metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.44C23, Dic6.15D10, Dic10.28D6, D60.16C22, (S3×Q8)⋊3D5, C5⋊Q16⋊4S3, C15⋊Q16⋊8C2, C5⋊2C8.10D6, C5⋊6(Q16⋊S3), (C3×Q8).8D10, (C5×Q8).40D6, Q8.13(S3×D5), Q8⋊2D15⋊8C2, (S3×C10).38D4, (C4×S3).12D10, C10.154(S3×D4), C30.206(C2×D4), Dic6⋊D5⋊7C2, D6.Dic5⋊8C2, D60⋊C2.1C2, D6.16(C5⋊D4), C3⋊3(C20.C23), C15⋊22(C8.C22), C20.44(C22×S3), (C5×Dic3).18D4, C12.44(C22×D5), (S3×C20).16C22, C15⋊3C8.18C22, (Q8×C15).14C22, Dic3.13(C5⋊D4), (C5×Dic6).16C22, (C3×Dic10).16C22, (C5×S3×Q8)⋊3C2, C4.44(C2×S3×D5), (C3×C5⋊Q16)⋊6C2, C6.57(C2×C5⋊D4), C2.35(S3×C5⋊D4), (C3×C5⋊2C8).14C22, SmallGroup(480,596)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.44C23
G = < a,b,c,d | a60=1, b2=c2=d2=a30, bab-1=a19, cac-1=a41, dad-1=a31, bc=cb, dbd-1=a45b, dcd-1=a30c >
Subgroups: 620 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C3×Q8, C3×Q8, C5×S3, D15, C30, C8.C22, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×Q8, C5×Q8, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, C5×Dic3, C5×Dic3, C3×Dic5, C60, C60, S3×C10, D30, C4.Dic5, Q8⋊D5, C5⋊Q16, C5⋊Q16, C4○D20, Q8×C10, Q16⋊S3, C3×C5⋊2C8, C15⋊3C8, D30.C2, C5⋊D12, C3×Dic10, C5×Dic6, C5×Dic6, S3×C20, S3×C20, D60, Q8×C15, C20.C23, D6.Dic5, Dic6⋊D5, C15⋊Q16, C3×C5⋊Q16, Q8⋊2D15, D60⋊C2, C5×S3×Q8, C60.44C23
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q16⋊S3, C2×S3×D5, C20.C23, S3×C5⋊D4, C60.44C23
►On 240 points
Generators in S240
(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)
(1 137 31 167)(2 156 32 126)(3 175 33 145)(4 134 34 164)(5 153 35 123)(6 172 36 142)(7 131 37 161)(8 150 38 180)(9 169 39 139)(10 128 40 158)(11 147 41 177)(12 166 42 136)(13 125 43 155)(14 144 44 174)(15 163 45 133)(16 122 46 152)(17 141 47 171)(18 160 48 130)(19 179 49 149)(20 138 50 168)(21 157 51 127)(22 176 52 146)(23 135 53 165)(24 154 54 124)(25 173 55 143)(26 132 56 162)(27 151 57 121)(28 170 58 140)(29 129 59 159)(30 148 60 178)(61 235 91 205)(62 194 92 224)(63 213 93 183)(64 232 94 202)(65 191 95 221)(66 210 96 240)(67 229 97 199)(68 188 98 218)(69 207 99 237)(70 226 100 196)(71 185 101 215)(72 204 102 234)(73 223 103 193)(74 182 104 212)(75 201 105 231)(76 220 106 190)(77 239 107 209)(78 198 108 228)(79 217 109 187)(80 236 110 206)(81 195 111 225)(82 214 112 184)(83 233 113 203)(84 192 114 222)(85 211 115 181)(86 230 116 200)(87 189 117 219)(88 208 118 238)(89 227 119 197)(90 186 120 216)
(1 16 31 46)(2 57 32 27)(3 38 33 8)(4 19 34 49)(5 60 35 30)(6 41 36 11)(7 22 37 52)(9 44 39 14)(10 25 40 55)(12 47 42 17)(13 28 43 58)(15 50 45 20)(18 53 48 23)(21 56 51 26)(24 59 54 29)(61 116 91 86)(62 97 92 67)(63 78 93 108)(64 119 94 89)(65 100 95 70)(66 81 96 111)(68 103 98 73)(69 84 99 114)(71 106 101 76)(72 87 102 117)(74 109 104 79)(75 90 105 120)(77 112 107 82)(80 115 110 85)(83 118 113 88)(121 126 151 156)(122 167 152 137)(123 148 153 178)(124 129 154 159)(125 170 155 140)(127 132 157 162)(128 173 158 143)(130 135 160 165)(131 176 161 146)(133 138 163 168)(134 179 164 149)(136 141 166 171)(139 144 169 174)(142 147 172 177)(145 150 175 180)(181 206 211 236)(182 187 212 217)(183 228 213 198)(184 209 214 239)(185 190 215 220)(186 231 216 201)(188 193 218 223)(189 234 219 204)(191 196 221 226)(192 237 222 207)(194 199 224 229)(195 240 225 210)(197 202 227 232)(200 205 230 235)(203 208 233 238)
