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G = C6.D4⋊D5order 480 = 25·3·5

7th semidirect product of C6.D4 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.68(C2×D20), (C2×C6).47D20, (C2×C30).77D4, C30.226(C2×D4), C6.D47D5, D304C427C2, C23.39(S3×D5), C6.Dic1031C2, (C22×Dic5)⋊7S3, (C22×C10).43D6, (C22×C6).89D10, C10.80(C4○D12), C30.142(C4○D4), C6.53(D42D5), (C2×C30).188C23, (C2×Dic5).192D6, (C2×Dic3).58D10, C34(C22.D20), C22.6(C3⋊D20), C53(C23.28D6), C1521(C22.D4), (C22×C30).50C22, C2.25(Dic3.D10), (C6×Dic5).221C22, (C22×D15).61C22, (C2×Dic15).129C22, (C10×Dic3).108C22, (C2×C6×Dic5)⋊4C2, C10.22(C2×C3⋊D4), C2.23(C2×C3⋊D20), C22.226(C2×S3×D5), (C5×C6.D4)⋊7C2, (C2×C157D4).10C2, (C2×C10).17(C3⋊D4), (C2×C6).200(C22×D5), (C2×C10).200(C22×S3), SmallGroup(480,622)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C6.D4⋊D5
C1C5C15C30C2×C30C6×Dic5C6.Dic10 — C6.D4⋊D5
C15C2×C30 — C6.D4⋊D5
C1C22C23

Generators and relations for C6.D4⋊D5
 G = < a,b,c,d,e | a6=b4=d5=e2=1, c2=a3, bab-1=cac-1=eae=a-1, ad=da, cbc-1=a3b-1, bd=db, ebe=a3b, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 828 in 156 conjugacy classes, 52 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C5, S3, C6, C6 [×2], C6 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10, C10 [×2], C10 [×2], Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×4], C22×S3, C22×C6, D15, C30, C30 [×2], C30 [×2], C22.D4, C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4, C2×C3⋊D4, C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, D30 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C23.28D6, C6×Dic5 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C157D4 [×2], C22×D15, C22×C30, C22.D20, D304C4 [×2], C6.Dic10 [×2], C5×C6.D4, C2×C6×Dic5, C2×C157D4, C6.D4⋊D5
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C22.D4, D20 [×2], C22×D5, C4○D12 [×2], C2×C3⋊D4, S3×D5, C2×D20, D42D5 [×2], C23.28D6, C3⋊D20 [×2], C2×S3×D5, C22.D20, Dic3.D10 [×2], C2×C3⋊D20, C6.D4⋊D5

