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G = C3×Dic5⋊D4order 480 = 25·3·5

Direct product of C3 and Dic5⋊D4

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

Aliases: C3×Dic5⋊D4, (C6×D4)⋊12D5, (D4×C10)⋊9C6, (C2×C30)⋊18D4, (D4×C30)⋊23C2, Dic53(C3×D4), C10.40(C6×D4), C6.194(D4×D5), C1537(C4⋊D4), (C3×Dic5)⋊18D4, C30.408(C2×D4), C23.D512C6, D10⋊C415C6, C23.23(C6×D5), C10.D415C6, (C2×C12).241D10, (C22×Dic5)⋊9C6, (C22×C6).10D10, C30.241(C4○D4), (C2×C60).424C22, (C2×C30).371C23, C6.123(D42D5), (C22×C30).107C22, (C6×Dic5).165C22, C55(C3×C4⋊D4), C2.27(C3×D4×D5), (C2×D4)⋊5(C3×D5), (C2×C10)⋊5(C3×D4), (C2×C5⋊D4)⋊6C6, (C6×C5⋊D4)⋊21C2, (C2×C6)⋊7(C5⋊D4), (C2×C6×Dic5)⋊17C2, (C2×C4).15(C6×D5), C2.15(C6×C5⋊D4), C222(C3×C5⋊D4), C22.61(D5×C2×C6), (C2×C20).62(C2×C6), C10.31(C3×C4○D4), C6.136(C2×C5⋊D4), (D5×C2×C6).83C22, C2.18(C3×D42D5), (C3×C23.D5)⋊28C2, (C3×D10⋊C4)⋊37C2, (C3×C10.D4)⋊37C2, (C22×C10).26(C2×C6), (C2×C10).54(C22×C6), (C2×Dic5).41(C2×C6), (C22×D5).13(C2×C6), (C2×C6).367(C22×D5), SmallGroup(480,732)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×Dic5⋊D4
C1C5C10C2×C10C2×C30D5×C2×C6C6×C5⋊D4 — C3×Dic5⋊D4
C5C2×C10 — C3×Dic5⋊D4
C1C2×C6C6×D4

Generators and relations for C3×Dic5⋊D4
 G = < a,b,c,d,e | a3=b10=d4=e2=1, c2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede=d-1 >

Subgroups: 608 in 188 conjugacy classes, 70 normal (58 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×2], C22 [×8], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, D5, C10 [×3], C10 [×3], C12 [×5], C2×C6, C2×C6 [×2], C2×C6 [×8], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], Dic5 [×2], Dic5 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C12, C2×C12 [×5], C3×D4 [×6], C22×C6 [×2], C22×C6, C3×D5, C30 [×3], C30 [×3], C4⋊D4, C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12, C6×D4, C6×D4 [×2], C3×Dic5 [×2], C3×Dic5 [×2], C60, C6×D5 [×3], C2×C30, C2×C30 [×2], C2×C30 [×5], C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4 [×2], D4×C10, C3×C4⋊D4, C6×Dic5 [×3], C6×Dic5 [×2], C3×C5⋊D4 [×4], C2×C60, D4×C15 [×2], D5×C2×C6, C22×C30 [×2], Dic5⋊D4, C3×C10.D4, C3×D10⋊C4, C3×C23.D5, C2×C6×Dic5, C6×C5⋊D4 [×2], D4×C30, C3×Dic5⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×4], C23, D5, C2×C6 [×7], C2×D4 [×2], C4○D4, D10 [×3], C3×D4 [×4], C22×C6, C3×D5, C4⋊D4, C5⋊D4 [×2], C22×D5, C6×D4 [×2], C3×C4○D4, C6×D5 [×3], D4×D5, D42D5, C2×C5⋊D4, C3×C4⋊D4, C3×C5⋊D4 [×2], D5×C2×C6, Dic5⋊D4, C3×D4×D5, C3×D42D5, C6×C5⋊D4, C3×Dic5⋊D4

