direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic5⋊D4, (C6×D4)⋊12D5, (D4×C10)⋊9C6, (C2×C30)⋊18D4, (D4×C30)⋊23C2, Dic5⋊3(C3×D4), C10.40(C6×D4), C6.194(D4×D5), C15⋊37(C4⋊D4), (C3×Dic5)⋊18D4, C30.408(C2×D4), C23.D5⋊12C6, D10⋊C4⋊15C6, C23.23(C6×D5), C10.D4⋊15C6, (C2×C12).241D10, (C22×Dic5)⋊9C6, (C22×C6).10D10, C30.241(C4○D4), (C2×C60).424C22, (C2×C30).371C23, C6.123(D4⋊2D5), (C22×C30).107C22, (C6×Dic5).165C22, C5⋊5(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), C22⋊2(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×D4⋊2D5), (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
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, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C4⋊D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C3×Dic5, C3×Dic5, C60, C6×D5, C2×C30, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, D4×C10, C3×C4⋊D4, C6×Dic5, C6×Dic5, C3×C5⋊D4, C2×C60, D4×C15, D5×C2×C6, C22×C30, Dic5⋊D4, C3×C10.D4, C3×D10⋊C4, C3×C23.D5, C2×C6×Dic5, C6×C5⋊D4, D4×C30, C3×Dic5⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C4⋊D4, C5⋊D4, C22×D5, C6×D4, C3×C4○D4, C6×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C3×C4⋊D4, C3×C5⋊D4, D5×C2×C6, Dic5⋊D4, C3×D4×D5, C3×D4⋊2D5, C6×C5⋊D4, C3×Dic5⋊D4
►On 240 points
Generators in S240
(1 56 36)(2 57 37)(3 58 38)(4 59 39)(5 60 40)(6 51 31)(7 52 32)(8 53 33)(9 54 34)(10 55 35)(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 56 16 51)(12 55 17 60)(13 54 18 59)(14 53 19 58)(15 52 20 57)(21 195 26 200)(22 194 27 199)(23 193 28 198)(24 192 29 197)(25 191 30 196)(31 228 36 223)(32 227 37 222)(33 226 38 221)(34 225 39 230)(35 224 40 229)(41 215 46 220)(42 214 47 219)(43 213 48 218)(44 212 49 217)(45 211 50 216)(61 235 66 240)(62 234 67 239)(63 233 68 238)(64 232 69 237)(65 231 70 236)(71 148 76 143)(72 147 77 142)(73 146 78 141)(74 145 79 150)(75 144 80 149)(81 135 86 140)(82 134 87 139)(83 133 88 138)(84 132 89 137)(85 131 90 136)(91 168 96 163)(92 167 97 162)(93 166 98 161)(94 165 99 170)(95 164 100 169)(101 155 106 160)(102 154 107 159)(103 153 108 158)(104 152 109 157)(105 151 110 156)(111 188 116 183)(112 187 117 182)(113 186 118 181)(114 185 119 190)(115 184 120 189)(121 178 126 173)(122 177 127 172)(123 176 128 171)(124 175 129 180)(125 174 130 179)
(1 76 27 87)(2 77 28 88)(3 78 29 89)(4 79 30 90)(5 80 21 81)(6 71 22 82)(7 72 23 83)(8 73 24 84)(9 74 25 85)(10 75 26 86)(11 183 234 179)(12 184 235 180)(13 185 236 171)(14 186 237 172)(15 187 238 173)(16 188 239 174)(17 189 240 175)(18 190 231 176)(19 181 232 177)(20 182 233 178)(31 91 42 102)(32 92 43 103)(33 93 44 104)(34 94 45 105)(35 95 46 106)(36 96 47 107)(37 97 48 108)(38 98 49 109)(39 99 50 110)(40 100 41 101)(51 111 62 125)(52 112 63 126)(53 113 64 127)(54 114 65 128)(55 115 66 129)(56 116 67 130)(57 117 68 121)(58 118 69 122)(59 119 70 123)(60 120 61 124)(131 210 145 196)(132 201 146 197)(133 202 147 198)(134 203 148 199)(135 204 149 200)(136 205 150 191)(137 206 141 192)(138 207 142 193)(139 208 143 194)(140 209 144 195)(151 230 165 216)(152 221 166 217)(153 222 167 218)(154 223 168 219)(155 224 169 220)(156 225 170 211)(157 226 161 212)(158 227 162 213)(159 228 163 214)(160 229 164 215)
(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 81)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)(99 110)(100 101)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 141)(138 142)(139 143)(140 144)(151 165)(152 166)(153 167)(154 168)(155 169)(156 170)(157 161)(158 162)(159 163)(160 164)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 181)(178 182)(179 183)(180 184)
