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G = C60.57C23  order 480 = 25·3·5

57th non-split extension by C60 of C23 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.57C23, C30.33C24, C15⋊62- (1+4), Dic6.31D10, Dic10.30D6, D30.17C23, D60.19C22, Dic15.39C23, (Q8×D5)⋊7S3, (S3×Q8)⋊5D5, D15⋊Q8⋊6C2, (C4×D5).17D6, Q8.27(S3×D5), (C5×Q8).42D6, D60⋊C2⋊6C2, C12.28D10⋊6C2, Q8⋊3D15⋊4C2, C15⋊Q8.6C22, (C4×S3).17D10, (C3×Q8).25D10, C6.33(C23×D5), D6.D10⋊6C2, C20.57(C22×S3), C10.33(S3×C23), C5⋊2(Q8.15D6), (C6×D5).47C23, D6.29(C22×D5), C12.57(C22×D5), C5⋊D12.3C22, C3⋊D20.4C22, C15⋊D4.5C22, (S3×C20).20C22, (S3×C10).32C23, C3⋊2(Q8.10D10), (D5×C12).20C22, (C4×D15).20C22, D10.43(C22×S3), D30.C2.4C22, (Q8×C15).20C22, Dic5.18(C22×S3), Dic3.18(C22×D5), (C5×Dic3).19C23, (C3×Dic5).16C23, (C5×Dic6).21C22, (C3×Dic10).20C22, (C5×S3×Q8)⋊5C2, (C3×Q8×D5)⋊4C2, C4.57(C2×S3×D5), C2.36(C22×S3×D5), SmallGroup(480,1105)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — C60.57C23
Chief series
C1 — C5 — C15 — C30 — C6×D5 — C15⋊D4 — D6.D10 — C60.57C23
Lower central
C15 — C30 — C60.57C23

Subgroups: 1388 in 292 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C4 [×7], C22 [×5], C5, S3 [×4], C6, C6, C2×C4 [×15], D4 [×10], Q8, Q8 [×9], D5 [×4], C10, C10, Dic3 [×3], Dic3, C12 [×3], C12 [×3], D6, D6 [×3], C2×C6, C15, C2×Q8 [×5], C4○D4 [×10], Dic5 [×3], Dic5, C20 [×3], C20 [×3], D10, D10 [×3], C2×C10, Dic6 [×3], Dic6 [×3], C4×S3 [×3], C4×S3 [×9], D12 [×6], C3⋊D4 [×4], C2×C12 [×3], C3×Q8, C3×Q8 [×3], C5×S3, C3×D5, D15 [×3], C30, 2- (1+4), Dic10 [×3], Dic10 [×3], C4×D5 [×3], C4×D5 [×9], D20 [×6], C5⋊D4 [×4], C2×C20 [×3], C5×Q8, C5×Q8 [×3], C4○D12 [×6], S3×Q8, S3×Q8 [×3], Q8⋊3S3 [×4], C6×Q8, C5×Dic3 [×3], C3×Dic5 [×3], Dic15, C60 [×3], C6×D5, S3×C10, D30 [×3], C4○D20 [×6], Q8×D5, Q8×D5 [×3], Q8⋊2D5 [×4], Q8×C10, Q8.15D6, D30.C2 [×6], C15⋊D4, C3⋊D20 [×3], C5⋊D12 [×3], C15⋊Q8 [×3], C3×Dic10 [×3], D5×C12 [×3], C5×Dic6 [×3], S3×C20 [×3], C4×D15 [×3], D60 [×3], Q8×C15, Q8.10D10, D60⋊C2 [×3], D15⋊Q8 [×3], D6.D10 [×3], C12.28D10 [×3], C3×Q8×D5, C5×S3×Q8, Q8⋊3D15, C60.57C23

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2- (1+4), C22×D5 [×7], S3×C23, S3×D5, C23×D5, Q8.15D6, C2×S3×D5 [×3], Q8.10D10, C22×S3×D5, C60.57C23

Generators and relations
 G = < a,b,c,d | a60=b2=c2=1, d2=a30, bab=a49, cac=a41, dad-1=a31, cbc=a30b, bd=db, cd=dc >

