direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic5.14D4, (C2×C30)⋊6Q8, C4⋊Dic5⋊2C6, C10.4(C6×Q8), (C2×C6)⋊4Dic10, C6.168(D4×D5), C10.14(C6×D4), C30.76(C2×Q8), (C2×Dic10)⋊2C6, C30.328(C2×D4), C10.D4⋊4C6, C15⋊26(C22⋊Q8), C2.6(C6×Dic10), C23.16(C6×D5), C23.D5.2C6, (C6×Dic10)⋊18C2, (C2×C12).230D10, Dic5.14(C3×D4), (C3×Dic5).75D4, C6.44(C2×Dic10), C22⋊2(C3×Dic10), (C22×C6).72D10, C30.228(C4○D4), (C2×C60).261C22, (C2×C30).336C23, C6.107(D4⋊2D5), (C22×Dic5).5C6, (C22×C30).94C22, (C6×Dic5).234C22, C2.6(C3×D4×D5), C5⋊1(C3×C22⋊Q8), (C2×C10)⋊2(C3×Q8), (C2×C4).5(C6×D5), (C2×C20).1(C2×C6), C22.39(D5×C2×C6), (C3×C4⋊Dic5)⋊20C2, C10.19(C3×C4○D4), C2.6(C3×D4⋊2D5), (C5×C22⋊C4).1C6, C22⋊C4.1(C3×D5), (C3×C22⋊C4).4D5, (C2×C6×Dic5).11C2, (C15×C22⋊C4).4C2, (C2×Dic5).5(C2×C6), (C3×C23.D5).8C2, (C3×C10.D4)⋊15C2, (C22×C10).13(C2×C6), (C2×C10).19(C22×C6), (C2×C6).332(C22×D5), SmallGroup(480,671)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic5.14D4
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=b5d-1 >
Subgroups: 416 in 148 conjugacy classes, 70 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×Q8, C22×C6, C30, C30, C22⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C3×Dic5, C3×Dic5, C60, C2×C30, C2×C30, C2×C30, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, C3×C22⋊Q8, C3×Dic10, C6×Dic5, C6×Dic5, C2×C60, C22×C30, Dic5.14D4, C3×C10.D4, C3×C4⋊Dic5, C3×C23.D5, C15×C22⋊C4, C6×Dic10, C2×C6×Dic5, C3×Dic5.14D4
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, D5, C2×C6, C2×D4, C2×Q8, C4○D4, D10, C3×D4, C3×Q8, C22×C6, C3×D5, C22⋊Q8, Dic10, C22×D5, C6×D4, C6×Q8, C3×C4○D4, C6×D5, C2×Dic10, D4×D5, D4⋊2D5, C3×C22⋊Q8, C3×Dic10, D5×C2×C6, Dic5.14D4, C6×Dic10, C3×D4×D5, C3×D4⋊2D5, C3×Dic5.14D4
►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)(11 228 208)(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)
(1 143 6 148)(2 142 7 147)(3 141 8 146)(4 150 9 145)(5 149 10 144)(11 112 16 117)(12 111 17 116)(13 120 18 115)(14 119 19 114)(15 118 20 113)(21 135 26 140)(22 134 27 139)(23 133 28 138)(24 132 29 137)(25 131 30 136)(31 164 36 169)(32 163 37 168)(33 162 38 167)(34 161 39 166)(35 170 40 165)(41 155 46 160)(42 154 47 159)(43 153 48 158)(44 152 49 157)(45 151 50 156)(51 184 56 189)(52 183 57 188)(53 182 58 187)(54 181 59 186)(55 190 60 185)(61 175 66 180)(62 174 67 179)(63 173 68 178)(64 172 69 177)(65 171 70 176)(71 204 76 209)(72 203 77 208)(73 202 78 207)(74 201 79 206)(75 210 80 205)(81 195 86 200)(82 194 87 199)(83 193 88 198)(84 192 89 197)(85 191 90 196)(91 224 96 229)(92 223 97 228)(93 222 98 227)(94 221 99 226)(95 230 100 225)(101 215 106 220)(102 214 107 219)(103 213 108 218)(104 212 109 217)(105 211 110 216)(121 238 126 233)(122 237 127 232)(123 236 128 231)(124 235 129 240)(125 234 130 239)
(1 77 22 87)(2 78 23 88)(3 79 24 89)(4 80 25 90)(5 71 26 81)(6 72 27 82)(7 73 28 83)(8 74 29 84)(9 75 30 85)(10 76 21 86)(11 179 239 188)(12 180 240 189)(13 171 231 190)(14 172 232 181)(15 173 233 182)(16 174 234 183)(17 175 235 184)(18 176 236 185)(19 177 237 186)(20 178 238 187)(31 96 41 106)(32 97 42 107)(33 98 43 108)(34 99 44 109)(35 100 45 110)(36 91 46 101)(37 92 47 102)(38 93 48 103)(39 94 49 104)(40 95 50 105)(51 116 61 129)(52 117 62 130)(53 118 63 121)(54 119 64 122)(55 120 65 123)(56 111 66 124)(57 112 67 125)(58 113 68 126)(59 114 69 127)(60 115 70 128)(131 191 150 210)(132 192 141 201)(133 193 142 202)(134 194 143 203)(135 195 144 204)(136 196 145 205)(137 197 146 206)(138 198 147 207)(139 199 148 208)(140 200 149 209)(151 211 170 230)(152 212 161 221)(153 213 162 222)(154 214 163 223)(155 215 164 224)(156 216 165 225)(157 217 166 226)(158 218 167 227)(159 219 168 228)(160 220 169 229)
