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G = C30.33C24  order 480 = 25·3·5

33rd non-split extension by C30 of C24 acting via C24/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.33C24, C60.57C23, C15⋊62- 1+4, Dic6.31D10, Dic10.30D6, D60.19C22, D30.17C23, Dic15.39C23, (S3×Q8)⋊5D5, (Q8×D5)⋊7S3, D15⋊Q8⋊6C2, (C4×D5).17D6, Q8.27(S3×D5), (C5×Q8).42D6, Q8⋊3D15⋊4C2, C12.28D10⋊6C2, D60⋊C2⋊6C2, C15⋊Q8.6C22, (C4×S3).17D10, (C3×Q8).25D10, C6.33(C23×D5), D6.D10⋊6C2, C10.33(S3×C23), C20.57(C22×S3), 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×C10).32C23, (S3×C20).20C22, C3⋊2(Q8.10D10), (C4×D15).20C22, (D5×C12).20C22, D10.43(C22×S3), D30.C2.4C22, (Q8×C15).20C22, (C5×Dic6).21C22, (C3×Dic5).16C23, Dic5.18(C22×S3), (C5×Dic3).19C23, Dic3.18(C22×D5), (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 — C30.33C24
Chief series
C1 — C5 — C15 — C30 — C6×D5 — C15⋊D4 — D6.D10 — C30.33C24
Lower central
C15 — C30 — C30.33C24
Upper central
C1 — C2 — Q8

Generators and relations for C30.33C24
 G = < a,b,c,d | a60=b2=c2=1, d2=a30, bab=a49, cac=a41, dad-1=a31, cbc=a30b, bd=db, cd=dc >

Subgroups: 1388 in 292 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×Q8, C5×S3, C3×D5, D15, C30, 2- 1+4, Dic10, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C2×C20, C5×Q8, C5×Q8, C4○D12, S3×Q8, S3×Q8, Q8⋊3S3, C6×Q8, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C4○D20, Q8×D5, Q8×D5, Q8⋊2D5, Q8×C10, Q8.15D6, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, C3×Dic10, D5×C12, C5×Dic6, S3×C20, C4×D15, D60, Q8×C15, Q8.10D10, D60⋊C2, D15⋊Q8, D6.D10, C12.28D10, C3×Q8×D5, C5×S3×Q8, Q8⋊3D15, C30.33C24
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2- 1+4, C22×D5, S3×C23, S3×D5, C23×D5, Q8.15D6, C2×S3×D5, Q8.10D10, C22×S3×D5, C30.33C24

Smallest permutation representation of C30.33C24
►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 78)(2 67)(3 116)(4 105)(5 94)(6 83)(7 72)(8 61)(9 110)(10 99)(11 88)(12 77)(13 66)(14 115)(15 104)(16 93)(17 82)(18 71)(19 120)(20 109)(21 98)(22 87)(23 76)(24 65)(25 114)(26 103)(27 92)(28 81)(29 70)(30 119)(31 108)(32 97)(33 86)(34 75)(35 64)(36 113)(37 102)(38 91)(39 80)(40 69)(41 118)(42 107)(43 96)(44 85)(45 74)(46 63)(47 112)(48 101)(49 90)(50 79)(51 68)(52 117)(53 106)(54 95)(55 84)(56 73)(57 62)(58 111)(59 100)(60 89)(121 182)(122 231)(123 220)(124 209)(125 198)(126 187)(127 236)(128 225)(129 214)(130 203)(131 192)(132 181)(133 230)(134 219)(135 208)(136 197)(137 186)(138 235)(139 224)(140 213)(141 202)(142 191)(143 240)(144 229)(145 218)(146 207)(147 196)(148 185)(149 234)(150 223)(151 212)(152 201)(153 190)(154 239)(155 228)(156 217)(157 206)(158 195)(159 184)(160 233)(161 222)(162 211)(163 200)(164 189)(165 238)(166 227)(167 216)(168 205)(169 194)(170 183)(171 232)(172 221)(173 210)(174 199)(175 188)(176 237)(177 226)(178 215)(179 204)(180 193)

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

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

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+4S3×D5Q8.15D6C2×S3×D5Q8.10D10C30.33C24
kernelC30.33C24D60⋊C2D15⋊Q8D6.D10C12.28D10C3×Q8×D5C5×S3×Q8Q8⋊3D15Q8×D5S3×Q8Dic10C4×D5C5×Q8Dic6C4×S3C3×Q8C15Q8C5C4C3C1
# reps1333311112331662122642

Matrix representation of C30.33C24 ►in 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] >;
 

C30.33C24 in GAP, Magma, Sage, TeX 

C_{30}._{33}C_2^4
 
% in TeX
 
G:=Group("C30.33C2^4");
 
// 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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