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G = C23.13(S3×D5)  order 480 = 25·3·5

6th non-split extension by C23 of S3×D5 acting via S3×D5/C15=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.13(S3×D5), C23.D5.1S3, C6.76(C4○D20), C30.Q8⋊28C2, C6.Dic10⋊28C2, Dic15⋊5C4⋊27C2, (C2×Dic5).53D6, C6.D4.1D5, (C22×C10).35D6, (C22×C6).20D10, C5⋊5(C23.8D6), C15⋊13(C42⋊2C2), (Dic3×Dic5)⋊29C2, C30.132(C4○D4), C10.77(C4○D12), C6.51(D4⋊2D5), (C2×C30).172C23, (C2×Dic3).52D10, C30.38D4.7C2, C3⋊5(C23.D10), C10.76(D4⋊2S3), (C22×C30).34C22, C2.22(Dic3.D10), C2.21(Dic5.D6), C2.21(C30.C23), (C6×Dic5).100C22, (C10×Dic3).100C22, (C2×Dic15).122C22, C22.217(C2×S3×D5), (C3×C23.D5).2C2, (C5×C6.D4).2C2, (C2×C6).184(C22×D5), (C2×C10).184(C22×S3), SmallGroup(480,606)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C30 — C23.13(S3×D5)
Chief series
C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — C23.13(S3×D5)
Lower central
C15 — C2×C30 — C23.13(S3×D5)
Upper central
C1 — C22 — C23

Generators and relations for C23.13(S3×D5)
 G = < a,b,c,d,e,f,g | a2=b2=c2=d3=f5=1, e2=b, g2=c, ab=ba, eae-1=ac=ca, ad=da, af=fa, gag-1=abc, 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: 492 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C30, C30, C42⋊2C2, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C6.D4, C3×C22⋊C4, C5×Dic3, C3×Dic5, Dic15, C2×C30, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C23.D5, C5×C22⋊C4, C23.8D6, C6×Dic5, C10×Dic3, C2×Dic15, C22×C30, C23.D10, Dic3×Dic5, C30.Q8, Dic15⋊5C4, C6.Dic10, C3×C23.D5, C5×C6.D4, C30.38D4, C23.13(S3×D5)
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C42⋊2C2, C22×D5, C4○D12, D4⋊2S3, S3×D5, C4○D20, D4⋊2D5, C23.8D6, C2×S3×D5, C23.D10, Dic5.D6, C30.C23, Dic3.D10, C23.13(S3×D5)

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

(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)

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

60 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A···20H30A···30N
order122223444444444556666610···101010101012121212151520···2030···30
size111142661010122030306022222442···24444202020204412···124···4

60 irreducible representations

dim111111112222222224444444
type++++++++++++++-+-+-
imageC1C2C2C2C2C2C2C2S3D5D6D6C4○D4D10D10C4○D12C4○D20D4⋊2S3S3×D5D4⋊2D5C2×S3×D5Dic5.D6C30.C23Dic3.D10
kernelC23.13(S3×D5)Dic3×Dic5C30.Q8Dic15⋊5C4C6.Dic10C3×C23.D5C5×C6.D4C30.38D4C23.D5C6.D4C2×Dic5C22×C10C30C2×Dic3C22×C6C10C6C10C23C6C22C2C2C2
# reps111111111221642482242444

Matrix representation of C23.13(S3×D5) ►in GL4(𝔽61) generated by

1000
516000
0010
005760
,
60000
06000
00600
00060
,
1000
0100
00600
00060
,
1000
0100
00130
00747
,
11000
01100
004020
004521
,
34000
58900
0010
0001
,
91400
35200
00500
00050
G:=sub<GL(4,GF(61))| [1,51,0,0,0,60,0,0,0,0,1,57,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,13,7,0,0,0,47],[11,0,0,0,0,11,0,0,0,0,40,45,0,0,20,21],[34,58,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,3,0,0,14,52,0,0,0,0,50,0,0,0,0,50] >;
 

C23.13(S3×D5) in GAP, Magma, Sage, TeX 

C_2^3._{13}(S_3\times D_5)
 
% in TeX
 
G:=Group("C2^3.13(S3xD5)");
 
// GroupNames label
 
G:=SmallGroup(480,606);
 
// by ID
 
G=gap.SmallGroup(480,606);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,590,219,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^3=f^5=1,e^2=b,g^2=c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,g*a*g^-1=a*b*c,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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