direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Q8.Dic5, C60.281C23, D4.(C3×Dic5), C15⋊28(C8○D4), (D4×C15).4C4, (C5×D4).2C12, C4.Dic5⋊8C6, (Q8×C15).4C4, (C5×Q8).5C12, C4.5(C6×Dic5), C20.36(C2×C12), C60.170(C2×C4), (C3×D4).2Dic5, (C3×Q8).2Dic5, Q8.2(C3×Dic5), (C2×C12).368D10, C20.41(C22×C6), C12.34(C2×Dic5), C22.1(C6×Dic5), C10.40(C22×C12), (C2×C60).301C22, C30.225(C22×C4), C12.248(C22×D5), C6.37(C22×Dic5), C5⋊5(C3×C8○D4), C4.42(D5×C2×C6), (C2×C5⋊2C8)⋊7C6, (C6×C5⋊2C8)⋊21C2, C2.8(C2×C6×Dic5), (C3×C4○D4).6D5, (C5×C4○D4).6C6, C4○D4.5(C3×D5), (C2×C4).58(C6×D5), (C2×C20).38(C2×C6), C5⋊2C8.17(C2×C6), (C15×C4○D4).5C2, (C2×C6).8(C2×Dic5), (C2×C30).124(C2×C4), (C2×C10).27(C2×C12), (C3×C4.Dic5)⋊20C2, (C3×C5⋊2C8).52C22, SmallGroup(480,741)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8.Dic5
G = < a,b,c,d,e | a3=b4=1, c2=d10=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >
Subgroups: 224 in 124 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, C6, C6 [×3], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C12, C12 [×3], C2×C6 [×3], C15, C2×C8 [×3], M4(2) [×3], C4○D4, C20, C20 [×3], C2×C10 [×3], C24 [×4], C2×C12 [×3], C3×D4 [×3], C3×Q8, C30, C30 [×3], C8○D4, C5⋊2C8, C5⋊2C8 [×3], C2×C20 [×3], C5×D4 [×3], C5×Q8, C2×C24 [×3], C3×M4(2) [×3], C3×C4○D4, C60, C60 [×3], C2×C30 [×3], C2×C5⋊2C8 [×3], C4.Dic5 [×3], C5×C4○D4, C3×C8○D4, C3×C5⋊2C8, C3×C5⋊2C8 [×3], C2×C60 [×3], D4×C15 [×3], Q8×C15, Q8.Dic5, C6×C5⋊2C8 [×3], C3×C4.Dic5 [×3], C15×C4○D4, C3×Q8.Dic5
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, D5, C12 [×4], C2×C6 [×7], C22×C4, Dic5 [×4], D10 [×3], C2×C12 [×6], C22×C6, C3×D5, C8○D4, C2×Dic5 [×6], C22×D5, C22×C12, C3×Dic5 [×4], C6×D5 [×3], C22×Dic5, C3×C8○D4, C6×Dic5 [×6], D5×C2×C6, Q8.Dic5, C2×C6×Dic5, C3×Q8.Dic5
►On 240 points
Generators in S240
(1 46 36)(2 47 37)(3 48 38)(4 49 39)(5 50 40)(6 51 21)(7 52 22)(8 53 23)(9 54 24)(10 55 25)(11 56 26)(12 57 27)(13 58 28)(14 59 29)(15 60 30)(16 41 31)(17 42 32)(18 43 33)(19 44 34)(20 45 35)(61 116 91)(62 117 92)(63 118 93)(64 119 94)(65 120 95)(66 101 96)(67 102 97)(68 103 98)(69 104 99)(70 105 100)(71 106 81)(72 107 82)(73 108 83)(74 109 84)(75 110 85)(76 111 86)(77 112 87)(78 113 88)(79 114 89)(80 115 90)(121 176 155)(122 177 156)(123 178 157)(124 179 158)(125 180 159)(126 161 160)(127 162 141)(128 163 142)(129 164 143)(130 165 144)(131 166 145)(132 167 146)(133 168 147)(134 169 148)(135 170 149)(136 171 150)(137 172 151)(138 173 152)(139 174 153)(140 175 154)(181 224 212)(182 225 213)(183 226 214)(184 227 215)(185 228 216)(186 229 217)(187 230 218)(188 231 219)(189 232 220)(190 233 201)(191 234 202)(192 235 203)(193 236 204)(194 237 205)(195 238 206)(196 239 207)(197 240 208)(198 221 209)(199 222 210)(200 223 211)
