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G = C22×D30.C2  order 480 = 25·3·5

Direct product of C22 and D30.C2

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C22×D30.C2, C30.45C24, D30.43C23, C15⋊7(C23×C4), D30⋊31(C2×C4), C30⋊7(C22×C4), (C2×Dic5)⋊26D6, D15⋊3(C22×C4), C6.45(C23×D5), C23.69(S3×D5), (C2×Dic3)⋊26D10, (C22×D15)⋊11C4, C10.45(S3×C23), (C3×Dic5)⋊7C23, Dic5⋊7(C22×S3), (C5×Dic3)⋊7C23, Dic3⋊7(C22×D5), (C23×D15).5C2, (C2×C30).248C23, (C22×Dic3)⋊10D5, (C6×Dic5)⋊30C22, (C22×Dic5)⋊13S3, (C22×C10).117D6, (C22×C6).100D10, (C10×Dic3)⋊30C22, (C22×C30).86C22, (C22×D15).123C22, C6⋊2(C2×C4×D5), C10⋊3(S3×C2×C4), C5⋊3(S3×C22×C4), C3⋊2(D5×C22×C4), (C2×C6)⋊9(C4×D5), (C2×C30)⋊25(C2×C4), (C2×C10)⋊18(C4×S3), C2.4(C22×S3×D5), (C2×C6×Dic5)⋊10C2, (Dic3×C2×C10)⋊10C2, C22.108(C2×S3×D5), (C2×C6).254(C22×D5), (C2×C10).252(C22×S3), SmallGroup(480,1117)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C22×D30.C2
Chief series
C1 — C5 — C15 — C30 — C3×Dic5 — D30.C2 — C2×D30.C2 — C22×D30.C2
Lower central
C15 — C22×D30.C2

Subgroups: 2204 in 472 conjugacy classes, 188 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×8], C22 [×7], C22 [×28], C5, S3 [×8], C6, C6 [×6], C2×C4 [×28], C23, C23 [×14], D5 [×8], C10, C10 [×6], Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×7], C15, C22×C4 [×14], C24, Dic5 [×4], C20 [×4], D10 [×28], C2×C10 [×7], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×14], C22×C6, D15 [×8], C30, C30 [×6], C23×C4, C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×14], C22×C10, S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, C5×Dic3 [×4], C3×Dic5 [×4], D30 [×28], C2×C30 [×7], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, S3×C22×C4, D30.C2 [×16], C6×Dic5 [×6], C10×Dic3 [×6], C22×D15 [×14], C22×C30, D5×C22×C4, C2×D30.C2 [×12], C2×C6×Dic5, Dic3×C2×C10, C23×D15, C22×D30.C2

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D5, D6 [×7], C22×C4 [×14], C24, D10 [×7], C4×S3 [×4], C22×S3 [×7], C23×C4, C4×D5 [×4], C22×D5 [×7], S3×C2×C4 [×6], S3×C23, S3×D5, C2×C4×D5 [×6], C23×D5, S3×C22×C4, D30.C2 [×4], C2×S3×D5 [×3], D5×C22×C4, C2×D30.C2 [×6], C22×S3×D5, C22×D30.C2

Generators and relations
 G = < a,b,c,d,e | a2=b2=c30=d2=1, e2=c15, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ece-1=c19, ede-1=c18d >

Smallest permutation representation
►On 240 points
Generators in S240
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 88)(32 89)(33 90)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(121 233)(122 234)(123 235)(124 236)(125 237)(126 238)(127 239)(128 240)(129 211)(130 212)(131 213)(132 214)(133 215)(134 216)(135 217)(136 218)(137 219)(138 220)(139 221)(140 222)(141 223)(142 224)(143 225)(144 226)(145 227)(146 228)(147 229)(148 230)(149 231)(150 232)(151 195)(152 196)(153 197)(154 198)(155 199)(156 200)(157 201)(158 202)(159 203)(160 204)(161 205)(162 206)(163 207)(164 208)(165 209)(166 210)(167 181)(168 182)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)

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

(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)

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

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

Matrix representation ►G ⊆ GL5(𝔽61)

600000
060000
006000
00010
00001
,
10000
060000
006000
00010
00001
,
10000
0606000
01000
0001716
00011
,
10000
01100
006000
000145
000060
,
600000
01000
00100
0005757
000504

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,1,0,0,0,60,0,0,0,0,0,0,17,1,0,0,0,16,1],[1,0,0,0,0,0,1,0,0,0,0,1,60,0,0,0,0,0,1,0,0,0,0,45,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,57,50,0,0,0,57,4] >;
 

96 conjugacy classes

class 1 2A···2G2H···2O 3 4A···4H4I···4P5A5B6A···6G10A···10N12A···12H15A15B20A···20P30A···30N
order12···22···234···44···4556···610···1012···12151520···2030···30
size11···115···1523···35···5222···22···210···10446···64···4

96 irreducible representations

dim11111122222222444
type++++++++++++++
imageC1C2C2C2C2C4S3D5D6D6D10D10C4×S3C4×D5S3×D5D30.C2C2×S3×D5
kernelC22×D30.C2C2×D30.C2C2×C6×Dic5Dic3×C2×C10C23×D15C22×D15C22×Dic5C22×Dic3C2×Dic5C22×C10C2×Dic3C22×C6C2×C10C2×C6C23C22C22
# reps112111161261122816286

In GAP, Magma, Sage, TeX 

C_2^2\times D_{30}.C_2
 
% in TeX
 
G:=Group("C2^2xD30.C2");
 
// GroupNames label
 
G:=SmallGroup(480,1117);
 
// by ID
 
G=gap.SmallGroup(480,1117);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^30=d^2=1,e^2=c^15,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,e*c*e^-1=c^19,e*d*e^-1=c^18*d>;
 
// generators/relations
 

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