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G = C23.26(S3×D5)  order 480 = 25·3·5

1st non-split extension by C23 of S3×D5 acting via S3×D5/C5×S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.26(S3×D5), C23.D5.6S3, (C2×Dic3)⋊3Dic5, (C10×Dic3)⋊10C4, C6.75(C4○D20), C6.Dic1027C2, (C22×C6).19D10, (C22×C10).98D6, C1519(C42⋊C2), (Dic3×Dic5)⋊28C2, C30.131(C4○D4), C22.7(S3×Dic5), (C2×C30).171C23, C30.135(C22×C4), (C2×Dic5).124D6, Dic3.8(C2×Dic5), C30.38D4.6C2, C56(C23.16D6), C10.49(D42S3), (C22×Dic3).2D5, C6.15(C22×Dic5), (C2×Dic3).186D10, C2.5(Dic5.D6), C33(C23.21D10), (C22×C30).33C22, (C6×Dic5).99C22, (C2×Dic15).121C22, (C10×Dic3).196C22, C10.122(S3×C2×C4), C2.15(C2×S3×Dic5), C22.75(C2×S3×D5), (C2×C10).77(C4×S3), (Dic3×C2×C10).3C2, (C2×C6).6(C2×Dic5), (C2×C30).110(C2×C4), (C3×C23.D5).1C2, (C5×Dic3).46(C2×C4), (C2×C6).183(C22×D5), (C2×C10).183(C22×S3), SmallGroup(480,605)

Series: Derived Chief Lower central Upper central

C1C30 — C23.26(S3×D5)
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — C23.26(S3×D5)
C15C30 — C23.26(S3×D5)
C1C22C23

Generators and relations for C23.26(S3×D5)
 G = < a,b,c,d,e,f,g | a2=b2=c2=d3=f5=1, e2=c, g2=b, ab=ba, gag-1=ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, ede-1=d-1, df=fd, dg=gd, ef=fe, eg=ge, gfg-1=f-1 >

Subgroups: 524 in 152 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C2×C4 [×10], C23, C10, C10 [×2], C10 [×2], Dic3 [×4], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C2×C12 [×2], C22×C6, C30, C30 [×2], C30 [×2], C42⋊C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×6], C22×C10, C4×Dic3 [×2], Dic3⋊C4 [×2], C6.D4, C3×C22⋊C4, C22×Dic3, C5×Dic3 [×4], C3×Dic5 [×2], Dic15 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5, C23.D5, C22×C20, C23.16D6, C6×Dic5 [×2], C10×Dic3 [×2], C10×Dic3 [×4], C2×Dic15 [×2], C22×C30, C23.21D10, Dic3×Dic5 [×2], C6.Dic10 [×2], C3×C23.D5, C30.38D4, Dic3×C2×C10, C23.26(S3×D5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], Dic5 [×4], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C2×Dic5 [×6], C22×D5, S3×C2×C4, D42S3 [×2], S3×D5, C4○D20 [×2], C22×Dic5, C23.16D6, S3×Dic5 [×2], C2×S3×D5, C23.21D10, Dic5.D6 [×2], C2×S3×Dic5, C23.26(S3×D5)

