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## G = C2×C30.C23order 480 = 25·3·5

### Direct product of C2 and C30.C23

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
 Upper central C1 — C22 — C23

Generators and relations for C2×C30.C23
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 >

Subgroups: 1404 in 328 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, C30, C30, C30, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, C6×D5, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C2×Dic10, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C2×D42S3, D5×Dic3, S3×Dic5, C15⋊D4, C15⋊Q8, C6×Dic5, C3×C5⋊D4, C10×Dic3, C5×C3⋊D4, C2×Dic15, C2×Dic15, D5×C2×C6, S3×C2×C10, C22×C30, C2×D42D5, C2×D5×Dic3, C30.C23, C2×S3×Dic5, C2×C15⋊D4, C2×C15⋊Q8, C6×C5⋊D4, C10×C3⋊D4, C22×Dic15, C2×C30.C23
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, D42S3, S3×C23, S3×D5, D42D5, C23×D5, C2×D42S3, C2×S3×D5, C2×D42D5, C30.C23, C22×S3×D5, C2×C30.C23

Smallest permutation representation of C2×C30.C23
On 240 points
Generators in S240
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 91)(57 92)(58 93)(59 94)(60 95)(121 191)(122 192)(123 193)(124 194)(125 195)(126 196)(127 197)(128 198)(129 199)(130 200)(131 201)(132 202)(133 203)(134 204)(135 205)(136 206)(137 207)(138 208)(139 209)(140 210)(141 181)(142 182)(143 183)(144 184)(145 185)(146 186)(147 187)(148 188)(149 189)(150 190)(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)
(1 136 16 121)(2 147 17 132)(3 128 18 143)(4 139 19 124)(5 150 20 135)(6 131 21 146)(7 142 22 127)(8 123 23 138)(9 134 24 149)(10 145 25 130)(11 126 26 141)(12 137 27 122)(13 148 28 133)(14 129 29 144)(15 140 30 125)(31 171 46 156)(32 152 47 167)(33 163 48 178)(34 174 49 159)(35 155 50 170)(36 166 51 151)(37 177 52 162)(38 158 53 173)(39 169 54 154)(40 180 55 165)(41 161 56 176)(42 172 57 157)(43 153 58 168)(44 164 59 179)(45 175 60 160)(61 196 76 181)(62 207 77 192)(63 188 78 203)(64 199 79 184)(65 210 80 195)(66 191 81 206)(67 202 82 187)(68 183 83 198)(69 194 84 209)(70 205 85 190)(71 186 86 201)(72 197 87 182)(73 208 88 193)(74 189 89 204)(75 200 90 185)(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 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 162)(128 163)(129 164)(130 165)(131 166)(132 167)(133 168)(134 169)(135 170)(136 171)(137 172)(138 173)(139 174)(140 175)(141 176)(142 177)(143 178)(144 179)(145 180)(146 151)(147 152)(148 153)(149 154)(150 155)(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)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G 10H 10I 10J 10K 10L 10M 10N 12A 12B 15A 15B 20A 20B 20C 20D 30A ··· 30N order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 10 10 10 10 10 10 10 12 12 15 15 20 20 20 20 30 ··· 30 size 1 1 1 1 2 2 6 6 10 10 2 6 6 10 10 15 15 15 15 30 30 2 2 2 2 2 4 4 20 20 2 ··· 2 4 4 4 4 12 12 12 12 20 20 4 4 12 12 12 12 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 C4○D4 D10 D10 D10 D10 D4⋊2S3 S3×D5 D4⋊2D5 C2×S3×D5 C30.C23 kernel C2×C30.C23 C2×D5×Dic3 C30.C23 C2×S3×Dic5 C2×C15⋊D4 C2×C15⋊Q8 C6×C5⋊D4 C10×C3⋊D4 C22×Dic15 C2×C5⋊D4 C2×C3⋊D4 C2×Dic5 C5⋊D4 C22×D5 C22×C10 C30 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C10 C23 C6 C22 C2 # reps 1 1 8 1 1 1 1 1 1 1 2 1 4 1 1 4 2 8 2 2 2 2 4 6 8

Matrix representation of C2×C30.C23 in GL6(𝔽61)

 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 60 0 0 0 0 1 0 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 1 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 17 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 17 0 0 0 0 53 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

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] >;

C2×C30.C23 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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