Smallest permutation representation
►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)

(1 195)(2 184)(3 233)(4 222)(5 211)(6 200)(7 189)(8 238)(9 227)(10 216)(11 205)(12 194)(13 183)(14 232)(15 221)(16 210)(17 199)(18 188)(19 237)(20 226)(21 215)(22 204)(23 193)(24 182)(25 231)(26 220)(27 209)(28 198)(29 187)(30 236)(31 225)(32 214)(33 203)(34 192)(35 181)(36 230)(37 219)(38 208)(39 197)(40 186)(41 235)(42 224)(43 213)(44 202)(45 191)(46 240)(47 229)(48 218)(49 207)(50 196)(51 185)(52 234)(53 223)(54 212)(55 201)(56 190)(57 239)(58 228)(59 217)(60 206)(61 167)(62 156)(63 145)(64 134)(65 123)(66 172)(67 161)(68 150)(69 139)(70 128)(71 177)(72 166)(73 155)(74 144)(75 133)(76 122)(77 171)(78 160)(79 149)(80 138)(81 127)(82 176)(83 165)(84 154)(85 143)(86 132)(87 121)(88 170)(89 159)(90 148)(91 137)(92 126)(93 175)(94 164)(95 153)(96 142)(97 131)(98 180)(99 169)(100 158)(101 147)(102 136)(103 125)(104 174)(105 163)(106 152)(107 141)(108 130)(109 179)(110 168)(111 157)(112 146)(113 135)(114 124)(115 173)(116 162)(117 151)(118 140)(119 129)(120 178)

(1 240)(2 221)(3 202)(4 183)(5 224)(6 205)(7 186)(8 227)(9 208)(10 189)(11 230)(12 211)(13 192)(14 233)(15 214)(16 195)(17 236)(18 217)(19 198)(20 239)(21 220)(22 201)(23 182)(24 223)(25 204)(26 185)(27 226)(28 207)(29 188)(30 229)(31 210)(32 191)(33 232)(34 213)(35 194)(36 235)(37 216)(38 197)(39 238)(40 219)(41 200)(42 181)(43 222)(44 203)(45 184)(46 225)(47 206)(48 187)(49 228)(50 209)(51 190)(52 231)(53 212)(54 193)(55 234)(56 215)(57 196)(58 237)(59 218)(60 199)(61 136)(62 177)(63 158)(64 139)(65 180)(66 161)(67 142)(68 123)(69 164)(70 145)(71 126)(72 167)(73 148)(74 129)(75 170)(76 151)(77 132)(78 173)(79 154)(80 135)(81 176)(82 157)(83 138)(84 179)(85 160)(86 141)(87 122)(88 163)(89 144)(90 125)(91 166)(92 147)(93 128)(94 169)(95 150)(96 131)(97 172)(98 153)(99 134)(100 175)(101 156)(102 137)(103 178)(104 159)(105 140)(106 121)(107 162)(108 143)(109 124)(110 165)(111 146)(112 127)(113 168)(114 149)(115 130)(116 171)(117 152)(118 133)(119 174)(120 155)

(1 144 31 174)(2 175 32 145)(3 146 33 176)(4 177 34 147)(5 148 35 178)(6 179 36 149)(7 150 37 180)(8 121 38 151)(9 152 39 122)(10 123 40 153)(11 154 41 124)(12 125 42 155)(13 156 43 126)(14 127 44 157)(15 158 45 128)(16 129 46 159)(17 160 47 130)(18 131 48 161)(19 162 49 132)(20 133 50 163)(21 164 51 134)(22 135 52 165)(23 166 53 136)(24 137 54 167)(25 168 55 138)(26 139 56 169)(27 170 57 140)(28 141 58 171)(29 172 59 142)(30 143 60 173)(61 182 91 212)(62 213 92 183)(63 184 93 214)(64 215 94 185)(65 186 95 216)(66 217 96 187)(67 188 97 218)(68 219 98 189)(69 190 99 220)(70 221 100 191)(71 192 101 222)(72 223 102 193)(73 194 103 224)(74 225 104 195)(75 196 105 226)(76 227 106 197)(77 198 107 228)(78 229 108 199)(79 200 109 230)(80 231 110 201)(81 202 111 232)(82 233 112 203)(83 204 113 234)(84 235 114 205)(85 206 115 236)(86 237 116 207)(87 208 117 238)(88 239 118 209)(89 210 119 240)(90 181 120 211)
 
