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G = C2×C30.C23  order 480 = 25·3·5

Direct product of C2 and C30.C23

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

Aliases: C2×C30.C23, C30.42C24, Dic15.43C23, C5⋊D4⋊10D6, C30⋊9(C4○D4), C3⋊D4⋊10D10, C15⋊Q8⋊16C22, C6⋊4(D4⋊2D5), C10⋊4(D4⋊2S3), C23.58(S3×D5), C6.42(C23×D5), C15⋊D4⋊18C22, C10.42(S3×C23), (C22×D5).71D6, D6.19(C22×D5), (C6×D5).19C23, (C22×C6).69D10, (C22×C10).83D6, (S3×C10).21C23, (C2×C30).245C23, (C2×Dic5).140D6, (S3×Dic5)⋊15C22, (D5×Dic3)⋊16C22, D10.22(C22×S3), (C22×S3).61D10, (C2×Dic3).132D10, (C2×Dic15)⋊36C22, (C22×Dic15)⋊18C2, (C22×C30).83C22, Dic5.23(C22×S3), Dic3.21(C22×D5), (C5×Dic3).22C23, (C3×Dic5).21C23, (C6×Dic5).133C22, (C10×Dic3).132C22, C15⋊18(C2×C4○D4), C5⋊5(C2×D4⋊2S3), C3⋊5(C2×D4⋊2D5), (C2×C15⋊Q8)⋊24C2, (C2×C5⋊D4)⋊11S3, (C2×C3⋊D4)⋊11D5, (C6×C5⋊D4)⋊12C2, (C2×D5×Dic3)⋊23C2, (C2×S3×Dic5)⋊22C2, (C10×C3⋊D4)⋊11C2, (C2×C15⋊D4)⋊21C2, C2.44(C22×S3×D5), C22.20(C2×S3×D5), (D5×C2×C6).62C22, (C3×C5⋊D4)⋊13C22, (C5×C3⋊D4)⋊12C22, (S3×C2×C10).61C22, (C2×C10).14(C22×S3), (C2×C6).251(C22×D5), SmallGroup(480,1114)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — C2×C30.C23
Chief series
C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — C2×D5×Dic3 — C2×C30.C23
Lower central
C15 — C30 — C2×C30.C23

Subgroups: 1404 in 328 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×8], C22, C22 [×2], C22 [×10], C5, S3 [×2], C6, C6 [×2], C6 [×4], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×4], Dic3 [×2], Dic3 [×4], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×6], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×6], Dic6 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×10], C3⋊D4 [×4], C3⋊D4 [×4], C2×C12, C3×D4 [×4], C22×S3, C22×C6, C22×C6, C5×S3 [×2], C3×D5 [×2], C30, C30 [×2], C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×10], C5⋊D4 [×4], C5⋊D4 [×4], C2×C20, C5×D4 [×4], C22×D5, C22×C10, C22×C10, C2×Dic6, S3×C2×C4, D4⋊2S3 [×8], C22×Dic3 [×2], C2×C3⋊D4, C2×C3⋊D4, C6×D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×4], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×Dic10, C2×C4×D5, D4⋊2D5 [×8], C22×Dic5 [×2], C2×C5⋊D4, C2×C5⋊D4, D4×C10, C2×D4⋊2S3, D5×Dic3 [×4], S3×Dic5 [×4], C15⋊D4 [×4], C15⋊Q8 [×4], C6×Dic5, C3×C5⋊D4 [×4], C10×Dic3, C5×C3⋊D4 [×4], C2×Dic15 [×2], C2×Dic15 [×4], D5×C2×C6, S3×C2×C10, C22×C30, C2×D4⋊2D5, C2×D5×Dic3, C30.C23 [×8], C2×S3×Dic5, C2×C15⋊D4, C2×C15⋊Q8, C6×C5⋊D4, C10×C3⋊D4, C22×Dic15, C2×C30.C23

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], D4⋊2S3 [×2], S3×C23, S3×D5, D4⋊2D5 [×2], C23×D5, C2×D4⋊2S3, C2×S3×D5 [×3], C2×D4⋊2D5, C30.C23 [×2], C22×S3×D5, C2×C30.C23

Generators and relations
 G = < a,b,c,d,e | a2=b30=c2=e2=1, d2=b15, ab=ba, ac=ca, ad=da, ae=ea, cbc=b19, dbd-1=b11, be=eb, cd=dc, ece=b15c, ede=b15d >

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

(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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽61)

100000
010000
0060000
0006000
0000600
0000060
,
6000000
0600000
00176000
001000
0000601
0000600
,
100000
0600000
001000
00176000
000010
000001
,
1100000
0500000
001000
000100
00003617
00005325
,
010000
100000
0060000
0006000
0000600
0000060

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,17,1,0,0,0,0,60,0,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,53,0,0,0,0,17,25],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
 

66 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J10K10L10M10N12A12B15A15B20A20B20C20D30A···30N
order12222222223444444444455666666610···101010101010101010121215152020202030···30
size1111226610102661010151515153030222224420202···2444412121212202044121212124···4

66 irreducible representations

dim1111111112222222222244444
type+++++++++++++++++++-+-+-
imageC1C2C2C2C2C2C2C2C2S3D5D6D6D6D6C4○D4D10D10D10D10D4⋊2S3S3×D5D4⋊2D5C2×S3×D5C30.C23
kernelC2×C30.C23C2×D5×Dic3C30.C23C2×S3×Dic5C2×C15⋊D4C2×C15⋊Q8C6×C5⋊D4C10×C3⋊D4C22×Dic15C2×C5⋊D4C2×C3⋊D4C2×Dic5C5⋊D4C22×D5C22×C10C30C2×Dic3C3⋊D4C22×S3C22×C6C10C23C6C22C2
# reps1181111111214114282222468

In GAP, Magma, Sage, TeX 

C_2\times C_{30}.C_2^3
 
% in TeX
 
G:=Group("C2xC30.C2^3");
 
// GroupNames label
 
G:=SmallGroup(480,1114);
 
// by ID
 
G=gap.SmallGroup(480,1114);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,675,346,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^30=c^2=e^2=1,d^2=b^15,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^19,d*b*d^-1=b^11,b*e=e*b,c*d=d*c,e*c*e=b^15*c,e*d*e=b^15*d>;
 
// generators/relations
 

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