(1 61 11 71)(2 62 12 72)(3 63 13 73)(4 64 14 74)(5 65 15 75)(6 66 16 76)(7 67 17 77)(8 68 18 78)(9 69 19 79)(10 70 20 80)(21 96 31 86)(22 97 32 87)(23 98 33 88)(24 99 34 89)(25 100 35 90)(26 81 36 91)(27 82 37 92)(28 83 38 93)(29 84 39 94)(30 85 40 95)(41 111 51 101)(42 112 52 102)(43 113 53 103)(44 114 54 104)(45 115 55 105)(46 116 56 106)(47 117 57 107)(48 118 58 108)(49 119 59 109)(50 120 60 110)(121 183 131 193)(122 184 132 194)(123 185 133 195)(124 186 134 196)(125 187 135 197)(126 188 136 198)(127 189 137 199)(128 190 138 200)(129 191 139 181)(130 192 140 182)(141 220 151 210)(142 201 152 211)(143 202 153 212)(144 203 154 213)(145 204 155 214)(146 205 156 215)(147 206 157 216)(148 207 158 217)(149 208 159 218)(150 209 160 219)(161 231 171 221)(162 232 172 222)(163 233 173 223)(164 234 174 224)(165 235 175 225)(166 236 176 226)(167 237 177 227)(168 238 178 228)(169 239 179 229)(170 240 180 230)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(81 96 91 86)(82 97 92 87)(83 98 93 88)(84 99 94 89)(85 100 95 90)(101 116 111 106)(102 117 112 107)(103 118 113 108)(104 119 114 109)(105 120 115 110)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)(141 146 151 156)(142 147 152 157)(143 148 153 158)(144 149 154 159)(145 150 155 160)(161 166 171 176)(162 167 172 177)(163 168 173 178)(164 169 174 179)(165 170 175 180)(181 196 191 186)(182 197 192 187)(183 198 193 188)(184 199 194 189)(185 200 195 190)(201 216 211 206)(202 217 212 207)(203 218 213 208)(204 219 214 209)(205 220 215 210)(221 236 231 226)(222 237 232 227)(223 238 233 228)(224 239 234 229)(225 240 235 230)
(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)
(1 132 6 137 11 122 16 127)(2 121 7 126 12 131 17 136)(3 130 8 135 13 140 18 125)(4 139 9 124 14 129 19 134)(5 128 10 133 15 138 20 123)(21 151 26 156 31 141 36 146)(22 160 27 145 32 150 37 155)(23 149 28 154 33 159 38 144)(24 158 29 143 34 148 39 153)(25 147 30 152 35 157 40 142)(41 162 46 167 51 172 56 177)(42 171 47 176 52 161 57 166)(43 180 48 165 53 170 58 175)(44 169 49 174 54 179 59 164)(45 178 50 163 55 168 60 173)(61 194 66 199 71 184 76 189)(62 183 67 188 72 193 77 198)(63 192 68 197 73 182 78 187)(64 181 69 186 74 191 79 196)(65 190 70 195 75 200 80 185)(81 215 86 220 91 205 96 210)(82 204 87 209 92 214 97 219)(83 213 88 218 93 203 98 208)(84 202 89 207 94 212 99 217)(85 211 90 216 95 201 100 206)(101 222 106 227 111 232 116 237)(102 231 107 236 112 221 117 226)(103 240 108 225 113 230 118 235)(104 229 109 234 114 239 119 224)(105 238 110 223 115 228 120 233)
G:=sub<Sym(240)| (1,46,36)(2,47,37)(3,48,38)(4,49,39)(5,50,40)(6,51,21)(7,52,22)(8,53,23)(9,54,24)(10,55,25)(11,56,26)(12,57,27)(13,58,28)(14,59,29)(15,60,30)(16,41,31)(17,42,32)(18,43,33)(19,44,34)(20,45,35)(61,116,91)(62,117,92)(63,118,93)(64,119,94)(65,120,95)(66,101,96)(67,102,97)(68,103,98)(69,104,99)(70,105,100)(71,106,81)(72,107,82)(73,108,83)(74,109,84)(75,110,85)(76,111,86)(77,112,87)(78,113,88)(79,114,89)(80,115,90)(121,176,155)(122,177,156)(123,178,157)(124,179,158)(125,180,159)(126,161,160)(127,162,141)(128,163,142)(129,164,143)(130,165,144)(131,166,145)(132,167,146)(133,168,147)(134,169,148)(135,170,149)(136,171