(1 87 31 117)(2 118 32 88)(3 89 33 119)(4 120 34 90)(5 91 35 61)(6 62 36 92)(7 93 37 63)(8 64 38 94)(9 95 39 65)(10 66 40 96)(11 97 41 67)(12 68 42 98)(13 99 43 69)(14 70 44 100)(15 101 45 71)(16 72 46 102)(17 103 47 73)(18 74 48 104)(19 105 49 75)(20 76 50 106)(21 107 51 77)(22 78 52 108)(23 109 53 79)(24 80 54 110)(25 111 55 81)(26 82 56 112)(27 113 57 83)(28 84 58 114)(29 115 59 85)(30 86 60 116)(121 188 151 218)(122 219 152 189)(123 190 153 220)(124 221 154 191)(125 192 155 222)(126 223 156 193)(127 194 157 224)(128 225 158 195)(129 196 159 226)(130 227 160 197)(131 198 161 228)(132 229 162 199)(133 200 163 230)(134 231 164 201)(135 202 165 232)(136 233 166 203)(137 204 167 234)(138 235 168 205)(139 206 169 236)(140 237 170 207)(141 208 171 238)(142 239 172 209)(143 210 173 240)(144 181 174 211)(145 212 175 182)(146 183 176 213)(147 214 177 184)(148 185 178 215)(149 216 179 186)(150 187 180 217)
G:=sub<Sym(240)| (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), (1,137,31,167)(2,156,32,126)(3,175,33,145)(4,134,34,164)(5,153,35,123)(6,172,36,142)(7,131,37,161)(8,150,38,180)(9,169,39,139)(10,128,40,158)(11,147,41,177)(12,166,42,136)(13,125,43,155)(14,144,44,174)(15,163,45,133)(16,122,46,152)(17,141,47,171)(18,160,48,130)(19,179,49,149)(20,138,50,168)(21,157,51,127)(22,176,52,146)(23,135,53,165)(24,154,54,124)(25,173,55,143)(26,132,56,162)(27,151,57,121)(28,170,58,140)(29,129,59,159)(30,148,60,178)(61,235,91,205)(62,194,92,224)(63,213,93,183)(64,232,94,202)(65,191,95,221)(66,210,96,240)(67,229,97,199)(68,188,98,218)(69,207,99,237)(70,226,100,196)(71,185,101,215)(72,204,102,234)(73,223,103,193)(74,182,104,212)(75,201,105,231)(76,220,106,190)(77,239,107,209)(78,198,108,228)(79,217,109,187)(80,236,110,206)(81,195,111,225)(82,214,112,184)(83,233,113,203)(84,192,114,222)(85,211,115,181)(86,230,116,200)(87,189,117,219)(88,208,118,238)(89,227,119,197)(90,186,120,216), (1,16,31,46)(2,57,32,27)(3,38,33,8)(4,19,34,49)(5,60,35,30)(6,41,36,11)(7,22,37,52)(9,44,39,14)(10,25,40,55)(12,47,42,17)(13,28,43,58)(15,50,45,20)(18,53,48,23)(21,56,51,26)(24,59,54,29)(61,116,91,86)(62,97,92,67)(63,78,93,108)(64,119,94,89)(65,100,95,70)(66,81,96,111)(68,103,98,73)(69,84,99,114)(71,106,101,76)(72,87,102,117)(74,109,104,79)(75,90,105,120)(77,112,107,82)(80,115,110,85)(83,118,113,88)(121,126,151,156)(122,167,152,137)(123,148,153,178)(124,129,154,159)(125,170,155,140)(127,132,157,162)(128,173,158,143)(130,135,160,165)(131,176,161,146)(133,138,163,168)(134,179,164,149)(136,141,166,171)(139,144,169,174)(142,147,172,177)(145,150,175,180)(181,206,211,236)(182,187,212,217)(183,228,213,198)(184,209,214,239)(185,190,215,220)(186,231,216,201)(188,193,218,223)(189,234,219,204)(191,196,221,226)(192,237,222,207)(194,199,224,229)(195,240,225,210)(197,202,227,232)(200,205,230,235)(203,208,233,238), (1,87,31,117)(2,118,32,88)(3,89,33,119)(4,120,34,90)(5,91,35,61)(6,62,36,92)(7,93,37,63)(8,64,38,94)(9,95,39,65)(10,66,40,96)(11,97,41,67)(12,68,42,98)(13,99,43,69)(14,70,44,100)(15,101,45,71)(16,72,46,102)(17,103,47,73)(18,74,