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), (1,195)(2,184)(3,233)(4,222)(5,211)(6,200)(7,189)(8,238)(9,227)(10,216)(11,205)(12,194)(13,183)(14,232)(15,221)(16,210)(17,199)(18,188)(19,237)(20,226)(21,215)(22,204)(23,193)(24,182)(25,231)(26,220)(27,209)(28,198)(29,187)(30,236)(31,225)(32,214)(33,203)(34,192)(35,181)(36,230)(37,219)(38,208)(39,197)(40,186)(41,235)(42,224)(43,213)(44,202)(45,191)(46,240)(47,229)(48,218)(49,207)(50,196)(51,185)(52,234)(53,223)(54,212)(55,201)(56,190)(57,239)(58,228)(59,217)(60,206)(61,167)(62,156)(63,145)(64,134)(65,123)(66,172)(67,161)(68,150)(69,139)(70,128)(71,177)(72,166)(73,155)(74,144)(75,133)(76,122)(77,171)(78,160)(79,149)(80,138)(81,127)(82,176)(83,165)(84,154)(85,143)(86,132)(87,121)(88,170)(89,159)(90,148)(91,137)(92,126)(93,175)(94,164)(95,153)(96,142)(97,131)(98,180)(99,169)(100,158)(101,147)(102,136)(103,125)(104,174)(105,163)(106,152)(107,141)(108,130)(109,179)(110,168)(111,157)(112,146)(113,135)(114,124)(115,173)(116,162)(117,151)(118,140)(119,129)(120,178), (1,240)(2,221)(3,202)(4,183)(5,224)(6,205)(7,186)(8,227)(9,208)(10,189)(11,230)(12,211)(13,192)(14,233)(15,214)(16,195)(17,236)(18,217)(19,198)(20,239)(21,220)(22,201)(23,182)(24,223)(25,204)(26,185)(27,226)(28,207)(29,188)(30,229)(31,210)(32,191)(33,232)(34,213)(35,194)(36,235)(37,216)(38,197)(39,238)(40,219)(41,200)(42,181)(43,222)(44,203)(45,184)(46,225)(47,206)(48,187)(49,228)(50,209)(51,190)(52,231)(53,212)(54,193)(55,234)(56,215)(57,196)(58,237)(59,218)(60,199)(61,136)(62,177)(63,158)(64,139)(65,180)(66,161)(67,142)(68,123)(69,164)(70,145)(71,126)(72,167)(73,148)(74,129)(75,170)(76,151)(77,132)(78,173)(79,154)(80,135)(81,176)(82,157)(83,138)(84,179)(85,160)(86,141)(87,122)(88,163)(89,144)(90,125)(91,166)(92,147)(93,128)(94,169)(95,150)(96,131)(97,172)(98,153)(99,134)(100,175)(101,156)(102,137)(103,178)(104,159)(105,140)(106,121)(107,162)(108,143)(109,124)(110,165)(111,146)(112,127)(113,168)(114,149)(115,130)(116,171)(117,152)(118,133)(119,174)(120,155), (1,144,31,174)(2,175,32,145)(3,146,33,176)(4,177,34,147)(5,148,35,178)(6,179,36,149)(7,150,37,180)(8,121,38,151)(9,152,39,122)(10,123,40,153)(11,154,41,124)(12,125,42,155)(13,156,43,126)(14,127,44,157)(15,158,45,128)(16,129,46,159)(17,160,47,130)(18,131,48,161)(19,162,49,132)(20,133,50,163)(21,164,51,134)(22,135,52,165)(23,166,53,136)(24,137,54,167)(25,168,55,138)(26,139,56,169)(27,170,57,140)(28,141,58,171)(29,172,59,142)(30,143,60,173)(61,182,91,212)(62,213,92,183)(63,184,93,214)(64,215,94,185)(65,186,95,216)(66,217,96,187)(67,188,97,218)(68,219,98,189)(69,190,99,220)(70,221,100,191)(71,192,101,222)(72,223,102,193)(73,194,103,224)(74,225,104,195)(75,196,105,226)(76,227,106,197)(77,198,107,228)(78,229,108,199)(79,200,109,230)(80,231,110,201)(81,202,111,232)(82,233,112,203)(83,204,113,234)(84,235,114,205)(85,206,115,236)(86,237,116,207)(87,208,117,238)(88,239,118,209)(89,210,119,240)(90,181,120,211)>;
 