Smallest permutation representation of C3×Dic5⋊D4
On 240 points
Generators in S240
(1 54 34)(2 55 35)(3 56 36)(4 57 37)(5 58 38)(6 59 39)(7 60 40)(8 51 31)(9 52 32)(10 53 33)(11 228 208)(12 229 209)(13 230 210)(14 221 201)(15 222 202)(16 223 203)(17 224 204)(18 225 205)(19 226 206)(20 227 207)(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)(81 124 101)(82 125 102)(83 126 103)(84 127 104)(85 128 105)(86 129 106)(87 130 107)(88 121 108)(89 122 109)(90 123 110)(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)
(1 203 6 208)(2 202 7 207)(3 201 8 206)(4 210 9 205)(5 209 10 204)(11 54 16 59)(12 53 17 58)(13 52 18 57)(14 51 19 56)(15 60 20 55)(21 197 26 192)(22 196 27 191)(23 195 28 200)(24 194 29 199)(25 193 30 198)(31 226 36 221)(32 225 37 230)(33 224 38 229)(34 223 39 228)(35 222 40 227)(41 217 46 212)(42 216 47 211)(43 215 48 220)(44 214 49 219)(45 213 50 218)(61 237 66 232)(62 236 67 231)(63 235 68 240)(64 234 69 239)(65 233 70 238)(71 146 76 141)(72 145 77 150)(73 144 78 149)(74 143 79 148)(75 142 80 147)(81 137 86 132)(82 136 87 131)(83 135 88 140)(84 134 89 139)(85 133 90 138)(91 166 96 161)(92 165 97 170)(93 164 98 169)(94 163 99 168)(95 162 100 167)(101 157 106 152)(102 156 107 151)(103 155 108 160)(104 154 109 159)(105 153 110 158)(111 186 116 181)(112 185 117 190)(113 184 118 189)(114 183 119 188)(115 182 120 187)(121 180 126 175)(122 179 127 174)(123 178 128 173)(124 177 129 172)(125 176 130 171)
(1 74 27 87)(2 75 28 88)(3 76 29 89)(4 77 30 90)(5 78 21 81)(6 79 22 82)(7 80 23 83)(8 71 24 84)(9 72 25 85)(10 73 26 86)(11 183 236 171)(12 184 237 172)(13 185 238 173)(14 186 239 174)(15 187 240 175)(16 188 231 176)(17 189 232 177)(18 190 233 178)(19 181 234 179)(20 182 235 180)(31 91 44 104)(32 92 45 105)(33 93 46 106)(34 94 47 107)(35 95 48 108)(36 96 49 109)(37 97 50 110)(38 98 41 101)(39 99 42 102)(40 100 43 103)(51 111 64 127)(52 112 65 128)(53 113 66 129)(54 114 67 130)(55 115 68 121)(56 116 69 122)(57 117 70 123)(58 118 61 124)(59 119 62 125)(60 120 63 126)(131 208 143 196)(132 209 144 197)(133 210 145 198)(134 201 146 199)(135 202 147 200)(136 203 148 191)(137 204 149 192)(138 205 150 193)(139 206 141 194)(140 207 142 195)(151 228 163 216)(152 229 164 217)(153 230 165 218)(154 221 166 219)(155 222 167 220)(156 223 168 211)(157 224 169 212)(158 225 170 213)(159 226 161 214)(160 227 162 215)
(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 81)(79 82)(80 83)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 101)(99 102)(100 103)(111 127)(112 128)(113 129)(114 130)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(131 143)(132 144)(133 145)(134 146)(135 147)(136 148)(137 149)(138 150)(139 141)(140 142)(151 163)(152 164)(153 165)(154 166)(155 167)(156 168)(157 169)(158 170)(159 161)(160 162)(171 183)(172 184)(173 185)(174 186)(175 187)(176 188)(177 189)(178 190)(179 181)(180 182)

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

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

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

102 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F5A5B6A···6F6G6H6I6J6K6L6M6N10A···10F10G···10N12A12B12C···12J12K12L15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order1222222233444444556···66666666610···1010···10121212···121212151515152020202030···3030···3060···60
size1111224201141010101020221···122224420202···24···44410···102020222244442···24···44···4

102 irreducible representations

dim11111111111111222222222222224444
type+++++++++++++-
imageC1C2C2C2C2C2C2C3C6C6C6C6C6C6D4D4D5C4○D4D10D10C3×D4C3×D4C3×D5C5⋊D4C3×C4○D4C6×D5C6×D5C3×C5⋊D4D4×D5D42D5C3×D4×D5C3×D42D5
kernelC3×Dic5⋊D4C3×C10.D4C3×D10⋊C4C3×C23.D5C2×C6×Dic5C6×C5⋊D4D4×C30Dic5⋊D4C10.D4D10⋊C4C23.D5C22×Dic5C2×C5⋊D4D4×C10C3×Dic5C2×C30C6×D4C30C2×C12C22×C6Dic5C2×C10C2×D4C2×C6C10C2×C4C23C22C6C6C2C2
# reps111112122222422222244448448162244

Matrix representation of C3×Dic5⋊D4 in GL4(𝔽61) generated by

1000
0100
00130
00013
,
1100
161700
00600
00060
,
11000
545000
00500
005711
,
144400
334700
004841
003913
,
1000
0100
0010
001760
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[1,16,0,0,1,17,0,0,0,0,60,0,0,0,0,60],[11,54,0,0,0,50,0,0,0,0,50,57,0,0,0,11],[14,33,0,0,44,47,0,0,0,0,48,39,0,0,41,13],[1,0,0,0,0,1,0,0,0,0,1,17,0,0,0,60] >;

C3×Dic5⋊D4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_5\rtimes D_4
% in TeX

G:=Group("C3xDic5:D4");
// GroupNames label

G:=SmallGroup(480,732);
// by ID

G=gap.SmallGroup(480,732);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,590,555,18822]);
// Polycyclic

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

׿
×
𝔽