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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 [×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, D42S3 [×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, D42D5 [×8], C22×Dic5 [×2], C2×C5⋊D4, C2×C5⋊D4, D4×C10, C2×D42S3, 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×D42D5, 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], D42S3 [×2], S3×C23, S3×D5, D42D5 [×2], C23×D5, C2×D42S3, C2×S3×D5 [×3], C2×D42D5, C30.C23 [×2], C22×S3×D5, C2×C30.C23

Smallest permutation representation of C2×C30.C23
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)(31 116)(32 117)(33 118)(34 119)(35 120)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(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)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)
(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)
(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)
(1 126 16 141)(2 137 17 122)(3 148 18 133)(4 129 19 144)(5 140 20 125)(6 121 21 136)(7 132 22 147)(8 143 23 128)(9 124 24 139)(10 135 25 150)(11 146 26 131)(12 127 27 142)(13 138 28 123)(14 149 29 134)(15 130 30 145)(31 161 46 176)(32 172 47 157)(33 153 48 168)(34 164 49 179)(35 175 50 160)(36 156 51 171)(37 167 52 152)(38 178 53 163)(39 159 54 174)(40 170 55 155)(41 151 56 166)(42 162 57 177)(43 173 58 158)(44 154 59 169)(45 165 60 180)(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)(91 216 106 231)(92 227 107 212)(93 238 108 223)(94 219 109 234)(95 230 110 215)(96 211 111 226)(97 222 112 237)(98 233 113 218)(99 214 114 229)(100 225 115 240)(101 236 116 221)(102 217 117 232)(103 228 118 213)(104 239 119 224)(105 220 120 235)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 113)(79 114)(80 115)(81 116)(82 117)(83 118)(84 119)(85 120)(86 91)(87 92)(88 93)(89 94)(90 95)(121 176)(122 177)(123 178)(124 179)(125 180)(126 151)(127 152)(128 153)(129 154)(130 155)(131 156)(132 157)(133 158)(134 159)(135 160)(136 161)(137 162)(138 163)(139 164)(140 165)(141 166)(142 167)(143 168)(144 169)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)

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

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

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

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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