,147,30,143),(4,146,25,142),(5,145,26,141),(6,150,27,140),(7,156,194,76),(8,155,195,75),(9,154,196,74),(10,153,197,73),(11,152,198,78),(12,151,193,77),(13,180,60,159),(14,179,55,158),(15,178,56,157),(16,177,57,162),(17,176,58,161),(18,175,59,160),(19,183,72,171),(20,182,67,170),(21,181,68,169),(22,186,69,174),(23,185,70,173),(24,184,71,172),(31,132,228,108),(32,131,223,107),(33,130,224,106),(34,129,225,105),(35,128,226,104),(36,127,227,103),(37,126,232,84),(38,125,233,83),(39,124,234,82),(40,123,229,81),(41,122,230,80),(42,121,231,79),(43,191,93,167),(44,190,94,166),(45,189,95,165),(46,188,96,164),(47,187,91,163),(48,192,92,168),(49,211,63,137),(50,216,64,136),(51,215,65,135),(52,214,66,134),(53,213,61,133),(54,212,62,138),(85,210,110,200),(86,209,111,199),(87,208,112,204),(88,207,113,203),(89,206,114,202),(90,205,109,201),(97,236,118,221),(98,235,119,220),(99,240,120,219),(100,239,115,218),(101,238,116,217),(102,237,117,222)], [(1,199),(2,204),(3,203),(4,202),(5,201),(6,200),(7,66),(8,65),(9,64),(10,63),(11,62),(12,61),(13,219),(14,218),(15,217),(16,222),(17,221),(18,220),(19,231),(20,230),(21,229),(22,234),(23,233),(24,232),(25,206),(26,205),(27,210),(28,209),(29,208),(30,207),(31,92),(32,91),(33,96),(34,95),(35,94),(36,93),(37,71),(38,70),(39,69),(40,68),(41,67),(42,72),(43,227),(44,226),(45,225),(46,224),(47,223),(48,228),(49,197),(50,196),(51,195),(52,194),(53,193),(54,198),(55,239),(56,238),(57,237),(58,236),(59,235),(60,240),(73,137),(74,136),(75,135),(76,134),(77,133),(78,138),(79,171),(80,170),(81,169),(82,174),(83,173),(84,172),(85,140),(86,139),(87,144),(88,143),(89,142),(90,141),(97,161),(98,160),(99,159),(100,158),(101,157),(102,162),(103,167),(104,166),(105,165),(106,164),(107,163),(108,168),(109,145),(110,150),(111,149),(112,148),(113,147),(114,146),(115,179),(116,178),(117,177),(118,176),(119,175),(120,180),(121,183),(122,182),(123,181),(124,186),(125,185),(126,184),(127,191),(128,190),(129,189),(130,188),(131,187),(132,192),(151,213),(152,212),(153,211),(154,216),(155,215),(156,214)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 12 | 2 | 6 | 6 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
60 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 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C3⋊D4 | C5⋊D4 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | C30.C23 | D5×C3⋊D4 | S3×C5⋊D4 |
| kernel | (C6×D5)⋊D4 | D6⋊Dic5 | C6.Dic10 | C30.38D4 | C2×D5×Dic3 | C2×C15⋊D4 | C6×C5⋊D4 | C10×C3⋊D4 | C2×C5⋊D4 | C5×Dic3 | C6×D5 | C2×C3⋊D4 | C2×Dic5 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×S3 | C22×C6 | D10 | Dic3 | C10 | C10 | C23 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of (C6×D5)⋊D4 ►in GL6(𝔽61)
| 60 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 60 | 0 | 0 |
| 0 | 0 | 19 | 60 | 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 | 0 | 44 | 0 | 0 |
| 0 | 0 | 43 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 1 | 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 | 50 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 45 | 0 | 0 |
| 0 | 0 | 60 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 50 | 0 |
G:=sub<GL(6,GF(61))| [60,60,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,19,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,43,0,0,0,0,44,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1],[60,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,50,0,0,0,0,0,0,11],[1,0,0,0,0,0,60,60,0,0,0,0,0,0,30,60,0,0,0,0,45,31,0,0,0,0,0,0,0,50,0,0,0,0,11,0] >;
(C6×D5)⋊D4 in GAP, Magma, Sage, TeX
(C_6\times D_5)\rtimes D_4
% in TeX
G:=Group("(C6xD5):D4"); // GroupNames label
G:=SmallGroup(480,625);
// by ID
G=gap.SmallGroup(480,625);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^5=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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^3*c,e*d*e=d^-1>;
// generators/relations