48,104)(19,105,49,75)(20,76,50,106)(21,107,51,77)(22,78,52,108)(23,109,53,79)(24,80,54,110)(25,111,55,81)(26,82,56,112)(27,113,57,83)(28,84,58,114)(29,115,59,85)(30,86,60,116)(121,188,151,218)(122,219,152,189)(123,190,153,220)(124,221,154,191)(125,192,155,222)(126,223,156,193)(127,194,157,224)(128,225,158,195)(129,196,159,226)(130,227,160,197)(131,198,161,228)(132,229,162,199)(133,200,163,230)(134,231,164,201)(135,202,165,232)(136,233,166,203)(137,204,167,234)(138,235,168,205)(139,206,169,236)(140,237,170,207)(141,208,171,238)(142,239,172,209)(143,210,173,240)(144,181,174,211)(145,212,175,182)(146,183,176,213)(147,214,177,184)(148,185,178,215)(149,216,179,186)(150,187,180,217)>;
G:=Group( (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), (1,137,31,167)(2,156,32,126)(3,175,33,145)(4,134,34,164)(5,153,35,123)(6,172,36,142)(7,131,37,161)(8,150,38,180)(9,169,39,139)(10,128,40,158)(11,147,41,177)(12,166,42,136)(13,125,43,155)(14,144,44,174)(15,163,45,133)(16,122,46,152)(17,141,47,171)(18,160,48,130)(19,179,49,149)(20,138,50,168)(21,157,51,127)(22,176,52,146)(23,135,53,165)(24,154,54,124)(25,173,55,143)(26,132,56,162)(27,151,57,121)(28,170,58,140)(29,129,59,159)(30,148,60,178)(61,235,91,205)(62,194,92,224)(63,213,93,183)(64,232,94,202)(65,191,95,221)(66,210,96,240)(67,229,97,199)(68,188,98,218)(69,207,99,237)(70,226,100,196)(71,185,101,215)(72,204,102,234)(73,223,103,193)(74,182,104,212)(75,201,105,231)(76,220,106,190)(77,239,107,209)(78,198,108,228)(79,217,109,187)(80,236,110,206)(81,195,111,225)(82,214,112,184)(83,233,113,203)(84,192,114,222)(85,211,115,181)(86,230,116,200)(87,189,117,219)(88,208,118,238)(89,227,119,197)(90,186,120,216), (1,16,31,46)(2,57,32,27)(3,38,33,8)(4,19,34,49)(5,60,35,30)(6,41,36,11)(7,22,37,52)(9,44,39,14)(10,25,40,55)(12,47,42,17)(13,28,43,58)(15,50,45,20)(18,53,48,23)(21,56,51,26)(24,59,54,29)(61,116,91,86)(62,97,92,67)(63,78,93,108)(64,119,94,89)(65,100,95,70)(66,81,96,111)(68,103,98,73)(69,84,99,114)(71,106,101,76)(72,87,102,117)(74,109,104,79)(75,90,105,120)(77,112,107,82)(80,115,110,85)(83,118,113,88)(121,126,151,156)(122,167,152,137)(123,148,153,178)(124,129,154,159)(125,170,155,140)(127,132,157,162)(128,173,158,143)(130,135,160,165)(131,176,161,146)(133,138,163,168)(134,179,164,149)(136,141,166,171)(139,144,169,174)(142,147,172,177)(145,150,175,180)(181,206,211,236)(182,187,212,217)(183,228,213,198)(184,209,214,239)(185,190,215,220)(186,231,216,201)(188,193,218,223)(189,234,219,204)(191,196,221,226)(192,237,222,207)(194,199,224,229)(195,240,225,210)(197,202,227,232)(200,205,230,235)(203,208,233,238), (1,87,31,117)(2,118,32,88)(3,89,33,119)(4,120,34,90)(5,91,35,61)(6,62,36,92)(7,93,37,63)(8,64,38,94)(9,95,39,65)(10,66,40,96)(11,97,41,67)(12,68,42,98)(13,99,43,69)(14,70,44,100)(15,101,45,71)(16,72,46,102)(17,103,47,73)(18,74,48,104)(19,105,49,75)(20,76,50,106)(21,107,51,77)(22,78,52,108)(23,109,53,79)(24,80,54,110)(25,111,55,81)(26,82,56,112)(27,113,57,83)(28,84,58,114)(29,115,59,85)(30,86,60,116)(121,188,151,218)(122,219,152,189)(123,190,153,220)(124,221,154,191)(125,192,155,222)(126,223,156,193)(127,194,157,224)(128,225,158,195)(129,196,159,226)(130,227,160,197)(131,198,161,228)(132,229,162,199)(133,200,163,230)(134,231,164,201)(135,202,165,232)(136,233,166,203)(137,204,167,234)(138,235,168,205)(139,206,169,236)(140,237,170,207)(141,208,171,238)(142,239,172,209)(143,210,173,240)(144,181,174,211)(145,212,175,182)(146,183,176,213)(147,214,177,184)(148,185,178,215)(149,216,179,186)(150,187,180,217) );