G:=sub<Sym(240)| (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,51,31)(7,52,32)(8,53,33)(9,54,34)(10,55,35)(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)(11,56,16,51)(12,55,17,60)(13,54,18,59)(14,53,19,58)(15,52,20,57)(21,195,26,200)(22,194,27,199)(23,193,28,198)(24,192,29,197)(25,191,30,196)(31,228,36,223)(32,227,37,222)(33,226,38,221)(34,225,39,230)(35,224,40,229)(41,215,46,220)(42,214,47,219)(43,213,48,218)(44,212,49,217)(45,211,50,216)(61,235,66,240)(62,234,67,239)(63,233,68,238)(64,232,69,237)(65,231,70,236)(71,148,76,143)(72,147,77,142)(73,146,78,141)(74,145,79,150)(75,144,80,149)(81,135,86,140)(82,134,87,139)(83,133,88,138)(84,132,89,137)(85,131,90,136)(91,168,96,163)(92,167,97,162)(93,166,98,161)(94,165,99,170)(95,164,100,169)(101,155,106,160)(102,154,107,159)(103,153,108,158)(104,152,109,157)(105,151,110,156)(111,188,116,183)(112,187,117,182)(113,186,118,181)(114,185,119,190)(115,184,120,189)(121,178,126,173)(122,177,127,172)(123,176,128,171)(124,175,129,180)(125,174,130,179), (1,76,27,87)(2,77,28,88)(3,78,29,89)(4,79,30,90)(5,80,21,81)(6,71,22,82)(7,72,23,83)(8,73,24,84)(9,74,25,85)(10,75,26,86)(11,183,234,179)(12,184,235,180)(13,185,236,171)(14,186,237,172)(15,187,238,173)(16,188,239,174)(17,189,240,175)(18,190,231,176)(19,181,232,177)(20,182,233,178)(31,91,42,102)(32,92,43,103)(33,93,44,104)(34,94,45,105)(35,95,46,106)(36,96,47,107)(37,97,48,108)(38,98,49,109)(39,99,50,110)(40,100,41,101)(51,111,62,125)(52,112,63,126)(53,113,64,127)(54,114,65,128)(55,115,66,129)(56,116,67,130)(57,117,68,121)(58,118,69,122)(59,119,70,123)(60,120,61,124)(131,210,145,196)(132,201,146,197)(133,202,147,198)(134,203,148,199)(135,204,149,200)(136,205,150,191)(137,206,141,192)(138,207,142,193)(139,208,143,194)(140,209,144,195)(151,230,165,216)(152,221,166,217)(153,222,167,218)(154,223,168,219)(155,224,169,220)(156,225,170,211)(157,226,161,212)(158,227,162,213)(159,228,163,214)(160,229,164,215), (71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,81)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(100,101)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,141)(138,142)(139,143)(140,144)(151,165)(152,166)(153,167)(154,168)(155,169)(156,170)(157,161)(158,162)(159,163)(160,164)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,181)(178,182)(179,183)(180,184)>;
G:=Group( (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,51,31)(7,52,32)(8,53,33)(9,54,34)(10,55,35)(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)(11,56,16,51)(12,55,17,60)(13,54,18,59)(14,53,19,58)(15,52,20,57)(21,195,26,200)(22,194,27,199)(23,193,28,198)(24,192,29,197)(25,191,30,196)(31,228,36,223)(32,227,37,222)(33,226,38,221)(34,225,39,230)(35,224,40,229)(41,215,46,220)(42,214,47,219)(43,213,48,218)(44,212,49,217)(45,211,50,216)(61,235,66,240)(62,234,67,239)(63,233,68,238)(64,232,69,237)(65,231,70,236)(71,148,76,143)(72,147,77,142)(73,146,78,141)(74,145,79,150)(75,144,80,149)(81,135,86,140)(82,134,87,139)(83,133,88,138)(84,132,89,137)(85,131,90,136)(91,168,96,163)(92,167,97,162)(93,166,98,161)(94,165,99,170)(95,164,100,169)(101,155,106,160)(102,154,107,159)(103,153,108,158)(104,152,109,157)(105,151,110,156)(111,188,116,183)(112,187,117,182)(113,186,118,181)(114,185,119,190)(115,184,120,189)(121,178,126,173)(122,177,127,172)(123,176,128,171)(124,175,129,180)(125,174,130,179), (1,76,27,87)(2,77,28,88)(3,78,29,89)(4,79,30,90)(5,80,21,81)(6,71,22,82)(7,72,23,83)(8,73,24,84)(9,74,25,85)(10,75,26,86)(11,183,234,179)(12,184,235,180)(13,185,236,171)(14,186,237,172)(15,187,238,173)(16,188,239,174)(17,189,240,175)(18,190,231,176)(19,181,232,177)(20,182,233,178)(31,91,42,102)(32,92,43,103)(33,93,44,104)(34,94,45,105)(35,95,46,106)(36,96,47,107)(37,97,48,108)(38,98,49,109)(39,99,50,110)(40,100,41,101)(51,111,62,125)(52,112,63,126)(53,113,64,127)(54,114,65,128)(55,115,66,129)(56,116,67,130)(57,117,68,121)(58,118,69,122)(59,119,70,123)(60,120,61,124)(131,210,145,196)(132,201,146,197)(133,202,147,198)(134,203,148,199)(135,204,149,200)(136,205,150,191)(137,206,141,192)(138,207,142,193)(139,208,143,194)(140,209,144,195)(151,230,165,216)(152,221,166,217)(153,222,167,218)(154,223,168,219)(155,224,169,220)(156,225,170,211)(157,226,161,212)(158,227,162,213)(159,228,163,214)(160,229,164,215), (71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,81)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(100,101)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,141)(138,142)(139,143)(140,144)(151,165)(152,166)(153,167)(154,168)(155,169)(156,170)(157,161)(158,162)(159,163)(160,164)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,181)(178,182)(179,183)(180,184) );