Smallest permutation representation of C6.D4⋊D5
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)
(1 116 13 104)(2 115 14 103)(3 120 15 108)(4 119 16 107)(5 118 17 106)(6 117 18 105)(7 133 232 216)(8 138 233 215)(9 137 234 214)(10 136 229 213)(11 135 230 212)(12 134 231 211)(19 139 90 78)(20 144 85 77)(21 143 86 76)(22 142 87 75)(23 141 88 74)(24 140 89 73)(25 149 80 72)(26 148 81 71)(27 147 82 70)(28 146 83 69)(29 145 84 68)(30 150 79 67)(31 153 91 111)(32 152 92 110)(33 151 93 109)(34 156 94 114)(35 155 95 113)(36 154 96 112)(37 161 97 64)(38 160 98 63)(39 159 99 62)(40 158 100 61)(41 157 101 66)(42 162 102 65)(43 173 198 210)(44 172 193 209)(45 171 194 208)(46 170 195 207)(47 169 196 206)(48 174 197 205)(49 121 188 204)(50 126 189 203)(51 125 190 202)(52 124 191 201)(53 123 192 200)(54 122 187 199)(55 131 184 217)(56 130 185 222)(57 129 186 221)(58 128 181 220)(59 127 182 219)(60 132 183 218)(163 227 175 239)(164 226 176 238)(165 225 177 237)(166 224 178 236)(167 223 179 235)(168 228 180 240)
(1 107 4 104)(2 106 5 103)(3 105 6 108)(7 133 10 136)(8 138 11 135)(9 137 12 134)(13 119 16 116)(14 118 17 115)(15 117 18 120)(19 75 22 78)(20 74 23 77)(21 73 24 76)(25 69 28 72)(26 68 29 71)(27 67 30 70)(31 114 34 111)(32 113 35 110)(33 112 36 109)(37 61 40 64)(38 66 41 63)(39 65 42 62)(43 173 46 170)(44 172 47 169)(45 171 48 174)(49 121 52 124)(50 126 53 123)(51 125 54 122)(55 131 58 128)(56 130 59 127)(57 129 60 132)(79 147 82 150)(80 146 83 149)(81 145 84 148)(85 141 88 144)(86 140 89 143)(87 139 90 142)(91 156 94 153)(92 155 95 152)(93 154 96 151)(97 158 100 161)(98 157 101 160)(99 162 102 159)(163 236 166 239)(164 235 167 238)(165 240 168 237)(175 224 178 227)(176 223 179 226)(177 228 180 225)(181 220 184 217)(182 219 185 222)(183 218 186 221)(187 199 190 202)(188 204 191 201)(189 203 192 200)(193 209 196 206)(194 208 197 205)(195 207 198 210)(211 234 214 231)(212 233 215 230)(213 232 216 229)
(1 101 88 95 29)(2 102 89 96 30)(3 97 90 91 25)(4 98 85 92 26)(5 99 86 93 27)(6 100 87 94 28)(7 44 56 223 187)(8 45 57 224 188)(9 46 58 225 189)(10 47 59 226 190)(11 48 60 227 191)(12 43 55 228 192)(13 41 23 35 84)(14 42 24 36 79)(15 37 19 31 80)(16 38 20 32 81)(17 39 21 33 82)(18 40 22 34 83)(49 233 194 186 236)(50 234 195 181 237)(51 229 196 182 238)(52 230 197 183 239)(53 231 198 184 240)(54 232 193 185 235)(61 75 114 146 117)(62 76 109 147 118)(63 77 110 148 119)(64 78 111 149 120)(65 73 112 150 115)(66 74 113 145 116)(67 103 162 140 154)(68 104 157 141 155)(69 105 158 142 156)(70 106 159 143 151)(71 107 160 144 152)(72 108 161 139 153)(121 215 208 221 166)(122 216 209 222 167)(123 211 210 217 168)(124 212 205 218 163)(125 213 206 219 164)(126 214 207 220 165)(127 176 202 136 169)(128 177 203 137 170)(129 178 204 138 171)(130 179 199 133 172)(131 180 200 134 173)(132 175 201 135 174)
(1 57)(2 56)(3 55)(4 60)(5 59)(6 58)(7 89)(8 88)(9 87)(10 86)(11 85)(12 90)(13 186)(14 185)(15 184)(16 183)(17 182)(18 181)(19 231)(20 230)(21 229)(22 234)(23 233)(24 232)(25 228)(26 227)(27 226)(28 225)(29 224)(30 223)(31 53)(32 52)(33 51)(34 50)(35 49)(36 54)(37 198)(38 197)(39 196)(40 195)(41 194)(42 193)(43 97)(44 102)(45 101)(46 100)(47 99)(48 98)(61 173)(62 172)(63 171)(64 170)(65 169)(66 174)(67 164)(68 163)(69 168)(70 167)(71 166)(72 165)(73 136)(74 135)(75 134)(76 133)(77 138)(78 137)(79 235)(80 240)(81 239)(82 238)(83 237)(84 236)(91 192)(92 191)(93 190)(94 189)(95 188)(96 187)(103 219)(104 218)(105 217)(106 222)(107 221)(108 220)(109 199)(110 204)(111 203)(112 202)(113 201)(114 200)(115 127)(116 132)(117 131)(118 130)(119 129)(120 128)(121 152)(122 151)(123 156)(124 155)(125 154)(126 153)(139 214)(140 213)(141 212)(142 211)(143 216)(144 215)(145 175)(146 180)(147 179)(148 178)(149 177)(150 176)(157 205)(158 210)(159 209)(160 208)(161 207)(162 206)

