direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic5.5D4, C10.20(C6×D4), C6.174(D4×D5), C23.D5⋊5C6, C23.6(C6×D5), (C4×Dic5)⋊12C6, (C2×Dic10)⋊3C6, C30.333(C2×D4), Dic5.5(C3×D4), D10⋊C4⋊11C6, (C22×C6).6D10, (C6×Dic10)⋊19C2, (C12×Dic5)⋊30C2, (C2×C12).275D10, C15⋊20(C4.4D4), (C3×Dic5).52D4, C30.230(C4○D4), C6.117(C4○D20), (C2×C30).343C23, (C2×C60).265C22, C6.111(D4⋊2D5), (C22×C30).101C22, (C6×Dic5).236C22, C2.10(C3×D4×D5), C5⋊2(C3×C4.4D4), (C5×C22⋊C4)⋊7C6, C22⋊C4⋊5(C3×D5), C10.9(C3×C4○D4), (C2×C4).25(C6×D5), (C2×C5⋊D4).4C6, C22.44(D5×C2×C6), (C2×C20).51(C2×C6), (C3×C22⋊C4)⋊13D5, C2.12(C3×C4○D20), C2.9(C3×D4⋊2D5), (C6×C5⋊D4).11C2, (D5×C2×C6).78C22, (C15×C22⋊C4)⋊16C2, (C3×C23.D5)⋊21C2, (C3×D10⋊C4)⋊17C2, (C22×D5).6(C2×C6), (C22×C10).20(C2×C6), (C2×C10).26(C22×C6), (C2×Dic5).28(C2×C6), (C2×C6).339(C22×D5), SmallGroup(480,678)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic5.5D4
G = < a,b,c,d,e | a3=b10=d4=1, c2=e2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=b5d-1 >
Subgroups: 512 in 152 conjugacy classes, 62 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C3×D5, C30, C30, C4.4D4, Dic10, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C6×D4, C6×Q8, C3×Dic5, C3×Dic5, C60, C6×D5, C2×C30, C2×C30, C4×Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C3×C4.4D4, C3×Dic10, C6×Dic5, C3×C5⋊D4, C2×C60, D5×C2×C6, C22×C30, Dic5.5D4, C12×Dic5, C3×D10⋊C4, C3×C23.D5, C15×C22⋊C4, C6×Dic10, C6×C5⋊D4, C3×Dic5.5D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C4.4D4, C22×D5, C6×D4, C3×C4○D4, C6×D5, C4○D20, D4×D5, D4⋊2D5, C3×C4.4D4, D5×C2×C6, Dic5.5D4, C3×C4○D20, C3×D4×D5, C3×D4⋊2D5, C3×Dic5.5D4
►On 240 points
Generators in S240
(1 62 42)(2 63 43)(3 64 44)(4 65 45)(5 66 46)(6 67 47)(7 68 48)(8 69 49)(9 70 50)(10 61 41)(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 53 33)(22 54 34)(23 55 35)(24 56 36)(25 57 37)(26 58 38)(27 59 39)(28 60 40)(29 51 31)(30 52 32)(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 76 6 71)(2 75 7 80)(3 74 8 79)(4 73 9 78)(5 72 10 77)(11 178 16 173)(12 177 17 172)(13 176 18 171)(14 175 19 180)(15 174 20 179)(21 85 26 90)(22 84 27 89)(23 83 28 88)(24 82 29 87)(25 81 30 86)(31 107 36 102)(32 106 37 101)(33 105 38 110)(34 104 39 109)(35 103 40 108)(41 97 46 92)(42 96 47 91)(43 95 48 100)(44 94 49 99)(45 93 50 98)(51 127 56 122)(52 126 57 121)(53 125 58 130)(54 124 59 129)(55 123 60 128)(61 117 66 112)(62 116 67 111)(63 115 68 120)(64 114 69 119)(65 113 70 118)(131 207 136 202)(132 206 137 201)(133 205 138 210)(134 204 139 209)(135 203 140 208)(141 197 146 192)(142 196 147 191)(143 195 148 200)(144 194 149 199)(145 193 150 198)(151 227 156 222)(152 226 157 221)(153 225 158 230)(154 224 159 229)(155 223 160 228)(161 217 166 212)(162 216 167 211)(163 215 168 220)(164 214 169 219)(165 213 170 218)(181 237 186 232)(182 236 187 231)(183 235 188 240)(184 234 189 239)(185 233 190 238)