(11 234)(12 235)(13 236)(14 237)(15 238)(16 239)(17 240)(18 231)(19 232)(20 233)(71 86)(72 87)(73 88)(74 89)(75 90)(76 81)(77 82)(78 83)(79 84)(80 85)(91 106)(92 107)(93 108)(94 109)(95 110)(96 101)(97 102)(98 103)(99 104)(100 105)(111 129)(112 130)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)(191 205)(192 206)(193 207)(194 208)(195 209)(196 210)(197 201)(198 202)(199 203)(200 204)(211 225)(212 226)(213 227)(214 228)(215 229)(216 230)(217 221)(218 222)(219 223)(220 224)
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)(11,228,208)(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), (1,143,6,148)(2,142,7,147)(3,141,8,146)(4,150,9,145)(5,149,10,144)(11,112,16,117)(12,111,17,116)(13,120,18,115)(14,119,19,114)(15,118,20,113)(21,135,26,140)(22,134,27,139)(23,133,28,138)(24,132,29,137)(25,131,30,136)(31,164,36,169)(32,163,37,168)(33,162,38,167)(34,161,39,166)(35,170,40,165)(41,155,46,160)(42,154,47,159)(43,153,48,158)(44,152,49,157)(45,151,50,156)(51,184,56,189)(52,183,57,188)(53,182,58,187)(54,181,59,186)(55,190,60,185)(61,175,66,180)(62,174,67,179)(63,173,68,178)(64,172,69,177)(65,171,70,176)(71,204,76,209)(72,203,77,208)(73,202,78,207)(74,201,79,206)(75,210,80,205)(81,195,86,200)(82,194,87,199)(83,193,88,198)(84,192,89,197)(85,191,90,196)(91,224,96,229)(92,223,97,228)(93,222,98,227)(94,221,99,226)(95,230,100,225)(101,215,106,220)(102,214,107,219)(103,213,108,218)(104,212,109,217)(105,211,110,216)(121,238,126,233)(122,237,127,232)(123,236,128,231)(124,235,129,240)(125,234,130,239), (1,77,22,87)(2,78,23,88)(3,79,24,89)(4,80,25,90)(5,71,26,81)(6,72,27,82)(7,73,28,83)(8,74,29,84)(9,75,30,85)(10,76,21,86)(11,179,239,188)(12,180,240,189)(13,171,231,190)(14,172,232,181)(15,173,233,182)(16,174,234,183)(17,175,235,184)(18,176,236,185)(19,177,237,186)(20,178,238,187)(31,96,41,106)(32,97,42,107)(33,98,43,108)(34,99,44,109)(35,100,45,110)(36,91,46,101)(37,92,47,102)(38,93,48,103)(39,94,49,104)(40,95,50,105)(51,116,61,129)(52,117,62,130)(53,118,63,121)(54,119,64,122)(55,120,65,123)(56,111,66,124)(57,112,67,125)(58,113,68,126)(59,114,69,127)(60,115,70,128)(131,191,150,210)(132,192,141,201)(133,193,142,202)(134,194,143,203)(135,195,144,204)(136,196,145,205)(137,197,146,206)(138,198,147,207)(139,199,148,208)(140,200,149,209)(151,211,170,230)(152,212,161,221)(153,213,162,222)(154,214,163,223)(155,215,164,224)(156,216,165,225)(157,217,166,226)(158,218,167,227)(159,219,168,228)(160,220,169,229), (11,234)(12,235)(13,236)(14,237)(15,238)(16,239)(17,240)(18,231)(19,232)(20,233)(71,86)(72,87)(73,88)(74,89)(75,90)(76,81)(77,82)(78,83)(79,84)(80,85)(91,106)(92,107)(93,108)(94,109)(95,110)(96,101)(97,102)(98,103)(99,104)(100,105)(111,129)(112,130)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(191,205)(192,206)(193,207)(194,208)(195,209)(196,210)(197,201)(198,202)(199,203)(200,204)(211,225)(212,226)(213,227)(214,228)(215,229)(216,230)(217,221)(218,222)(219,223)(220,224)>;