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), (1,116,13,104)(2,115,14,103)(3,120,15,108)(4,119,16,107)(5,118,17,106)(6,117,18,105)(7,133,232,216)(8,138,233,215)(9,137,234,214)(10,136,229,213)(11,135,230,212)(12,134,231,211)(19,139,90,78)(20,144,85,77)(21,143,86,76)(22,142,87,75)(23,141,88,74)(24,140,89,73)(25,149,80,72)(26,148,81,71)(27,147,82,70)(28,146,83,69)(29,145,84,68)(30,150,79,67)(31,153,91,111)(32,152,92,110)(33,151,93,109)(34,156,94,114)(35,155,95,113)(36,154,96,112)(37,161,97,64)(38,160,98,63)(39,159,99,62)(40,158,100,61)(41,157,101,66)(42,162,102,65)(43,173,198,210)(44,172,193,209)(45,171,194,208)(46,170,195,207)(47,169,196,206)(48,174,197,205)(49,121,188,204)(50,126,189,203)(51,125,190,202)(52,124,191,201)(53,123,192,200)(54,122,187,199)(55,131,184,217)(56,130,185,222)(57,129,186,221)(58,128,181,220)(59,127,182,219)(60,132,183,218)(163,227,175,239)(164,226,176,238)(165,225,177,237)(166,224,178,236)(167,223,179,235)(168,228,180,240), (1,107,4,104)(2,106,5,103)(3,105,6,108)(7,133,10,136)(8,138,11,135)(9,137,12,134)(13,119,16,116)(14,118,17,115)(15,117,18,120)(19,75,22,78)(20,74,23,77)(21,73,24,76)(25,69,28,72)(26,68,29,71)(27,67,30,70)(31,114,34,111)(32,113,35,110)(33,112,36,109)(37,61,40,64)(38,66,41,63)(39,65,42,62)(43,173,46,170)(44,172,47,169)(45,171,48,174)(49,121,52,124)(50,126,53,123)(51,125,54,122)(55,131,58,128)(56,130,59,127)(57,129,60,132)(79,147,82,150)(80,146,83,149)(81,145,84,148)(85,141,88,144)(86,140,89,143)(87,139,90,142)(91,156,94,153)(92,155,95,152)(93,154,96,151)(97,158,100,161)(98,157,101,160)(99,162,102,159)(163,236,166,239)(164,235,167,238)(165,240,168,237)(175,224,178,227)(176,223,179,226)(177,228,180,225)(181,220,184,217)(182,219,185,222)(183,218,186,221)(187,199,190,202)(188,204,191,201)(189,203,192,200)(193,209,196,206)(194,208,197,205)(195,207,198,210)(211,234,214,231)(212,233,215,230)(213,232,216,229), (1,101,88,95,29)(2,102,89,96,30)(3,97,90,91,25)(4,98,85,92,26)(5,99,86,93,27)(6,100,87,94,28)(7,44,56,223,187)(8,45,57,224,188)(9,46,58,225,189)(10,47,59,226,190)(11,48,60,227,191)(12,43,55,228,192)(13,41,23,35,84)(14,42,24,36,79)(15,37,19,31,80)(16,38,20,32,81)(17,39,21,33,82)(18,40,22,34,83)(49,233,194,186,236)(50,234,195,181,237)(51,229,196,182,238)(52,230,197,183,239)(53,231,198,184,240)(54,232,193,185,235)(61,75,114,146,117)(62,76,109,147,118)(63,77,110,148,119)(64,78,111,149,120)(65,73,112,150,115)(66,74,113,145,116)(67,103,162,140,154)(68,104,157,141,155)(69,105,158,142,156)(70,106,159,143,151)(71,107,160,144,152)(72,108,161,139,153)(121,215,208,221,166)(122,216,209,222,167)(123,211,210,217,168)(124,212,205,218,163)(125,213,206,219,164)(126,214,207,220,165)(127,176,202,136,169)(128,177,203,137,170)(129,178,204,138,171)(130,179,199,133,172)(131,180,200,134,173)(132,175,201,135,174), (1,57)(2,56)(3,55)(4,60)(5,59)(6,58)(7,89)(8,88)(9,87)(10,86)(11,85)(12,90)(13,186)(14,185)(15,184)(16,183)(17,182)(18,181)(19,231)(20,230)(21,229)(22,234)(23,233)(24,232)(25,228)(26,227)(27,226)(28,225)(29,224)(30,223)(31,53)(32,52)(33,51)(34,50)(35,49)(36,54)(37,198)(38,197)(39,196)(40,195)(41,194)(42,193)(43,97)(44,102)(45,101)(46,100)(47,99)(48,98)(61,173)(62,172)(63,171)(64,170)(65,169)(66,174)(67,164)(68,163)(69,168)(70,167)(71,166)(72,165)(73,136)(74,135)(75,134)(76,133)(77,138)(78,137)(79,235)(80,240)(81,239)(82,238)(83,237)(84,236)(91,192)(92,191)(93,190)(94,189)(95,188)(96,187)(103,219)(104,218)(105,217)(106,222)(107,221)(108,220)(109,199)(110,204)(111,203)(112,202)(113,201)(114,200)(115,127)(116,132)(117,131)(118,130)(119,129)(120,128)(121,152)(122,151)(123,156)(124,155)(125,154)(126,153)(139,214)(140,213)(141,212)(142,211)(143,216)(144,215)(145,175)(146,180)(147,179)(148,178)(149,177)(150,176)(157,205)(158,210)(159,209)(160,208)(161,207)(162,206)>;