(1 131 29 142)(2 132 30 143)(3 133 21 144)(4 134 22 145)(5 135 23 146)(6 136 24 147)(7 137 25 148)(8 138 26 149)(9 139 27 150)(10 140 28 141)(11 125 234 114)(12 126 235 115)(13 127 236 116)(14 128 237 117)(15 129 238 118)(16 130 239 119)(17 121 240 120)(18 122 231 111)(19 123 232 112)(20 124 233 113)(31 162 42 151)(32 163 43 152)(33 164 44 153)(34 165 45 154)(35 166 46 155)(36 167 47 156)(37 168 48 157)(38 169 49 158)(39 170 50 159)(40 161 41 160)(51 182 62 171)(52 183 63 172)(53 184 64 173)(54 185 65 174)(55 186 66 175)(56 187 67 176)(57 188 68 177)(58 189 69 178)(59 190 70 179)(60 181 61 180)(71 202 82 191)(72 203 83 192)(73 204 84 193)(74 205 85 194)(75 206 86 195)(76 207 87 196)(77 208 88 197)(78 209 89 198)(79 210 90 199)(80 201 81 200)(91 222 102 211)(92 223 103 212)(93 224 104 213)(94 225 105 214)(95 226 106 215)(96 227 107 216)(97 228 108 217)(98 229 109 218)(99 230 110 219)(100 221 101 220)
(1 136 6 131)(2 135 7 140)(3 134 8 139)(4 133 9 138)(5 132 10 137)(11 113 16 118)(12 112 17 117)(13 111 18 116)(14 120 19 115)(15 119 20 114)(21 145 26 150)(22 144 27 149)(23 143 28 148)(24 142 29 147)(25 141 30 146)(31 167 36 162)(32 166 37 161)(33 165 38 170)(34 164 39 169)(35 163 40 168)(41 157 46 152)(42 156 47 151)(43 155 48 160)(44 154 49 159)(45 153 50 158)(51 187 56 182)(52 186 57 181)(53 185 58 190)(54 184 59 189)(55 183 60 188)(61 177 66 172)(62 176 67 171)(63 175 68 180)(64 174 69 179)(65 173 70 178)(71 202 76 207)(72 201 77 206)(73 210 78 205)(74 209 79 204)(75 208 80 203)(81 192 86 197)(82 191 87 196)(83 200 88 195)(84 199 89 194)(85 198 90 193)(91 222 96 227)(92 221 97 226)(93 230 98 225)(94 229 99 224)(95 228 100 223)(101 212 106 217)(102 211 107 216)(103 220 108 215)(104 219 109 214)(105 218 110 213)(121 232 126 237)(122 231 127 236)(123 240 128 235)(124 239 129 234)(125 238 130 233)
G:=sub<Sym(240)| (1,62,42)(2,63,43)(3,64,44)(4,65,45)(5,66,46)(6,67,47)(7,68,48)(8,69,49)(9,70,50)(10,61,41)(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)(21,53,33)(22,54,34)(23,55,35)(24,56,36)(25,57,37)(26,58,38)(27,59,39)(28,60,40)(29,51,31)(30,52,32)(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), (1,76,6,71)(2,75,7,80)(3,74,8,79)(4,73,9,78)(5,72,10,77)(11,178,16,173)(12,177,17,172)(13,176,18,171)(14,175,19,180)(15,174,20,179)(21,85,26,90)(22,84,27,89)(23,83,28,88)(24,82,29,87)(25,81,30,86)(31,107,36,102)(32,106,37,101)(33,105,38,110)(34,104,39,109)(35,103,40,108)(41,97,46,92)(42,96,47,91)(43,95,48,100)(44,94,49,99)(45,93,50,98)(51,127,56,122)(52,126,57,121)(53,125,58,130)(54,124,59,129)(55,123,60,128)(61,117,66,112)(62,116,67,111)(63,115,68,120)(64,114,69,119)(65,113,70,118)(131,207,136,202)(132,206,137,201)(133,205,138,210)(134,204,139,209)(135,203,140,208)(141,197,146,192)(142,196,147,191)(143,195,148,200)(144,194,149,199)(145,193,150,198)(151,227,156,222)(152,226,157,221)(153,225,158,230)(154,224,159,229)(155,223,160,228)(161,217,166,212)(162,216,167,211)(163,215,168,220)(164,214,169,219)(165,213,170,218)(181,237,186,232)(182,236,187,231)(183,235,188,240)(184,234,189,239)(185,233,190,238), (1,131,29,142)(2,132,30,143)(3,133,21,144)(4,134,22,145)(5,135,23,146)(6,136,24,147)(7,137,25,148)(8,138,26,149)(9,139,27,150)(10,140,28,141)(11,125,234,114)(12,126,235,115)(13,127,236,116)(14,128,237,117)(15,129,238,118)(16,130,239,119)(17,121,240,120)(18,122,231,111)(19,123,232,112)(20,124,233,113)(31,162,42,151)(32,163,43,152)(33,164,44,153