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)(11,228,208)(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), (1,143,6,148)(2,142,7,147)(3,141,8,146)(4,150,9,145)(5,149,10,144)(11,112,16,117)(12,111,17,116)(13,120,18,115)(14,119,19,114)(15,118,20,113)(21,135,26,140)(22,134,27,139)(23,133,28,138)(24,132,29,137)(25,131,30,136)(31,164,36,169)(32,163,37,168)(33,162,38,167)(34,161,39,166)(35,170,40,165)(41,155,46,160)(42,154,47,159)(43,153,48,158)(44,152,49,157)(45,151,50,156)(51,184,56,189)(52,183,57,188)(53,182,58,187)(54,181,59,186)(55,190,60,185)(61,175,66,180)(62,174,67,179)(63,173,68,178)(64,172,69,177)(65,171,70,176)(71,204,76,209)(72,203,77,208)(73,202,78,207)(74,201,79,206)(75,210,80,205)(81,195,86,200)(82,194,87,199)(83,193,88,198)(84,192,89,197)(85,191,90,196)(91,224,96,229)(92,223,97,228)(93,222,98,227)(94,221,99,226)(95,230,100,225)(101,215,106,220)(102,214,107,219)(103,213,108,218)(104,212,109,217)(105,211,110,216)(121,238,126,233)(122,237,127,232)(123,236,128,231)(124,235,129,240)(125,234,130,239), (1,77,22,87)(2,78,23,88)(3,79,24,89)(4,80,25,90)(5,71,26,81)(6,72,27,82)(7,73,28,83)(8,74,29,84)(9,75,30,85)(10,76,21,86)(11,179,239,188)(12,180,240,189)(13,171,231,190)(14,172,232,181)(15,173,233,182)(16,174,234,183)(17,175,235,184)(18,176,236,185)(19,177,237,186)(20,178,238,187)(31,96,41,106)(32,97,42,107)(33,98,43,108)(34,99,44,109)(35,100,45,110)(36,91,46,101)(37,92,47,102)(38,93,48,103)(39,94,49,104)(40,95,50,105)(51,116,61,129)(52,117,62,130)(53,118,63,121)(54,119,64,122)(55,120,65,123)(56,111,66,124)(57,112,67,125)(58,113,68,126)(59,114,69,127)(60,115,70,128)(131,191,150,210)(132,192,141,201)(133,193,142,202)(134,194,143,203)(135,195,144,204)(136,196,145,205)(137,197,146,206)(138,198,147,207)(139,199,148,208)(140,200,149,209)(151,211,170,230)(152,212,161,221)(153,213,162,222)(154,214,163,223)(155,215,164,224)(156,216,165,225)(157,217,166,226)(158,218,167,227)(159,219,168,228)(160,220,169,229), (11,234)(12,235)(13,236)(14,237)(15,238)(16,239)(17,240)(18,231)(19,232)(20,233)(71,86)(72,87)(73,88)(74,89)(75,90)(76,81)(77,82)(78,83)(79,84)(80,85)(91,106)(92,107)(93,108)(94,109)(95,110)(96,101)(97,102)(98,103)(99,104)(100,105)(111,129)(112,130)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(191,205)(192,206)(193,207)(194,208)(195,209)(196,210)(197,201)(198,202)(199,203)(200,204)(211,225)(212,226)(213,227)(214,228)(215,229)(216,230)(217,221)(218,222)(219,223)(220,224) );
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),(11,228,208),(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)], [(1,143,6,148),(2,142,7,147),(3,141,8,146),(4,150,9,145),(5,149,10,144),(11,112,16,117),(12,111,17,116),(13,120,18,115),(14,119,19,114),(15,118,20,113),(21,135,26,140),(22,134,27,139),(23,133,28,138),(24,132,29,137),(25,131,30,136),(31,164,36,169),(32,163,37,168),(33,162,38,167),(34,161,39,166),(35,170,40,165),(41,155,46,160),(42,154,47,159),(43,153,48,158),(44,152,49,157),(45,151,50,156),(51,184,56,189),(52,183,57,188),(53,182,58,187),(54,181,59,186),(55,190,60,185),(61,175,66,180),(62,174,67,179),(63,173,68,178),(64,172,69,177),(65,171,70,176),(71,204,76,209),(72,203,77,208),(73,202,78,207),(74,201,79,206),(75,210,80,205),(81,195,86,200),(82,194,87,199),(83,193,88,198),(84,192,89,197),(85,191,90,196),(91,224,96,229),(92,223,97,228),(93,222,98,227),(94,221,99,226),(95,230,100,225),(101,215,106,220),(102,214,107,219