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), (1,116,13,104)(2,115,14,103)(3,120,15,108)(4,119,16,107)(5,118,17,106)(6,117,18,105)(7,133,232,216)(8,138,233,215)(9,137,234,214)(10,136,229,213)(11,135,230,212)(12,134,231,211)(19,139,90,78)(20,144,85,77)(21,143,86,76)(22,142,87,75)(23,141,88,74)(24,140,89,73)(25,149,80,72)(26,148,81,71)(27,147,82,70)(28,146,83,69)(29,145,84,68)(30,150,79,67)(31,153,91,111)(32,152,92,110)(33,151,93,109)(34,156,94,114)(35,155,95,113)(36,154,96,112)(37,161,97,64)(38,160,98,63)(39,159,99,62)(40,158,100,61)(41,157,101,66)(42,162,102,65)(43,173,198,210)(44,172,193,209)(45,171,194,208)(46,170,195,207)(47,169,196,206)(48,174,197,205)(49,121,188,204)(50,126,189,203)(51,125,190,202)(52,124,191,201)(53,123,192,200)(54,122,187,199)(55,131,184,217)(56,130,185,222)(57,129,186,221)(58,128,181,220)(59,127,182,219)(60,132,183,218)(163,227,175,239)(164,226,176,238)(165,225,177,237)(166,224,178,236)(167,223,179,235)(168,228,180,240), (1,107,4,104)(2,106,5,103)(3,105,6,108)(7,133,10,136)(8,138,11,135)(9,137,12,134)(13,119,16,116)(14,118,17,115)(15,117,18,120)(19,75,22,78)(20,74,23,77)(21,73,24,76)(25,69,28,72)(26,68,29,71)(27,67,30,70)(31,114,34,111)(32,113,35,110)(33,112,36,109)(37,61,40,64)(38,66,41,63)(39,65,42,62)(43,173,46,170)(44,172,47,169)(45,171,48,174)(49,121,52,124)(50,126,53,123)(51,125,54,122)(55,131,58,128)(56,130,59,127)(57,129,60,132)(79,147,82,150)(80,146,83,149)(81,145,84,148)(85,141,88,144)(86,140,89,143)(87,139,90,142)(91,156,94,153)(92,155,95,152)(93,154,96,151)(97,158,100,161)(98,157,101,160)(99,162,102,159)(163,236,166,239)(164,235,167,238)(165,240,168,237)(175,224,178,227)(176,223,179,226)(177,228,180,225)(181,220,184,217)(182,219,185,222)(183,218,186,221)(187,199,190,202)(188,204,191,201)(189,203,192,200)(193,209,196,206)(194,208,197,205)(195,207,198,210)(211,234,214,231)(212,233,215,230)(213,232,216,229), (1,101,88,95,29)(2,102,89,96,30)(3,97,90,91,25)(4,98,85,92,26)(5,99,86,93,27)(6,100,87,94,28)(7,44,56,223,187)(8,45,57,224,188)(9,46,58,225,189)(10,47,59,226,190)(11,48,60,227,191)(12,43,55,228,192)(13,41,23,35,84)(14,42,24,36,79)(15,37,19,31,80)(16,38,20,32,81)(17,39,21,33,82)(18,40,22,34,83)(49,233,194,186,236)(50,234,195,181,237)(51,229,196,182,238)(52,230,197,183,239)(53,231,198,184,240)(54,232,193,185,235)(61,75,114,146,117)(62,76,109,147,118)(63,77,110,148,119)(64,78,111,149,120)(65,73,112,150,115)(66,74,113,145,116)(67,103,162,140,154)(68,104,157,141,155)(69,105,158,142,156)(70,106,159,143,151)(71,107,160,144,152)(72,108,161,139,153)(121,215,208,221,166)(122,216,209,222,167)(123,211,210,217,168)(124,212,205,218,163)(125,213,206,219,164)(126,214,207,220,165)(127,176,202,136,169)(128,177,203,137,170)(129,178,204,138,171)(130,179,199,133,172)(131,180,200,134,173)(132,175,201,135,174), (1,57)(2,56)(3,55)(4,60)(5,59)(6,58)(7,89)(8,88)(9,87)(10,86)(11,85)(12,90)(13,186)(14,185)(15,184)(16,183)(17,182)(18,181)(19,231)(20,230)(21,229)(22,234)(23,233)(24,232)(25,228)(26,227)(27,226)(28,225)(29,224)(30,223)(31,53)(32,52)(33,51)(34,50)(35,49)(36,54)(37,198)(38,197)(39,196)(40,195)(41,194)(42,193)(43,97)(44,102)(45,101)(46,100)(47,99)(48,98)(61,173)(62,172)(63,171)(64,170)(65,169)(66,174)(67,164)(68,163)(69,168)(70,167)(71,166)(72,165)(73,136)(74,135)(75,134)(76,133)(77,138)(78,137)(79,235)(80,240)(81,239)(82,238)(83,237)(84,236)(91,192)(92,191)(93,190)(94,189)(95,188)(96,187)(103,219)(104,218)(105,217)(106,222)(107,221)(108,220)(109,199)(110,204)(111,203)(112,202)(113,201)(114,200)(115,127)(116,132)(117,131)(118,130)(119,129)(120,128)(121,152)(122,151)(123,156)(124,155)(125,154)(126,153)(139,214)(140,213)(141,212)(142,211)(143,216)(144,215)(145,175)(146,180)(147,179)(148,178)(149,177)(150,176)(157,205)(158,210)(159,209)(160,208)(161,207)(162,206) );