)(34,165,45,154)(35,166,46,155)(36,167,47,156)(37,168,48,157)(38,169,49,158)(39,170,50,159)(40,161,41,160)(51,182,62,171)(52,183,63,172)(53,184,64,173)(54,185,65,174)(55,186,66,175)(56,187,67,176)(57,188,68,177)(58,189,69,178)(59,190,70,179)(60,181,61,180)(71,202,82,191)(72,203,83,192)(73,204,84,193)(74,205,85,194)(75,206,86,195)(76,207,87,196)(77,208,88,197)(78,209,89,198)(79,210,90,199)(80,201,81,200)(91,222,102,211)(92,223,103,212)(93,224,104,213)(94,225,105,214)(95,226,106,215)(96,227,107,216)(97,228,108,217)(98,229,109,218)(99,230,110,219)(100,221,101,220), (1,136,6,131)(2,135,7,140)(3,134,8,139)(4,133,9,138)(5,132,10,137)(11,113,16,118)(12,112,17,117)(13,111,18,116)(14,120,19,115)(15,119,20,114)(21,145,26,150)(22,144,27,149)(23,143,28,148)(24,142,29,147)(25,141,30,146)(31,167,36,162)(32,166,37,161)(33,165,38,170)(34,164,39,169)(35,163,40,168)(41,157,46,152)(42,156,47,151)(43,155,48,160)(44,154,49,159)(45,153,50,158)(51,187,56,182)(52,186,57,181)(53,185,58,190)(54,184,59,189)(55,183,60,188)(61,177,66,172)(62,176,67,171)(63,175,68,180)(64,174,69,179)(65,173,70,178)(71,202,76,207)(72,201,77,206)(73,210,78,205)(74,209,79,204)(75,208,80,203)(81,192,86,197)(82,191,87,196)(83,200,88,195)(84,199,89,194)(85,198,90,193)(91,222,96,227)(92,221,97,226)(93,230,98,225)(94,229,99,224)(95,228,100,223)(101,212,106,217)(102,211,107,216)(103,220,108,215)(104,219,109,214)(105,218,110,213)(121,232,126,237)(122,231,127,236)(123,240,128,235)(124,239,129,234)(125,238,130,233)>;
G:=Group( (1,62,42)(2,63,43)(3,64,44)(4,65,45)(5,66,46)(6,67,47)(7,68,48)(8,69,49)(9,70,50)(10,61,41)(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)(21,53,33)(22,54,34)(23,55,35)(24,56,36)(25,57,37)(26,58,38)(27,59,39)(28,60,40)(29,51,31)(30,52,32)(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), (1,76,6,71)(2,75,7,80)(3,74,8,79)(4,73,9,78)(5,72,10,77)(11,178,16,173)(12,177,17,172)(13,176,18,171)(14,175,19,180)(15,174,20,179)(21,85,26,90)(22,84,27,89)(23,83,28,88)(24,82,29,87)(25,81,30,86)(31,107,36,102)(32,106,37,101)(33,105,38,110)(34,104,39,109)(35,103,40,108)(41,97,46,92)(42,96,47,91)(43,95,48,100)(44,94,49,99)(45,93,50,98)(51,127,56,122)(52,126,57,121)(53,125,58,130)(54,124,59,129)(55,123,60,128)(61,117,66,112)(62,116,67,111)(63,115,68,120)(64,114,69,119)(65,113,70,118)(131,207,136,202)(132,206,137,201)(133,205,138,210)(134,204,139,209)(135,203,140,208)(141,197,146,192)(142,196,147,191)(143,195,148,200)(144,194,149,199)(145,193,150,198)(151,227,156,222)(152,226,157,221)(153,225,158,230)(154,224,159,229)(155,223,160,228)(161,217,166,212)(162,216,167,211)(163,215,168,220)(164,214,169,219)(165,213,170,218)(181,237,186,232)(182,236,187,231)(183,235,188,240)(184,234,189,239)(185,233,190,238), (1,131,29,142)(2,132,30,143)(3,133,21,144)(4,134,22,145)(5,135,23,146)(6,136,24,147)(7,137,25,148)(8,138,26,149)(9,139,27,150)(10,140,28,141)(11,125,234,114)(12,126,235,115)(13,127,236,116)(14,128,237,117)(15,129,238,118)(16,130,239,119)(17,121,240,120)(18,122,231,111)(19,123,232,112)(20,124,233,113)(31,162,42,151)(32,163,43,152)(33,164,44,153)(34,165,45,154)(35,166,46,155)(36,167,47,156)(37,168,48,157)(38,169,49,158)(39,170,50