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G = (C6×D5)⋊D4order 480 = 25·3·5

6th semidirect product of C6×D5 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C6×D5)⋊6D4, C34(C202D4), (C5×Dic3)⋊3D4, C6.157(D4×D5), D103(C3⋊D4), C1517(C4⋊D4), D6⋊Dic528C2, C10.156(S3×D4), C30.229(C2×D4), C23.18(S3×D5), Dic32(C5⋊D4), C55(C23.14D6), C6.Dic1033C2, (C2×Dic5).60D6, (C22×D5).62D6, (C22×C10).46D6, (C22×C6).30D10, C30.144(C4○D4), C30.38D422C2, C6.82(D42D5), (C2×C30).191C23, (C22×S3).27D10, C10.81(D42S3), (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

C1C2×C30 — (C6×D5)⋊D4
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — (C6×D5)⋊D4
C15C2×C30 — (C6×D5)⋊D4
C1C22C23

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 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3, C6 [×3], C6 [×3], C2×C4 [×6], D4 [×6], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], Dic3 [×2], C12, D6 [×3], C2×C6, C2×C6 [×7], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C2×Dic3, C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6, C22×C6, C5×S3, C3×D5 [×2], C30 [×3], C30, C4⋊D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C6×D5 [×2], C6×D5 [×2], S3×C10 [×3], C2×C30, C2×C30 [×3], C4⋊Dic5, C23.D5 [×2], C2×C4×D5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23.14D6, D5×Dic3 [×2], C15⋊D4 [×2], C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3, C5×C3⋊D4 [×2], C2×Dic15 [×2], D5×C2×C6, S3×C2×C10, C22×C30, C202D4, D6⋊Dic5, C6.Dic10, C30.38D4, C2×D5×Dic3, C2×C15⋊D4, C6×C5⋊D4, C10×C3⋊D4, (C6×D5)⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊D4, C5⋊D4 [×2], C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, D4×D5, D42D5, C2×C5⋊D4, C23.14D6, C2×S3×D5, C202D4, C30.C23, D5×C3⋊D4, S3×C5⋊D4, (C6×D5)⋊D4

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J10K10L10M10N12A12B15A15B20A20B20C20D30A···30N
order12222222344444455666666610···101010101010101010121215152020202030···30
size1111410101226620303060222224420202···2444412121212202044121212124···4

60 irreducible representations

dim111111112222222222222444444444
type+++++++++++++++++++-++-+-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6C4○D4D10D10D10C3⋊D4C5⋊D4S3×D4D42S3S3×D5D4×D5D42D5C2×S3×D5C30.C23D5×C3⋊D4S3×C5⋊D4
kernel(C6×D5)⋊D4D6⋊Dic5C6.Dic10C30.38D4C2×D5×Dic3C2×C15⋊D4C6×C5⋊D4C10×C3⋊D4C2×C5⋊D4C5×Dic3C6×D5C2×C3⋊D4C2×Dic5C22×D5C22×C10C30C2×Dic3C22×S3C22×C6D10Dic3C10C10C23C6C6C22C2C2C2
# reps111111111222111222248112222444

Matrix representation of (C6×D5)⋊D4 in GL6(𝔽61)

6010000
6000000
0060000
0006000
0000600
0000060
,
100000
010000
00186000
00196000
000010
000001
,
100000
010000
0004400
0043000
0000600
000001
,
6010000
010000
001000
000100
0000500
0000011
,
1600000
0600000
00304500
00603100
0000011
0000500

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

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