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)], [(1,116,13,104),(2,115,14,103),(3,120,15,108),(4,119,16,107),(5,118,17,106),(6,117,18,105),(7,133,232,216),(8,138,233,215),(9,137,234,214),(10,136,229,213),(11,135,230,212),(12,134,231,211),(19,139,90,78),(20,144,85,77),(21,143,86,76),(22,142,87,75),(23,141,88,74),(24,140,89,73),(25,149,80,72),(26,148,81,71),(27,147,82,70),(28,146,83,69),(29,145,84,68),(30,150,79,67),(31,153,91,111),(32,152,92,110),(33,151,93,109),(34,156,94,114),(35,155,95,113),(36,154,96,112),(37,161,97,64),(38,160,98,63),(39,159,99,62),(40,158,100,61),(41,157,101,66),(42,162,102,65),(43,173,198,210),(44,172,193,209),(45,171,194,208),(46,170,195,207),(47,169,196,206),(48,174,197,205),(49,121,188,204),(50,126,189,203),(51,125,190,202),(52,124,191,201),(53,123,192,200),(54,122,187,199),(55,131,184,217),(56,130,185,222),(57,129,186,221),(58,128,181,220),(59,127,182,219),(60,132,183,218),(163,227,175,239),(164,226,176,238),(165,225,177,237),(166,224,178,236),(167,223,179,235),(168,228,180,240)], [(1,107,4,104),(2,106,5,103),(3,105,6,108),(7,133,10,136),(8,138,11,135),(9,137,12,134),(13,119,16,116),(14,118,17,115),(15,117,18,120),(19,75,22,78),(20,74,23,77),(21,73,24,76),(25,69,28,72),(26,68,29,71),(27,67,30,70),(31,114,34,111),(32,113,35,110),(33,112,36,109),(37,61,40,64),(38,66,41,63),(39,65,42,62),(43,173,46,170),(44,172,47,169),(45,171,48,174),(49,121,52,124),(50,126,53,123),(51,125,54,122),(55,131,58,128),(56,130,59,127),(57,129,60,132),(79,147,82,150),(80,146,83,149),(81,145,84,148),(85,141,88,144),(86,140,89,143),(87,139,90,142),(91,156,94,153),(92,155,95,152),(93,154,96,151),(97,158,100,161),(98,157,101,160),(99,162,102,159),(163,236,166,239),(164,235,167,238),(165,240,168,237),(175,224,178,227),(176,223,179,226),(177,228,180,225),(181,220,184,217),(182,219,185,222),(183,218,186,221),(187,199,190,202),(188,204,191,201),(189,203,192,200),(193,209,196,206),(194,208,197,205),(195,207,198,210),(211,234,214,231),(212,233,215,230),(213,232,216,229)], [(1,101,88,95,29),(2,102,89,96,30),(3,97,90,91,25),(4,98,85,92,26),(5,99,86,93,27),(6,100,87,94,28),(7,44,56,223,187),(8,45,57,224,188),(9,46,58,225,189),(10,47,59,226,190),(11,48,60,227,191),(12,43,55,228,192),(13,41,23,35,84),(14,42,24,36,79),(15,37,19,31,80),(16,38,20,32,81),(17,39,21,33,82),(18,40,22,34,83),(49,233,194,186,236),(50,234,195,181,237),(51,229,196,182,238),(52,230,197,183,239),(53,231,198,184,240),(54,232,193,185,235),(61,75,114,146,117),(62,76,109,147,118),(63,77,110,148,119),(64,78,111,149,120),(65,73,112,150,115),(66,74,113,145,116),(67,103,162,140,154),(68,104,157,141,155),(69,105,158,142,156),(70,106,159,143,151),(71,107,160,144,152),(72,108,161,139,153),(121,215,208,221,166),(122,216,209,222,167),(123,211,210,217,168),(124,212,205,218,163),(125,213,206,219,164),(126,214,207,220,165),(127,176,202,136,169),(128,177,203,137,170),(129,178,204,138,171),(130,179,199,133,172),(131,180,200,134,173),(132,175,201,135,174)], [(1,57),(2,56),(3,55),(4,60),(5,59),(6,58),(7,89),(8,88),(9,87),(10,86),(11,85),(12,90),(13,186),(14,185),(15,184),(16,183),(17,182),(18,181),(19,231),(20,230),(21,229),(22,234),(23,233),(24,232),(25,228),(26,227),(27,226),(28,225),(29,224),(30,223),(31,53),(32,52),(33,51),(34,50),(35,49),(36,54),(37,198),(38,197),(39,196),(40,195),(41,194),(42,193),(43,97),(44,102),(45,101),(46,100),(47,99),(48,98),(61,173),(62,172),(63,171),(64,170),(65,169),(66,174),(67,164),(68,163),(69,168),(70,167),(71,166),(72,165),(73,136),(74,135),(75,134),(76,133),(77,138),(78,137),(79,235),(80,240),(81,239),(82,238),(83,237),(84,236),(91,192),(92,191),(93,190),(94,189),(95,188),(96,187),(103,219),(104,218),(105,217),(106,222),(107,221),(108,220),(109,199),(110,204),(111,203),(112,202),(113,201),(114,200),(115,127),(116,132),(117,131),(118,130),(119,129),(120,128),(121,152),(122,151),(123,156),(124,155),(125,154),(126,153),(139,214),(140,213),(141,212),(142,211),(143,216),(144,215),(145,175),(146,180),(147,179),(148,178),(149,177),(150,176),(157,205),(158,210),(159,209),(160,208),(161,207),(162,206)])

