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G = C3×D10.12D4order 480 = 25·3·5

Direct product of C3 and D10.12D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3×D10.12D4, C4⋊Dic54C6, (C6×D5).69D4, C6.172(D4×D5), C10.18(C6×D4), C23.D54C6, D10⋊C45C6, C23.4(C6×D5), D10.11(C3×D4), C30.331(C2×D4), (C22×C6).4D10, C10.D410C6, (C2×C12).273D10, C30.187(C4○D4), C6.115(C4○D20), (C2×C60).263C22, (C2×C30).341C23, C6.110(D42D5), C1527(C22.D4), (C22×C30).99C22, (C6×Dic5).155C22, C2.8(C3×D4×D5), (C2×C4×D5)⋊10C6, (D5×C2×C12)⋊26C2, (C5×C22⋊C4)⋊5C6, C22⋊C43(C3×D5), C10.7(C3×C4○D4), (C2×C4).24(C6×D5), (C2×C5⋊D4).3C6, C22.42(D5×C2×C6), (C2×C20).50(C2×C6), (C3×C4⋊Dic5)⋊22C2, (C3×C22⋊C4)⋊11D5, C2.10(C3×C4○D20), C2.8(C3×D42D5), (C6×C5⋊D4).10C2, (C15×C22⋊C4)⋊14C2, C51(C3×C22.D4), (C3×C23.D5)⋊20C2, (C2×Dic5).7(C2×C6), (C3×D10⋊C4)⋊16C2, (D5×C2×C6).126C22, (C3×C10.D4)⋊26C2, (C22×C10).18(C2×C6), (C2×C10).24(C22×C6), (C22×D5).23(C2×C6), (C2×C6).337(C22×D5), SmallGroup(480,676)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×D10.12D4
C1C5C10C2×C10C2×C30D5×C2×C6D5×C2×C12 — C3×D10.12D4
C5C2×C10 — C3×D10.12D4
C1C2×C6C3×C22⋊C4

Generators and relations for C3×D10.12D4
 G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=b5d-1 >

Subgroups: 512 in 156 conjugacy classes, 62 normal (58 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, C6 [×3], C6 [×3], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, D5 [×2], C10 [×3], C10, C12 [×5], C2×C6, C2×C6 [×7], C15, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C2×C12 [×2], C2×C12 [×5], C3×D4 [×2], C22×C6, C22×C6, C3×D5 [×2], C30 [×3], C30, C22.D4, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C3×C22⋊C4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C22×C12, C6×D4, C3×Dic5 [×3], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×3], C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C3×C22.D4, D5×C12 [×2], C6×Dic5 [×3], C3×C5⋊D4 [×2], C2×C60 [×2], D5×C2×C6, C22×C30, D10.12D4, C3×C10.D4, C3×C4⋊Dic5, C3×D10⋊C4, C3×C23.D5, C15×C22⋊C4, D5×C2×C12, C6×C5⋊D4, C3×D10.12D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, C4○D4 [×2], D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C22.D4, C22×D5, C6×D4, C3×C4○D4 [×2], C6×D5 [×3], C4○D20, D4×D5, D42D5, C3×C22.D4, D5×C2×C6, D10.12D4, C3×C4○D20, C3×D4×D5, C3×D42D5, C3×D10.12D4