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), (1,195)(2,184)(3,233)(4,222)(5,211)(6,200)(7,189)(8,238)(9,227)(10,216)(11,205)(12,194)(13,183)(14,232)(15,221)(16,210)(17,199)(18,188)(19,237)(20,226)(21,215)(22,204)(23,193)(24,182)(25,231)(26,220)(27,209)(28,198)(29,187)(30,236)(31,225)(32,214)(33,203)(34,192)(35,181)(36,230)(37,219)(38,208)(39,197)(40,186)(41,235)(42,224)(43,213)(44,202)(45,191)(46,240)(47,229)(48,218)(49,207)(50,196)(51,185)(52,234)(53,223)(54,212)(55,201)(56,190)(57,239)(58,228)(59,217)(60,206)(61,167)(62,156)(63,145)(64,134)(65,123)(66,172)(67,161)(68,150)(69,139)(70,128)(71,177)(72,166)(73,155)(74,144)(75,133)(76,122)(77,171)(78,160)(79,149)(80,138)(81,127)(82,176)(83,165)(84,154)(85,143)(86,132)(87,121)(88,170)(89,159)(90,148)(91,137)(92,126)(93,175)(94,164)(95,153)(96,142)(97,131)(98,180)(99,169)(100,158)(101,147)(102,136)(103,125)(104,174)(105,163)(106,152)(107,141)(108,130)(109,179)(110,168)(111,157)(112,146)(113,135)(114,124)(115,173)(116,162)(117,151)(118,140)(119,129)(120,178), (1,240)(2,221)(3,202)(4,183)(5,224)(6,205)(7,186)(8,227)(9,208)(10,189)(11,230)(12,211)(13,192)(14,233)(15,214)(16,195)(17,236)(18,217)(19,198)(20,239)(21,220)(22,201)(23,182)(24,223)(25,204)(26,185)(27,226)(28,207)(29,188)(30,229)(31,210)(32,191)(33,232)(34,213)(35,194)(36,235)(37,216)(38,197)(39,238)(40,219)(41,200)(42,181)(43,222)(44,203)(45,184)(46,225)(47,206)(48,187)(49,228)(50,209)(51,190)(52,231)(53,212)(54,193)(55,234)(56,215)(57,196)(58,237)(59,218)(60,199)(61,136)(62,177)(63,158)(64,139)(65,180)(66,161)(67,142)(68,123)(69,164)(70,145)(71,126)(72,167)(73,148)(74,129)(75,170)(76,151)(77,132)(78,173)(79,154)(80,135)(81,176)(82,157)(83,138)(84,179)(85,160)(86,141)(87,122)(88,163)(89,144)(90,125)(91,166)(92,147)(93,128)(94,169)(95,150)(96,131)(97,172)(98,153)(99,134)(100,175)(101,156)(102,137)(103,178)(104,159)(105,140)(106,121)(107,162)(108,143)(109,124)(110,165)(111,146)(112,127)(113,168)(114,149)(115,130)(116,171)(117,152)(118,133)(119,174)(120,155), (1,144,31,174)(2,175,32,145)(3,146,33,176)(4,177,34,147)(5,148,35,178)(6,179,36,149)(7,150,37,180)(8,121,38,151)(9,152,39,122)(10,123,40,153)(11,154,41,124)(12,125,42,155)(13,156,43,126)(14,127,44,157)(15,158,45,128)(16,129,46,159)(17,160,47,130)(18,131,48,161)(19,162,49,132)(20,133,50,163)(21,164,51,134)(22,135,52,165)(23,166,53,136)(24,137,54,167)(25,168,55,138)(26,139,56,169)(27,170,57,140)(28,141,58,171)(29,172,59,142)(30,143,60,173)(61,182,91,212)(62,213,92,183)(63,184,93,214)(64,215,94,185)(65,186,95,216)(66,217,96,187)(67,188,97,218)(68,219,98,189)(69,190,99,220)(70,221,100,191)(71,192,101,222)(72,223,102,193)(73,194,103,224)(74,225,104,195)(75,196,105,226)(76,227,106,197)(77,198,107,228)(78,229,108,199)(79,200,109,230)(80,231,110,201)(81,202,111,232)(82,233,112,203)(83,204,113,234)(84,235,114,205)(85,206,115,236)(86,237,116,207)(87,208,117,238)(88,239,118,209)(89,210,119,240)(90,181,120,211) );
 