66 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A···6G10A···10F10G10H10I10J12A···12H15A15B20A···20H30A···30N
order122222234444444556···610···101010101012···12151520···2030···30
size11112260210101010121260222···22···2444410···104412···124···4

66 irreducible representations

dim1111112222222222244444
type+++++++++++++++-++
imageC1C2C2C2C2C2S3D4D5D6D6C4○D4D10D10C3⋊D4D20C4○D12S3×D5D42D5C3⋊D20C2×S3×D5Dic3.D10
kernelC6.D4⋊D5D304C4C6.Dic10C5×C6.D4C2×C6×Dic5C2×C157D4C22×Dic5C2×C30C6.D4C2×Dic5C22×C10C30C2×Dic3C22×C6C2×C10C2×C6C10C23C6C22C22C2
# reps1221111222144248824428

Matrix representation of C6.D4⋊D5 in GL4(𝔽61) generated by

14000
04800
00600
00060
,
0100
1000
002957
005832
,
0100
60000
002957
005832
,
1000
0100
00601
001644
,
05000
11000
0086
002053
G:=sub<GL(4,GF(61))| [14,0,0,0,0,48,0,0,0,0,60,0,0,0,0,60],[0,1,0,0,1,0,0,0,0,0,29,58,0,0,57,32],[0,60,0,0,1,0,0,0,0,0,29,58,0,0,57,32],[1,0,0,0,0,1,0,0,0,0,60,16,0,0,1,44],[0,11,0,0,50,0,0,0,0,0,8,20,0,0,6,53] >;

C6.D4⋊D5 in GAP, Magma, Sage, TeX

C_6.D_4\rtimes D_5
% in TeX

G:=Group("C6.D4:D5");
// GroupNames label

G:=SmallGroup(480,622);
// by ID

G=gap.SmallGroup(480,622);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,64,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=d^5=e^2=1,c^2=a^3,b*a*b^-1=c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=a^3*b^-1,b*d=d*b,e*b*e=a^3*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