Smallest permutation representation of C3×D10.12D4
On 240 points
Generators in S240
(1 52 32)(2 53 33)(3 54 34)(4 55 35)(5 56 36)(6 57 37)(7 58 38)(8 59 39)(9 60 40)(10 51 31)(11 225 205)(12 226 206)(13 227 207)(14 228 208)(15 229 209)(16 230 210)(17 221 201)(18 222 202)(19 223 203)(20 224 204)(21 61 41)(22 62 42)(23 63 43)(24 64 44)(25 65 45)(26 66 46)(27 67 47)(28 68 48)(29 69 49)(30 70 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 121 101)(82 122 102)(83 123 103)(84 124 104)(85 125 105)(86 126 106)(87 127 107)(88 128 108)(89 129 109)(90 130 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)
(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 24)(2 23)(3 22)(4 21)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 236)(12 235)(13 234)(14 233)(15 232)(16 231)(17 240)(18 239)(19 238)(20 237)(31 45)(32 44)(33 43)(34 42)(35 41)(36 50)(37 49)(38 48)(39 47)(40 46)(51 65)(52 64)(53 63)(54 62)(55 61)(56 70)(57 69)(58 68)(59 67)(60 66)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(111 130)(112 129)(113 128)(114 127)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)(131 145)(132 144)(133 143)(134 142)(135 141)(136 150)(137 149)(138 148)(139 147)(140 146)(151 165)(152 164)(153 163)(154 162)(155 161)(156 170)(157 169)(158 168)(159 167)(160 166)(171 185)(172 184)(173 183)(174 182)(175 181)(176 190)(177 189)(178 188)(179 187)(180 186)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(211 230)(212 229)(213 228)(214 227)(215 226)(216 225)(217 224)(218 223)(219 222)(220 221)
(1 210 30 192)(2 201 21 193)(3 202 22 194)(4 203 23 195)(5 204 24 196)(6 205 25 197)(7 206 26 198)(8 207 27 199)(9 208 28 200)(10 209 29 191)(11 65 237 57)(12 66 238 58)(13 67 239 59)(14 68 240 60)(15 69 231 51)(16 70 232 52)(17 61 233 53)(18 62 234 54)(19 63 235 55)(20 64 236 56)(31 229 49 211)(32 230 50 212)(33 221 41 213)(34 222 42 214)(35 223 43 215)(36 224 44 216)(37 225 45 217)(38 226 46 218)(39 227 47 219)(40 228 48 220)(71 149 89 131)(72 150 90 132)(73 141 81 133)(74 142 82 134)(75 143 83 135)(76 144 84 136)(77 145 85 137)(78 146 86 138)(79 147 87 139)(80 148 88 140)(91 169 109 151)(92 170 110 152)(93 161 101 153)(94 162 102 154)(95 163 103 155)(96 164 104 156)(97 165 105 157)(98 166 106 158)(99 167 107 159)(100 168 108 160)(111 189 129 171)(112 190 130 172)(113 181 121 173)(114 182 122 174)(115 183 123 175)(116 184 124 176)(117 185 125 177)(118 186 126 178)(119 187 127 179)(120 188 128 180)
(1 137 6 132)(2 138 7 133)(3 139 8 134)(4 140 9 135)(5 131 10 136)(11 125 16 130)(12 126 17 121)(13 127 18 122)(14 128 19 123)(15 129 20 124)(21 146 26 141)(22 147 27 142)(23 148 28 143)(24 149 29 144)(25 150 30 145)(31 156 36 151)(32 157 37 152)(33 158 38 153)(34 159 39 154)(35 160 40 155)(41 166 46 161)(42 167 47 162)(43 168 48 163)(44 169 49 164)(45 170 50 165)(51 176 56 171)(52 177 57 172)(53 178 58 173)(54 179 59 174)(55 180 60 175)(61 186 66 181)(62 187 67 182)(63 188 68 183)(64 189 69 184)(65 190 70 185)(71 196 76 191)(72 197 77 192)(73 198 78 193)(74 199 79 194)(75 200 80 195)(81 206 86 201)(82 207 87 202)(83 208 88 203)(84 209 89 204)(85 210 90 205)(91 216 96 211)(92 217 97 212)(93 218 98 213)(94 219 99 214)(95 220 100 215)(101 226 106 221)(102 227 107 222)(103 228 108 223)(104 229 109 224)(105 230 110 225)(111 236 116 231)(112 237 117 232)(113 238 118 233)(114 239 119 234)(115 240 120 235)

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

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

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

102 conjugacy classes

class 1 2A2B2C2D2E2F3A3B4A4B4C4D4E4F4G5A5B6A···6F6G6H6I6J6K6L10A···10F10G10H10I10J12A12B12C12D12E12F12G12H12I12J12K12L12M12N15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order1222222334444444556···666666610···101010101012121212121212121212121212121515151520···2030···3030···3060···60
size1111410101122410102020221···144101010102···24444222244101010102020202022224···42···24···44···4

102 irreducible representations

dim11111111111111112222222222224444
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6D4D5C4○D4D10D10C3×D4C3×D5C3×C4○D4C6×D5C6×D5C4○D20C3×C4○D20D4×D5D42D5C3×D4×D5C3×D42D5
kernelC3×D10.12D4C3×C10.D4C3×C4⋊Dic5C3×D10⋊C4C3×C23.D5C15×C22⋊C4D5×C2×C12C6×C5⋊D4D10.12D4C10.D4C4⋊Dic5D10⋊C4C23.D5C5×C22⋊C4C2×C4×D5C2×C5⋊D4C6×D5C3×C22⋊C4C30C2×C12C22×C6D10C22⋊C4C10C2×C4C23C6C2C6C6C2C2
# reps111111112222222222442448848162244

Matrix representation of C3×D10.12D4 in GL4(𝔽61) generated by

13000
01300
0010
0001
,
04300
171800
00600
00060
,
60000
1100
0010
00060
,
255000
73600
00011
00500
,
11000
01100
00500
00011
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,17,0,0,43,18,0,0,0,0,60,0,0,0,0,60],[60,1,0,0,0,1,0,0,0,0,1,0,0,0,0,60],[25,7,0,0,50,36,0,0,0,0,0,50,0,0,11,0],[11,0,0,0,0,11,0,0,0,0,50,0,0,0,0,11] >;

C3×D10.12D4 in GAP, Magma, Sage, TeX

C_3\times D_{10}._{12}D_4
% in TeX

G:=Group("C3xD10.12D4");
// GroupNames label

G:=SmallGroup(480,676);
// by ID

G=gap.SmallGroup(480,676);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,176,590,555,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^10=c^2=d^4=1,e^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=b^5*d^-1>;
// generators/relations

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