),(103,213,108,218),(104,212,109,217),(105,211,110,216),(121,238,126,233),(122,237,127,232),(123,236,128,231),(124,235,129,240),(125,234,130,239)], [(1,77,22,87),(2,78,23,88),(3,79,24,89),(4,80,25,90),(5,71,26,81),(6,72,27,82),(7,73,28,83),(8,74,29,84),(9,75,30,85),(10,76,21,86),(11,179,239,188),(12,180,240,189),(13,171,231,190),(14,172,232,181),(15,173,233,182),(16,174,234,183),(17,175,235,184),(18,176,236,185),(19,177,237,186),(20,178,238,187),(31,96,41,106),(32,97,42,107),(33,98,43,108),(34,99,44,109),(35,100,45,110),(36,91,46,101),(37,92,47,102),(38,93,48,103),(39,94,49,104),(40,95,50,105),(51,116,61,129),(52,117,62,130),(53,118,63,121),(54,119,64,122),(55,120,65,123),(56,111,66,124),(57,112,67,125),(58,113,68,126),(59,114,69,127),(60,115,70,128),(131,191,150,210),(132,192,141,201),(133,193,142,202),(134,194,143,203),(135,195,144,204),(136,196,145,205),(137,197,146,206),(138,198,147,207),(139,199,148,208),(140,200,149,209),(151,211,170,230),(152,212,161,221),(153,213,162,222),(154,214,163,223),(155,215,164,224),(156,216,165,225),(157,217,166,226),(158,218,167,227),(159,219,168,228),(160,220,169,229)], [(11,234),(12,235),(13,236),(14,237),(15,238),(16,239),(17,240),(18,231),(19,232),(20,233),(71,86),(72,87),(73,88),(74,89),(75,90),(76,81),(77,82),(78,83),(79,84),(80,85),(91,106),(92,107),(93,108),(94,109),(95,110),(96,101),(97,102),(98,103),(99,104),(100,105),(111,129),(112,130),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,127),(120,128),(191,205),(192,206),(193,207),(194,208),(195,209),(196,210),(197,201),(198,202),(199,203),(200,204),(211,225),(212,226),(213,227),(214,228),(215,229),(216,230),(217,221),(218,222),(219,223),(220,224)]])
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 | ··· | 12L | 12M | 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 | 2 | 2 | 1 | 1 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×Q8 | C3×D5 | Dic10 | C3×C4○D4 | C6×D5 | C6×D5 | C3×Dic10 | D4×D5 | D4⋊2D5 | C3×D4×D5 | C3×D4⋊2D5 |
| kernel | C3×Dic5.14D4 | C3×C10.D4 | C3×C4⋊Dic5 | C3×C23.D5 | C15×C22⋊C4 | C6×Dic10 | C2×C6×Dic5 | Dic5.14D4 | C10.D4 | C4⋊Dic5 | C23.D5 | C5×C22⋊C4 | C2×Dic10 | C22×Dic5 | C3×Dic5 | C2×C30 | C3×C22⋊C4 | C30 | C2×C12 | C22×C6 | Dic5 | C2×C10 | C22⋊C4 | C2×C6 | C10 | C2×C4 | C23 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 4 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×Dic5.14D4 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 18 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 23 | 55 | 0 | 0 |
| 7 | 38 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 36 | 4 | 0 | 0 |
| 57 | 25 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 60 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 60 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,47,0,0,0,0,47],[18,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[23,7,0,0,55,38,0,0,0,0,60,0,0,0,0,60],[36,57,0,0,4,25,0,0,0,0,1,60,0,0,2,60],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60] >;
C3×Dic5.14D4 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_5._{14}D_4 % in TeX
G:=Group("C3xDic5.14D4"); // GroupNames label
G:=SmallGroup(480,671);
// by ID
G=gap.SmallGroup(480,671);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,590,555,142,18822]);
// 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=b^5*d^-1>;
// generators/relations