G=PermutationGroup([[(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)], [(1,137,31,167),(2,156,32,126),(3,175,33,145),(4,134,34,164),(5,153,35,123),(6,172,36,142),(7,131,37,161),(8,150,38,180),(9,169,39,139),(10,128,40,158),(11,147,41,177),(12,166,42,136),(13,125,43,155),(14,144,44,174),(15,163,45,133),(16,122,46,152),(17,141,47,171),(18,160,48,130),(19,179,49,149),(20,138,50,168),(21,157,51,127),(22,176,52,146),(23,135,53,165),(24,154,54,124),(25,173,55,143),(26,132,56,162),(27,151,57,121),(28,170,58,140),(29,129,59,159),(30,148,60,178),(61,235,91,205),(62,194,92,224),(63,213,93,183),(64,232,94,202),(65,191,95,221),(66,210,96,240),(67,229,97,199),(68,188,98,218),(69,207,99,237),(70,226,100,196),(71,185,101,215),(72,204,102,234),(73,223,103,193),(74,182,104,212),(75,201,105,231),(76,220,106,190),(77,239,107,209),(78,198,108,228),(79,217,109,187),(80,236,110,206),(81,195,111,225),(82,214,112,184),(83,233,113,203),(84,192,114,222),(85,211,115,181),(86,230,116,200),(87,189,117,219),(88,208,118,238),(89,227,119,197),(90,186,120,216)], [(1,16,31,46),(2,57,32,27),(3,38,33,8),(4,19,34,49),(5,60,35,30),(6,41,36,11),(7,22,37,52),(9,44,39,14),(10,25,40,55),(12,47,42,17),(13,28,43,58),(15,50,45,20),(18,53,48,23),(21,56,51,26),(24,59,54,29),(61,116,91,86),(62,97,92,67),(63,78,93,108),(64,119,94,89),(65,100,95,70),(66,81,96,111),(68,103,98,73),(69,84,99,114),(71,106,101,76),(72,87,102,117),(74,109,104,79),(75,90,105,120),(77,112,107,82),(80,115,110,85),(83,118,113,88),(121,126,151,156),(122,167,152,137),(123,148,153,178),(124,129,154,159),(125,170,155,140),(127,132,157,162),(128,173,158,143),(130,135,160,165),(131,176,161,146),(133,138,163,168),(134,179,164,149),(136,141,166,171),(139,144,169,174),(142,147,172,177),(145,150,175,180),(181,206,211,236),(182,187,212,217),(183,228,213,198),(184,209,214,239),(185,190,215,220),(186,231,216,201),(188,193,218,223),(189,234,219,204),(191,196,221,226),(192,237,222,207),(194,199,224,229),(195,240,225,210),(197,202,227,232),(200,205,230,235),(203,208,233,238)], [(1,87,31,117),(2,118,32,88),(3,89,33,119),(4,120,34,90),(5,91,35,61),(6,62,36,92),(7,93,37,63),(8,64,38,94),(9,95,39,65),(10,66,40,96),(11,97,41,67),(12,68,42,98),(13,99,43,69),(14,70,44,100),(15,101,45,71),(16,72,46,102),(17,103,47,73),(18,74,48,104),(19,105,49,75),(20,76,50,106),(21,107,51,77),(22,78,52,108),(23,109,53,79),(24,80,54,110),(25,111,55,81),(26,82,56,112),(27,113,57,83),(28,84,58,114),(29,115,59,85),(30,86,60,116),(121,188,151,218),(122,219,152,189),(123,190,153,220),(124,221,154,191),(125,192,155,222),(126,223,156,193),(127,194,157,224),(128,225,158,195),(129,196,159,226),(130,227,160,197),(131,198,161,228),(132,229,162,199),(133,200,163,230),(134,231,164,201),(135,202,165,232),(136,233,166,203),(137,204,167,234),(138,235,168,205),(139,206,169,236),(140,237,170,207),(141,208,171,238),(142,239,172,209),(143,210,173,240),(144,181,174,211),(145,212,175,182),(146,183,176,213),(147,214,177,184),(148,185,178,215),(149,216,179,186),(150,187,180,217)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 