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)], [(1,195),(2,184),(3,233),(4,222),(5,211),(6,200),(7,189),(8,238),(9,227),(10,216),(11,205),(12,194),(13,183),(14,232),(15,221),(16,210),(17,199),(18,188),(19,237),(20,226),(21,215),(22,204),(23,193),(24,182),(25,231),(26,220),(27,209),(28,198),(29,187),(30,236),(31,225),(32,214),(33,203),(34,192),(35,181),(36,230),(37,219),(38,208),(39,197),(40,186),(41,235),(42,224),(43,213),(44,202),(45,191),(46,240),(47,229),(48,218),(49,207),(50,196),(51,185),(52,234),(53,223),(54,212),(55,201),(56,190),(57,239),(58,228),(59,217),(60,206),(61,167),(62,156),(63,145),(64,134),(65,123),(66,172),(67,161),(68,150),(69,139),(70,128),(71,177),(72,166),(73,155),(74,144),(75,133),(76,122),(77,171),(78,160),(79,149),(80,138),(81,127),(82,176),(83,165),(84,154),(85,143),(86,132),(87,121),(88,170),(89,159),(90,148),(91,137),(92,126),(93,175),(94,164),(95,153),(96,142),(97,131),(98,180),(99,169),(100,158),(101,147),(102,136),(103,125),(104,174),(105,163),(106,152),(107,141),(108,130),(109,179),(110,168),(111,157),(112,146),(113,135),(114,124),(115,173),(116,162),(117,151),(118,140),(119,129),(120,178)], [(1,240),(2,221),(3,202),(4,183),(5,224),(6,205),(7,186),(8,227),(9,208),(10,189),(11,230),(12,211),(13,192),(14,233),(15,214),(16,195),(17,236),(18,217),(19,198),(20,239),(21,220),(22,201),(23,182),(24,223),(25,204),(26,185),(27,226),(28,207),(29,188),(30,229),(31,210),(32,191),(33,232),(34,213),(35,194),(36,235),(37,216),(38,197),(39,238),(40,219),(41,200),(42,181),(43,222),(44,203),(45,184),(46,225),(47,206),(48,187),(49,228),(50,209),(51,190),(52,231),(53,212),(54,193),(55,234),(56,215),(57,196),(58,237),(59,218),(60,199),(61,136),(62,177),(63,158),(64,139),(65,180),(66,161),(67,142),(68,123),(69,164),(70,145),(71,126),(72,167),(73,148),(74,129),(75,170),(76,151),(77,132),(78,173),(79,154),(80,135),(81,176),(82,157),(83,138),(84,179),(85,160),(86,141),(87,122),(88,163),(89,144),(90,125),(91,166),(92,147),(93,128),(94,169),(95,150),(96,131),(97,172),(98,153),(99,134),(100,175),(101,156),(102,137),(103,178),(104,159),(105,140),(106,121),(107,162),(108,143),(109,124),(110,165),(111,146),(112,127),(113,168),(114,149),(115,130),(116,171),(117,152),(118,133),(119,174),(120,155)], [(1,144,31,174),(2,175,32,145),(3,146,33,176),(4,177,34,147),(5,148,35,178),(6,179,36,149),(7,150,37,180),(8,121,38,151),(9,152,39,122),(10,123,40,153),(11,154,41,124),(12,125,42,155),(13,156,43,126),(14,127,44,157),(15,158,45,128),(16,129,46,159),(17,160,47,130),(18,131,48,161),(19,162,49,132),(20,133,50,163),(21,164,51,134),(22,135,52,165),(23,166,53,136),(24,137,54,167),(25,168,55,138),(26,139,56,169),(27,170,57,140),(28,141,58,171),(29,172,59,142),(30,143,60,173),(61,182,91,212),(62,213,92,183),(63,184,93,214),(64,215,94,185),(65,186,95,216),(66,217,96,187),(67,188,97,218),(68,219,98,189),(69,190,99,220),(70,221,100,191),(71,192,101,222),(72,223,102,193),(73,194,103,224),(74,225,104,195),(75,196,105,226),(76,227,106,197),(77,198,107,228),(78,229,108,199),(79,200,109,230),(80,231,110,201),(81,202,111,232),(82,233,112,203),(83,204,113,234),(84,235,114,205),(85,206,115,236),(86,237,116,207),(87,208,117,238),(88,239,118,209),(89,210,119,240),(90,181,120,211)])
 