,150)(137,172,151)(138,173,152)(139,174,153)(140,175,154)(181,224,212)(182,225,213)(183,226,214)(184,227,215)(185,228,216)(186,229,217)(187,230,218)(188,231,219)(189,232,220)(190,233,201)(191,234,202)(192,235,203)(193,236,204)(194,237,205)(195,238,206)(196,239,207)(197,240,208)(198,221,209)(199,222,210)(200,223,211), (1,61,11,71)(2,62,12,72)(3,63,13,73)(4,64,14,74)(5,65,15,75)(6,66,16,76)(7,67,17,77)(8,68,18,78)(9,69,19,79)(10,70,20,80)(21,96,31,86)(22,97,32,87)(23,98,33,88)(24,99,34,89)(25,100,35,90)(26,81,36,91)(27,82,37,92)(28,83,38,93)(29,84,39,94)(30,85,40,95)(41,111,51,101)(42,112,52,102)(43,113,53,103)(44,114,54,104)(45,115,55,105)(46,116,56,106)(47,117,57,107)(48,118,58,108)(49,119,59,109)(50,120,60,110)(121,183,131,193)(122,184,132,194)(123,185,133,195)(124,186,134,196)(125,187,135,197)(126,188,136,198)(127,189,137,199)(128,190,138,200)(129,191,139,181)(130,192,140,182)(141,220,151,210)(142,201,152,211)(143,202,153,212)(144,203,154,213)(145,204,155,214)(146,205,156,215)(147,206,157,216)(148,207,158,217)(149,208,159,218)(150,209,160,219)(161,231,171,221)(162,232,172,222)(163,233,173,223)(164,234,174,224)(165,235,175,225)(166,236,176,226)(167,237,177,227)(168,238,178,228)(169,239,179,229)(170,240,180,230), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(81,96,91,86)(82,97,92,87)(83,98,93,88)(84,99,94,89)(85,100,95,90)(101,116,111,106)(102,117,112,107)(103,118,113,108)(104,119,114,109)(105,120,115,110)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160)(161,166,171,176)(162,167,172,177)(163,168,173,178)(164,169,174,179)(165,170,175,180)(181,196,191,186)(182,197,192,187)(183,198,193,188)(184,199,194,189)(185,200,195,190)(201,216,211,206)(202,217,212,207)(203,218,213,208)(204,219,214,209)(205,220,215,210)(221,236,231,226)(222,237,232,227)(223,238,233,228)(224,239,234,229)(225,240,235,230), (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), (1,132,6,137,11,122,16,127)(2,121,7,126,12,131,17,136)(3,130,8,135,13,140,18,125)(4,139,9,124,14,129,19,134)(5,128,10,133,15,138,20,123)(21,151,26,156,31,141,36,146)(22,160,27,145,32,150,37,155)(23,149,28,154,33,159,38,144)(24,158,29,143,34,148,39,153)(25,147,30,152,35,157,40,142)(41,162,46,167,51,172,56,177)(42,171,47,176,52,161,57,166)(43,180,48,165,53,170,58,175)(44,169,49,174,54,179,59,164)(45,178,50,163,55,168,60,173)(61,194,66,199,71,184,76,189)(62,183,67,188,72,193,77,198)(63,192,68,197,73,182,78,187)(64,181,69,186,74,191,79,196)(65,190,70,195,75,200,80,185)(81,215,86,220,91,205,96,210)(82,204,87,209,92,214,97,219)(83,213,88,218,93,203,98,208)(84,202,89,207,94,212,99,217)(85,211,90,216,95,201,100,206)(101,222,106,227,111,232,116,237)(102,231,107,236,112,221,117,226)(103,240,108,225,113,230,118,235)(104,229,109,234,114,239,119,224)(105,238,110,223,115,228,120,233)>;