,159)(40,161,41,160)(51,182,62,171)(52,183,63,172)(53,184,64,173)(54,185,65,174)(55,186,66,175)(56,187,67,176)(57,188,68,177)(58,189,69,178)(59,190,70,179)(60,181,61,180)(71,202,82,191)(72,203,83,192)(73,204,84,193)(74,205,85,194)(75,206,86,195)(76,207,87,196)(77,208,88,197)(78,209,89,198)(79,210,90,199)(80,201,81,200)(91,222,102,211)(92,223,103,212)(93,224,104,213)(94,225,105,214)(95,226,106,215)(96,227,107,216)(97,228,108,217)(98,229,109,218)(99,230,110,219)(100,221,101,220), (1,136,6,131)(2,135,7,140)(3,134,8,139)(4,133,9,138)(5,132,10,137)(11,113,16,118)(12,112,17,117)(13,111,18,116)(14,120,19,115)(15,119,20,114)(21,145,26,150)(22,144,27,149)(23,143,28,148)(24,142,29,147)(25,141,30,146)(31,167,36,162)(32,166,37,161)(33,165,38,170)(34,164,39,169)(35,163,40,168)(41,157,46,152)(42,156,47,151)(43,155,48,160)(44,154,49,159)(45,153,50,158)(51,187,56,182)(52,186,57,181)(53,185,58,190)(54,184,59,189)(55,183,60,188)(61,177,66,172)(62,176,67,171)(63,175,68,180)(64,174,69,179)(65,173,70,178)(71,202,76,207)(72,201,77,206)(73,210,78,205)(74,209,79,204)(75,208,80,203)(81,192,86,197)(82,191,87,196)(83,200,88,195)(84,199,89,194)(85,198,90,193)(91,222,96,227)(92,221,97,226)(93,230,98,225)(94,229,99,224)(95,228,100,223)(101,212,106,217)(102,211,107,216)(103,220,108,215)(104,219,109,214)(105,218,110,213)(121,232,126,237)(122,231,127,236)(123,240,128,235)(124,239,129,234)(125,238,130,233) );
G=PermutationGroup([[(1,62,42),(2,63,43),(3,64,44),(4,65,45),(5,66,46),(6,67,47),(7,68,48),(8,69,49),(9,70,50),(10,61,41),(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),(21,53,33),(22,54,34),(23,55,35),(24,56,36),(25,57,37),(26,58,38),(27,59,39),(28,60,40),(29,51,31),(30,52,32),(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)], [(1,76,6,71),(2,75,7,80),(3,74,8,79),(4,73,9,78),(5,72,10,77),(11,178,16,173),(12,177,17,172),(13,176,18,171),(14,175,19,180),(15,174,20,179),(21,85,26,90),(22,84,27,89),(23,83,28,88),(24,82,29,87),(25,81,30,86),(31,107,36,102),(32,106,37,101),(33,105,38,110),(34,104,39,109),(35,103,40,108),(41,97,46,92),(42,96,47,91),(43,95,48,100),(44,94,49,99),(45,93,50,98),(51,127,56,122),(52,126,57,121),(53,125,58,130),(54,124,59,129),(55,123,60,128),(61,117,66,112),(62,116,67,111),(63,115,68,120),(64,114,69,119),(65,113,70,118),(131,207,136,202),(132,206,137,201),(133,205,138,210),(134,204,139,209),(135,203,140,208),(141,197,146,192),(142,196,147,191),(143,195,148,200),(144,194,149,199),(145,193,150,198),(151,227,156,222),(152,226,157,221),(153,225,158,230),(154,224,159,229),(155,223,160,228),(161,217,166,212),(162,216,167,211),(163,215,168,220),(164,214,169,219),(165,213,170,218),(181,237,186,232),(182,236,187,231),(183,235,188,240),(184,234,189,239),(185,233,190,238)], [(1,131,29,142),(2,132,30,143),(3,133,21,144),(4,134,22,145),(5,135,23,146),(6,136,24,147),(7,137,25,148),(8,138,26,149),(9,139,27,150),(10,140,28,141),(11,125,234,114),(12,126,235,115),(13,127,236,116),(14,128,237,117),(15,129,238,118),(16,130,239,119),(17,121,240,120),(18,122,231,111),(19,123,232,112),(20,124,233,113),(31,162,42,151),(32,163,43,152),(33,164,44,153),(34,165,45,154),(35,166,46,155),(36,167,47,156),(