Matrix representation ►G ⊆ GL8(𝔽61)

14355350000
184340550000
503836360000
385025590000
0000060059
00001020
00000001
000000600
,
45600390000
161639310000
545827600000
5801340000
00002517500
00004436011
00003603617
00000254425
,
346030220000
11630300000
23584510000
162360270000
00004425270
00003617034
00001701725
00000443644
,
600000000
060000000
006000000
000600000
0000600590
0000060059
00001010
00000101

G:=sub<GL(8,GF(61))| [1,18,50,38,0,0,0,0,43,43,38,50,0,0,0,0,55,40,36,25,0,0,0,0,35,55,36,59,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,2,0,60,0,0,0,0,59,0,1,0],[45,16,54,58,0,0,0,0,60,16,58,0,0,0,0,0,0,39,27,1,0,0,0,0,39,31,60,34,0,0,0,0,0,0,0,0,25,44,36,0,0,0,0,0,17,36,0,25,0,0,0,0,50,0,36,44,0,0,0,0,0,11,17,25],[34,1,23,16,0,0,0,0,60,16,58,23,0,0,0,0,30,30,45,60,0,0,0,0,22,30,1,27,0,0,0,0,0,0,0,0,44,36,17,0,0,0,0,0,25,17,0,44,0,0,0,0,27,0,17,36,0,0,0,0,0,34,25,44],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,1,0,0,0,0,0,0,60,0,1,0,0,0,0,59,0,1,0,0,0,0,0,0,59,0,1] >;
 

57 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A10B10C10D10E10F12A12B12C12D12E12F15A15B20A···20F20G···20L30A30B60A···60F
order12222223444444444455666101010101010121212121212151520···2020···20303060···60
size116103030302222666101010302221010226666444202020444···412···12448···8

57 irreducible representations

dim1111111122222222444448
type++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2S3D5D6D6D6D10D10D102- (1+4)S3×D5Q8.15D6C2×S3×D5Q8.10D10C60.57C23
kernelC60.57C23D60⋊C2D15⋊Q8D6.D10C12.28D10C3×Q8×D5C5×S3×Q8Q8⋊3D15Q8×D5S3×Q8Dic10C4×D5C5×Q8Dic6C4×S3C3×Q8C15Q8C5C4C3C1
# reps1333311112331662122642

In GAP, Magma, Sage, TeX 

C_{60}._{57}C_2^3
 
% in TeX
 
G:=Group("C60.57C2^3");
 
// GroupNames label
 
G:=SmallGroup(480,1105);
 
// by ID
 
G=gap.SmallGroup(480,1105);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,100,185,80,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^60=b^2=c^2=1,d^2=a^30,b*a*b=a^49,c*a*c=a^41,d*a*d^-1=a^31,c*b*c=a^30*b,b*d=d*b,c*d=d*c>;
 
// generators/relations
 

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