G:=Group( (1,46,36)(2,47,37)(3,48,38)(4,49,39)(5,50,40)(6,51,21)(7,52,22)(8,53,23)(9,54,24)(10,55,25)(11,56,26)(12,57,27)(13,58,28)(14,59,29)(15,60,30)(16,41,31)(17,42,32)(18,43,33)(19,44,34)(20,45,35)(61,116,91)(62,117,92)(63,118,93)(64,119,94)(65,120,95)(66,101,96)(67,102,97)(68,103,98)(69,104,99)(70,105,100)(71,106,81)(72,107,82)(73,108,83)(74,109,84)(75,110,85)(76,111,86)(77,112,87)(78,113,88)(79,114,89)(80,115,90)(121,176,155)(122,177,156)(123,178,157)(124,179,158)(125,180,159)(126,161,160)(127,162,141)(128,163,142)(129,164,143)(130,165,144)(131,166,145)(132,167,146)(133,168,147)(134,169,148)(135,170,149)(136,171,150)(137,172,151)(138,173,152)(139,174,153)(140,175,154)(181,224,212)(182,225,213)(183,226,214)(184,227,215)(185,228,216)(186,229,217)(187,230,218)(188,231,219)(189,232,220)(190,233,201)(191,234,202)(192,235,203)(193,236,204)(194,237,205)(195,238,206)(196,239,207)(197,240,208)(198,221,209)(199,222,210)(200,223,211), (1,61,11,71)(2,62,12,72)(3,63,13,73)(4,64,14,74)(5,65,15,75)(6,66,16,76)(7,67,17,77)(8,68,18,78)(9,69,19,79)(10,70,20,80)(21,96,31,86)(22,97,32,87)(23,98,33,88)(24,99,34,89)(25,100,35,90)(26,81,36,91)(27,82,37,92)(28,83,38,93)(29,84,39,94)(30,85,40,95)(41,111,51,101)(42,112,52,102)(43,113,53,103)(44,114,54,104)(45,115,55,105)(46,116,56,106)(47,117,57,107)(48,118,58,108)(49,119,59,109)(50,120,60,110)(121,183,131,193)(122,184,132,194)(123,185,133,195)(124,186,134,196)(125,187,135,197)(126,188,136,198)(127,189,137,199)(128,190,138,200)(129,191,139,181)(130,192,140,182)(141,220,151,210)(142,201,152,211)(143,202,153,212)(144,203,154,213)(145,204,155,214)(146,205,156,215)(147,206,157,216)(148,207,158,217)(149,208,159,218)(150,209,160,219)(161,231,171,221)(162,232,172,222)(163,233,173,223)(164,234,174,224)(165,235,175,225)(166,236,176,226)(167,237,177,227)(168,238,178,228)(169,239,179,229)(170,240,180,230), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(81,96,91,86)(82,97,92,87)(83,98,93,88)(84,99,94,89)(85,100,95,90)(101,116,111,106)(102,117,112,107)(103,118,113,108)(104,119,114,109)(105,120,115,110)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160)(161,166,171,176)(162,167,172,177)(163,168,173,178)(164,169,174,179)(165,170,175,180)(181,196,191,186)(182,197,192,187)(183,198,193,188)(184,199,194,189)(185,200,195,190)(201,216,211,206)(202,217,212,207)(203,218,213,208)(204,219,214,209)(205,220,215,210)(221,236,231,226)(222,237,232,227)(223,238,233,228)(224,239,234,229)(225,240,235,230), (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), (1,132,6,137,11,122,16,127)(2,121,7,126,12,131,17,136)(3,130,8,135,13,140,18,125)(4,139,9,124,14,129,19,134)(5,128,10,133,15,138,20,123)(21,151,26,156,31,141,36,146)(22,160,27,145,32,150,37,155)(23,149,28,154,33,159,38,144)(24,158,29,143,34,148,39,153)(25,147,30,152,35,157,40,142)(41,162,46,167,51,172,56,177)(42,171,47,176,52,161,57,166)(43,180,48,165,53,170,58,175)(44,169,49,174,54,179,59,164)(45,178,50,163,55,168,60,173)(61,194,66,199,71,184,76,189)(62,183,67,188,72,193,77,198)(63,192,68,197,73,182,78,187)(64,181,69,186,74,191,79,196)(65,190,70,195,75,200,80,185)(81,215,86,220,91,205,96,210)(82,204,87,209,92,214,97,219)(83,213,88,218,93,203,98,208)(84,202,89,207,94,212,99,217)(85,211,90,216,95,201,100,206)(101,222,106,227,111,232,116,237)(102,231,107,236,112,221,117,226)(103,240,108,225,113,230,118,235)(104,229,109,234,114,239,119,224)(105,238,110,223,115,228,120,233) );