Smallest permutation representation of C23.26(S3×D5)
On 240 points
Generators in S240
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(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)(121 166)(122 167)(123 168)(124 169)(125 170)(126 171)(127 172)(128 173)(129 174)(130 175)(131 176)(132 177)(133 178)(134 179)(135 180)(136 151)(137 152)(138 153)(139 154)(140 155)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 162)(148 163)(149 164)(150 165)(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 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(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)(121 151)(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)(181 211)(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)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(121 136)(122 137)(123 138)(124 139)(125 140)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(151 166)(152 167)(153 168)(154 169)(155 170)(156 171)(157 172)(158 173)(159 174)(160 175)(161 176)(162 177)(163 178)(164 179)(165 180)(181 196)(182 197)(183 198)(184 199)(185 200)(186 201)(187 202)(188 203)(189 204)(190 205)(191 206)(192 207)(193 208)(194 209)(195 210)(211 226)(212 227)(213 228)(214 229)(215 230)(216 231)(217 232)(218 233)(219 234)(220 235)(221 236)(222 237)(223 238)(224 239)(225 240)
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)(121 131 126)(122 132 127)(123 133 128)(124 134 129)(125 135 130)(136 146 141)(137 147 142)(138 148 143)(139 149 144)(140 150 145)(151 161 156)(152 162 157)(153 163 158)(154 164 159)(155 165 160)(166 176 171)(167 177 172)(168 178 173)(169 179 174)(170 180 175)(181 186 191)(182 187 192)(183 188 193)(184 189 194)(185 190 195)(196 201 206)(197 202 207)(198 203 208)(199 204 209)(200 205 210)(211 216 221)(212 217 222)(213 218 223)(214 219 224)(215 220 225)(226 231 236)(227 232 237)(228 233 238)(229 234 239)(230 235 240)
(1 76 16 61)(2 77 17 62)(3 78 18 63)(4 79 19 64)(5 80 20 65)(6 81 21 66)(7 82 22 67)(8 83 23 68)(9 84 24 69)(10 85 25 70)(11 86 26 71)(12 87 27 72)(13 88 28 73)(14 89 29 74)(15 90 30 75)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 111 51 96)(37 112 52 97)(38 113 53 98)(39 114 54 99)(40 115 55 100)(41 116 56 101)(42 117 57 102)(43 118 58 103)(44 119 59 104)(45 120 60 105)(121 196 136 181)(122 197 137 182)(123 198 138 183)(124 199 139 184)(125 200 140 185)(126 201 141 186)(127 202 142 187)(128 203 143 188)(129 204 144 189)(130 205 145 190)(131 206 146 191)(132 207 147 192)(133 208 148 193)(134 209 149 194)(135 210 150 195)(151 226 166 211)(152 227 167 212)(153 228 168 213)(154 229 169 214)(155 230 170 215)(156 231 171 216)(157 232 172 217)(158 233 173 218)(159 234 174 219)(160 235 175 220)(161 236 176 221)(162 237 177 222)(163 238 178 223)(164 239 179 224)(165 240 180 225)
(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)
(1 153 31 123)(2 152 32 122)(3 151 33 121)(4 155 34 125)(5 154 35 124)(6 158 36 128)(7 157 37 127)(8 156 38 126)(9 160 39 130)(10 159 40 129)(11 163 41 133)(12 162 42 132)(13 161 43 131)(14 165 44 135)(15 164 45 134)(16 168 46 138)(17 167 47 137)(18 166 48 136)(19 170 49 140)(20 169 50 139)(21 173 51 143)(22 172 52 142)(23 171 53 141)(24 175 54 145)(25 174 55 144)(26 178 56 148)(27 177 57 147)(28 176 58 146)(29 180 59 150)(30 179 60 149)(61 213 91 183)(62 212 92 182)(63 211 93 181)(64 215 94 185)(65 214 95 184)(66 218 96 188)(67 217 97 187)(68 216 98 186)(69 220 99 190)(70 219 100 189)(71 223 101 193)(72 222 102 192)(73 221 103 191)(74 225 104 195)(75 224 105 194)(76 228 106 198)(77 227 107 197)(78 226 108 196)(79 230 109 200)(80 229 110 199)(81 233 111 203)(82 232 112 202)(83 231 113 201)(84 235 114 205)(85 234 115 204)(86 238 116 208)(87 237 117 207)(88 236 118 206)(89 240 119 210)(90 239 120 209)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B6A6B6C6D6E10A···10N12A12B12C12D15A15B20A···20P30A···30N
order122222344444444444444556666610···1012121212151520···2030···30
size1111222333366101010103030303022222442···220202020446···64···4

78 irreducible representations

dim1111111222222222244444
type++++++++++-++-+-+
imageC1C2C2C2C2C2C4S3D5D6D6C4○D4Dic5D10D10C4×S3C4○D20D42S3S3×D5S3×Dic5C2×S3×D5Dic5.D6
kernelC23.26(S3×D5)Dic3×Dic5C6.Dic10C3×C23.D5C30.38D4Dic3×C2×C10C10×Dic3C23.D5C22×Dic3C2×Dic5C22×C10C30C2×Dic3C2×Dic3C22×C6C2×C10C6C10C23C22C22C2
# reps12211181221484241622428

Matrix representation of C23.26(S3×D5) in GL4(𝔽61) generated by

60600
0100
00600
00060
,
60000
06000
00600
00060
,
60000
06000
0010
0001
,
1000
0100
00601
00600
,
11000
01100
00399
004822
,
58800
02000
0010
0001
,
13300
454800
00500
00050
G:=sub<GL(4,GF(61))| [60,0,0,0,6,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,60,0,0,1,0],[11,0,0,0,0,11,0,0,0,0,39,48,0,0,9,22],[58,0,0,0,8,20,0,0,0,0,1,0,0,0,0,1],[13,45,0,0,3,48,0,0,0,0,50,0,0,0,0,50] >;

C23.26(S3×D5) in GAP, Magma, Sage, TeX

C_2^3._{26}(S_3\times D_5)
% in TeX

G:=Group("C2^3.26(S3xD5)");
// GroupNames label

G:=SmallGroup(480,605);
// by ID

G=gap.SmallGroup(480,605);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^3=f^5=1,e^2=c,g^2=b,a*b=b*a,g*a*g^-1=a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*d*e^-1=d^-1,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,g*f*g^-1=f^-1>;
// generators/relations

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