37,168,48,157),(38,169,49,158),(39,170,50,159),(40,161,41,160),(51,182,62,171),(52,183,63,172),(53,184,64,173),(54,185,65,174),(55,186,66,175),(56,187,67,176),(57,188,68,177),(58,189,69,178),(59,190,70,179),(60,181,61,180),(71,202,82,191),(72,203,83,192),(73,204,84,193),(74,205,85,194),(75,206,86,195),(76,207,87,196),(77,208,88,197),(78,209,89,198),(79,210,90,199),(80,201,81,200),(91,222,102,211),(92,223,103,212),(93,224,104,213),(94,225,105,214),(95,226,106,215),(96,227,107,216),(97,228,108,217),(98,229,109,218),(99,230,110,219),(100,221,101,220)], [(1,136,6,131),(2,135,7,140),(3,134,8,139),(4,133,9,138),(5,132,10,137),(11,113,16,118),(12,112,17,117),(13,111,18,116),(14,120,19,115),(15,119,20,114),(21,145,26,150),(22,144,27,149),(23,143,28,148),(24,142,29,147),(25,141,30,146),(31,167,36,162),(32,166,37,161),(33,165,38,170),(34,164,39,169),(35,163,40,168),(41,157,46,152),(42,156,47,151),(43,155,48,160),(44,154,49,159),(45,153,50,158),(51,187,56,182),(52,186,57,181),(53,185,58,190),(54,184,59,189),(55,183,60,188),(61,177,66,172),(62,176,67,171),(63,175,68,180),(64,174,69,179),(65,173,70,178),(71,202,76,207),(72,201,77,206),(73,210,78,205),(74,209,79,204),(75,208,80,203),(81,192,86,197),(82,191,87,196),(83,200,88,195),(84,199,89,194),(85,198,90,193),(91,222,96,227),(92,221,97,226),(93,230,98,225),(94,229,99,224),(95,228,100,223),(101,212,106,217),(102,211,107,216),(103,220,108,215),(104,219,109,214),(105,218,110,213),(121,232,126,237),(122,231,127,236),(123,240,128,235),(124,239,129,234),(125,238,130,233)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12N | 12O | 12P | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 1 | ··· | 1 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D5 | C3×C4○D4 | C6×D5 | C6×D5 | C4○D20 | C3×C4○D20 | D4×D5 | D4⋊2D5 | C3×D4×D5 | C3×D4⋊2D5 |
| kernel | C3×Dic5.5D4 | C12×Dic5 | C3×D10⋊C4 | C3×C23.D5 | C15×C22⋊C4 | C6×Dic10 | C6×C5⋊D4 | Dic5.5D4 | C4×Dic5 | D10⋊C4 | C23.D5 | C5×C22⋊C4 | C2×Dic10 | C2×C5⋊D4 | C3×Dic5 | C3×C22⋊C4 | C30 | C2×C12 | C22×C6 | Dic5 | C22⋊C4 | C10 | C2×C4 | C23 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×Dic5.5D4 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 1 | 1 | 0 | 0 |
| 42 | 43 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 8 | 22 | 0 | 0 |
| 22 | 53 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 59 | 60 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 39 | 50 |
| 46 | 57 | 0 | 0 |
| 26 | 15 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 22 | 11 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,47,0,0,0,0,47],[1,42,0,0,1,43,0,0,0,0,60,0,0,0,0,60],[8,22,0,0,22,53,0,0,0,0,1,59,0,0,1,60],[50,0,0,0,0,50,0,0,0,0,11,39,0,0,11,50],[46,26,0,0,57,15,0,0,0,0,50,22,0,0,0,11] >;
C3×Dic5.5D4 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_5._5D_4 % in TeX
G:=Group("C3xDic5.5D4"); // GroupNames label
G:=SmallGroup(480,678);
// by ID
G=gap.SmallGroup(480,678);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,176,1598,555,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^10=d^4=1,c^2=e^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=b^5*d^-1>;
// generators/relations