G=PermutationGroup([(1,46,36),(2,47,37),(3,48,38),(4,49,39),(5,50,40),(6,51,21),(7,52,22),(8,53,23),(9,54,24),(10,55,25),(11,56,26),(12,57,27),(13,58,28),(14,59,29),(15,60,30),(16,41,31),(17,42,32),(18,43,33),(19,44,34),(20,45,35),(61,116,91),(62,117,92),(63,118,93),(64,119,94),(65,120,95),(66,101,96),(67,102,97),(68,103,98),(69,104,99),(70,105,100),(71,106,81),(72,107,82),(73,108,83),(74,109,84),(75,110,85),(76,111,86),(77,112,87),(78,113,88),(79,114,89),(80,115,90),(121,176,155),(122,177,156),(123,178,157),(124,179,158),(125,180,159),(126,161,160),(127,162,141),(128,163,142),(129,164,143),(130,165,144),(131,166,145),(132,167,146),(133,168,147),(134,169,148),(135,170,149),(136,171,150),(137,172,151),(138,173,152),(139,174,153),(140,175,154),(181,224,212),(182,225,213),(183,226,214),(184,227,215),(185,228,216),(186,229,217),(187,230,218),(188,231,219),(189,232,220),(190,233,201),(191,234,202),(192,235,203),(193,236,204),(194,237,205),(195,238,206),(196,239,207),(197,240,208),(198,221,209),(199,222,210),(200,223,211)], [(1,61,11,71),(2,62,12,72),(3,63,13,73),(4,64,14,74),(5,65,15,75),(6,66,16,76),(7,67,17,77),(8,68,18,78),(9,69,19,79),(10,70,20,80),(21,96,31,86),(22,97,32,87),(23,98,33,88),(24,99,34,89),(25,100,35,90),(26,81,36,91),(27,82,37,92),(28,83,38,93),(29,84,39,94),(30,85,40,95),(41,111,51,101),(42,112,52,102),(43,113,53,103),(44,114,54,104),(45,115,55,105),(46,116,56,106),(47,117,57,107),(48,118,58,108),(49,119,59,109),(50,120,60,110),(121,183,131,193),(122,184,132,194),(123,185,133,195),(124,186,134,196),(125,187,135,197),(126,188,136,198),(127,189,137,199),(128,190,138,200),(129,191,139,181),(130,192,140,182),(141,220,151,210),(142,201,152,211),(143,202,153,212),(144,203,154,213),(145,204,155,214),(146,205,156,215),(147,206,157,216),(148,207,158,217),(149,208,159,218),(150,209,160,219),(161,231,171,221),(162,232,172,222),(163,233,173,223),(164,234,174,224),(165,235,175,225),(166,236,176,226),(167,237,177,227),(168,238,178,228),(169,239,179,229),(170,240,180,230)], [(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,26,31,36),(22,27,32,37),(23,28,33,38),(24,29,34,39),(25,30,35,40),(41,46,51,56),(42,47,52,57),(43,48,53,58),(44,49,54,59),(45,50,55,60),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70),(81,96,91,86),(82,97,92,87),(83,98,93,88),(84,99,94,89),(85,100,95,90),(101,116,111,106),(102,117,112,107),(103,118,113,108),(104,119,114,109),(105,120,115,110),(121,126,131,136),(122,127,132,137),(123,128,133,138),(124,129,134,139),(125,130,135,140),(141,146,151,156),(142,147,152,157),(143,148,153,158),(144,149,154,159),(145,150,155,160),(161,166,171,176),(162,167,172,177),(163,168,173,178),(164,169,174