G=PermutationGroup([[(1,56,36),(2,57,37),(3,58,38),(4,59,39),(5,60,40),(6,51,31),(7,52,32),(8,53,33),(9,54,34),(10,55,35),(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),(11,56,16,51),(12,55,17,60),(13,54,18,59),(14,53,19,58),(15,52,20,57),(21,195,26,200),(22,194,27,199),(23,193,28,198),(24,192,29,197),(25,191,30,196),(31,228,36,223),(32,227,37,222),(33,226,38,221),(34,225,39,230),(35,224,40,229),(41,215,46,220),(42,214,47,219),(43,213,48,218),(44,212,49,217),(45,211,50,216),(61,235,66,240),(62,234,67,239),(63,233,68,238),(64,232,69,237),(65,231,70,236),(71,148,76,143),(72,147,77,142),(73,146,78,141),(74,145,79,150),(75,144,80,149),(81,135,86,140),(82,134,87,139),(83,133,88,138),(84,132,89,137),(85,131,90,136),(91,168,96,163),(92,167,97,162),(93,166,98,161),(94,165,99,170),(95,164,100,169),(101,155,106,160),(102,154,107,159),(103,153,108,158),(104,152,109,157),(105,151,110,156),(111,188,116,183),(112,187,117,182),(113,186,118,181),(114,185,119,190),(115,184,120,189),(121,178,126,173),(122,177,127,172),(123,176,128,171),(124,175,129,180),(125,174,130,179)], [(1,76,27,87),(2,77,28,88),(3,78,29,89),(4,79,30,90),(5,80,21,81),(6,71,22,82),(7,72,23,83),(8,73,24,84),(9,74,25,85),(10,75,26,86),(11,183,234,179),(12,184,235,180),(13,185,236,171),(14,186,237,172),(15,187,238,173),(16,188,239,174),(17,189,240,175),(18,190,231,176),(19,181,232,177),(20,182,233,178),(31,91,42,102),(32,92,43,103),(33,93,44,104),(34,94,45,105),(35,95,46,106),(36,96,47,107),(37,97,48,108),(38,98,49,109),(39,99,50,110),(40,100,41,101),(51,111,62,125),(52,112,63,126),(53,113,64,127),(54,114,65,128),(55,115,66,129),(56,116,67,130),(57,117,68,121),(58,118,69,122),(59,119,70,123),(60,120,61,124),(131,210,145,196),(132,201,146,197),(133,202,147,198),(134,203,148,199),(135,204,149,200),(136,205,150,191),(137,206,141,192),(138,207,142,193),(139,208,143,194),(140,209,144,195),(151,230,165,216),(152,221,166,217),(153,222,167,218),(154,223,168,219),(155,224,169,220),(156,225,170,211),(157,226,161,212),(158,227,162,213),(159,228,163,214),(160,229,164,215)], [(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,81),(91,102),(92,103),(93,104),(94,105),(95,106),(96,107),(97,108),(98,109),(99,110),(100,101),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,121),(118,122),(119,123),(120,124),(131,145),(132,146),(133,147),(134,148),(135,149),(136,150),(137,141),(138,142),(139,143),(140,144),(151,165),(152,166),(153,167),(154,168),(155,169),(156,170),(157,161),(158,162),(159,163),(160,164),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,181),(178,182),(179,183),(180,184)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | ··· | 12J | 12K | 12L | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 20 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C5⋊D4 | C3×C4○D4 | C6×D5 | C6×D5 | C3×C5⋊D4 | D4×D5 | D4⋊2D5 | C3×D4×D5 | C3×D4⋊2D5 |
| kernel | C3×Dic5⋊D4 | C3×C10.D4 | C3×D10⋊C4 | C3×C23.D5 | C2×C6×Dic5 | C6×C5⋊D4 | D4×C30 | Dic5⋊D4 | C10.D4 | D10⋊C4 | C23.D5 | C22×Dic5 | C2×C5⋊D4 | D4×C10 | C3×Dic5 | C2×C30 | C6×D4 | C30 | C2×C12 | C22×C6 | Dic5 | C2×C10 | C2×D4 | C2×C6 | C10 | C2×C4 | C23 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 4 | 8 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×Dic5⋊D4 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 1 | 0 | 0 |
| 16 | 17 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 11 | 0 | 0 | 0 |
| 54 | 50 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 57 | 11 |
| 14 | 44 | 0 | 0 |
| 33 | 47 | 0 | 0 |
| 0 | 0 | 48 | 41 |
| 0 | 0 | 39 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 17 | 60 |
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