15A | 15B | 20A | ··· | 20F | 20G | ··· | 20L | 24A | 24B | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | 24 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 60 | 2 | 2 | 4 | 6 | 12 | 20 | 2 | 2 | 2 | 20 | 60 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 8 | 40 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 20 | 20 | 4 | 4 | 8 | ··· | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | S3×D4 | S3×D5 | Q16⋊S3 | C2×S3×D5 | C20.C23 | S3×C5⋊D4 | C60.44C23 |
| kernel | C60.44C23 | D6.Dic5 | Dic6⋊D5 | C15⋊Q16 | C3×C5⋊Q16 | Q8⋊2D15 | D60⋊C2 | C5×S3×Q8 | C5⋊Q16 | C5×Dic3 | S3×C10 | S3×Q8 | C5⋊2C8 | Dic10 | C5×Q8 | Dic6 | C4×S3 | C3×Q8 | Dic3 | D6 | C15 | C10 | Q8 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of C60.44C23 ►in GL8(𝔽241)
| 0 | 0 | 1 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 190 | 51 | 0 | 0 | 0 | 0 |
| 240 | 190 | 240 | 190 | 0 | 0 | 0 | 0 |
| 51 | 190 | 51 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 239 |
| 0 | 0 | 0 | 0 | 240 | 240 | 2 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 240 | 240 | 1 | 1 |
| 127 | 170 | 0 | 0 | 0 | 0 | 0 | 0 |
| 200 | 114 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 127 | 170 | 0 | 0 | 0 | 0 |
| 0 | 0 | 200 | 114 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 52 | 150 | 59 |
| 0 | 0 | 0 | 0 | 189 | 215 | 182 | 91 |
| 0 | 0 | 0 | 0 | 101 | 202 | 215 | 189 |
| 0 | 0 | 0 | 0 | 39 | 140 | 52 | 26 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 239 | 239 |
| 0 | 0 | 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 240 | 240 |
| 165 | 49 | 0 | 0 | 0 | 0 | 0 | 0 |
| 192 | 76 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 165 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 192 | 76 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 0 | 139 | 0 |
| 0 | 0 | 0 | 0 | 0 | 44 | 0 | 139 |
| 0 | 0 | 0 | 0 | 234 | 0 | 197 | 0 |
| 0 | 0 | 0 | 0 | 0 | 234 | 0 | 197 |
G:=sub<GL(8,GF(241))| [0,0,240,51,0,0,0,0,0,0,190,190,0,0,0,0,1,190,240,51,0,0,0,0,51,51,190,190,0,0,0,0,0,0,0,0,0,240,0,240,0,0,0,0,1,240,1,240,0,0,0,0,0,2,0,1,0,0,0,0,239,2,240,1],[127,200,0,0,0,0,0,0,170,114,0,0,0,0,0,0,0,0,127,200,0,0,0,0,0,0,170,114,0,0,0,0,0,0,0,0,26,189,101,39,0,0,0,0,52,215,202,140,0,0,0,0,150,182,215,52,0,0,0,0,59,91,189,26],[240,0,1,0,0,0,0,0,0,240,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,1,240,1,0,0,0,0,0,1,0,1,0,0,0,0,2,239,1,240,0,0,0,0,0,239,0,240],[165,192,0,0,0,0,0,0,49,76,0,0,0,0,0,0,0,0,165,192,0,0,0,0,0,0,49,76,0,0,0,0,0,0,0,0,44,0,234,0,0,0,0,0,0,44,0,234,0,0,0,0,139,0,197,0,0,0,0,0,0,139,0,197] >;
C60.44C23 in GAP, Magma, Sage, TeX
C_{60}._{44}C_2^3 % in TeX
G:=Group("C60.44C2^3"); // GroupNames label
G:=SmallGroup(480,596);
// by ID
G=gap.SmallGroup(480,596);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,100,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^60=1,b^2=c^2=d^2=a^30,b*a*b^-1=a^19,c*a*c^-1=a^41,d*a*d^-1=a^31,b*c=c*b,d*b*d^-1=a^45*b,d*c*d^-1=a^30*c>;
// generators/relations