,179),(165,170,175,180),(181,196,191,186),(182,197,192,187),(183,198,193,188),(184,199,194,189),(185,200,195,190),(201,216,211,206),(202,217,212,207),(203,218,213,208),(204,219,214,209),(205,220,215,210),(221,236,231,226),(222,237,232,227),(223,238,233,228),(224,239,234,229),(225,240,235,230)], [(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)], [(1,132,6,137,11,122,16,127),(2,121,7,126,12,131,17,136),(3,130,8,135,13,140,18,125),(4,139,9,124,14,129,19,134),(5,128,10,133,15,138,20,123),(21,151,26,156,31,141,36,146),(22,160,27,145,32,150,37,155),(23,149,28,154,33,159,38,144),(24,158,29,143,34,148,39,153),(25,147,30,152,35,157,40,142),(41,162,46,167,51,172,56,177),(42,171,47,176,52,161,57,166),(43,180,48,165,53,170,58,175),(44,169,49,174,54,179,59,164),(45,178,50,163,55,168,60,173),(61,194,66,199,71,184,76,189),(62,183,67,188,72,193,77,198),(63,192,68,197,73,182,78,187),(64,181,69,186,74,191,79,196),(65,190,70,195,75,200,80,185),(81,215,86,220,91,205,96,210),(82,204,87,209,92,214,97,219),(83,213,88,218,93,203,98,208),(84,202,89,207,94,212,99,217),(85,211,90,216,95,201,100,206),(101,222,106,227,111,232,116,237),(102,231,107,236,112,221,117,226),(103,240,108,225,113,230,118,235),(104,229,109,234,114,239,119,224),(105,238,110,223,115,228,120,233)])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | ··· | 6H | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 24A | ··· | 24H | 24I | ··· | 24T | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 24 | ··· | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D5 | D10 | Dic5 | Dic5 | C3×D5 | C8○D4 | C6×D5 | C3×Dic5 | C3×Dic5 | C3×C8○D4 | Q8.Dic5 | C3×Q8.Dic5 |
| kernel | C3×Q8.Dic5 | C6×C5⋊2C8 | C3×C4.Dic5 | C15×C4○D4 | Q8.Dic5 | D4×C15 | Q8×C15 | C2×C5⋊2C8 | C4.Dic5 | C5×C4○D4 | C5×D4 | C5×Q8 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C15 | C2×C4 | D4 | Q8 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 2 | 6 | 6 | 2 | 12 | 4 | 2 | 6 | 6 | 2 | 4 | 4 | 12 | 12 | 4 | 8 | 4 | 8 |
Matrix representation of C3×Q8.Dic5 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 64 | 113 |
| 0 | 0 | 0 | 177 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 64 | 177 |
| 0 | 1 | 0 | 0 |
| 240 | 190 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 34 | 240 | 0 | 0 |
| 193 | 207 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,225,0,0,0,0,225],[240,0,0,0,0,240,0,0,0,0,64,0,0,0,113,177],[240,0,0,0,0,240,0,0,0,0,64,64,0,0,0,177],[0,240,0,0,1,190,0,0,0,0,64,0,0,0,0,64],[34,193,0,0,240,207,0,0,0,0,8,0,0,0,0,8] >;
C3×Q8.Dic5 in GAP, Magma, Sage, TeX
C_3\times Q_8.{\rm Dic}_5 % in TeX
G:=Group("C3xQ8.Dic5"); // GroupNames label
G:=SmallGroup(480,741);
// by ID
G=gap.SmallGroup(480,741);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,555,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=1,c^2=d^10=b^2,e^2=d^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,c*d=d*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations