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G = C23.D5⋊S3  order 480 = 25·3·5

3rd semidirect product of C23.D5 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.D5⋊3S3, D6⋊Dic5⋊25C2, C30.211(C2×D4), C10.155(S3×D4), C23.11(S3×D5), C3⋊2(C20.17D4), C15⋊13(C4.4D4), (C5×Dic3).19D4, (C22×C10).32D6, (C22×C6).17D10, (Dic3×Dic5)⋊25C2, C10.74(C4○D12), C30.128(C4○D4), C30.38D4⋊16C2, C6.75(D4⋊2D5), (C2×C30).167C23, (C2×Dic5).123D6, Dic3.7(C5⋊D4), C5⋊7(C23.11D6), (C22×S3).24D10, C10.75(D4⋊2S3), (C2×Dic3).115D10, (C22×C30).29C22, (C6×Dic5).97C22, C2.20(Dic3.D10), C2.20(C30.C23), (C10×Dic3).97C22, (C2×Dic15).118C22, (C2×C15⋊Q8)⋊13C2, (C2×C3⋊D4).2D5, C2.36(S3×C5⋊D4), C6.58(C2×C5⋊D4), (C10×C3⋊D4).2C2, (C3×C23.D5)⋊3C2, C22.215(C2×S3×D5), (S3×C2×C10).44C22, (C2×C6).179(C22×D5), (C2×C10).179(C22×S3), SmallGroup(480,601)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C23.D5⋊S3
 G = < a,b,c,d,e,f,g | a2=b2=c2=d5=f3=g2=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, af=fa, gag=abc, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, geg=ce=ec, cf=fc, cg=gc, ede-1=d-1, df=fd, dg=gd, ef=fe, gfg=f-1 >

Subgroups: 684 in 152 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, Q8, C23, C23, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C4.4D4, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C22×C10, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, S3×C10, C2×C30, C2×C30, C4×Dic5, C23.D5, C23.D5, C2×Dic10, D4×C10, C23.11D6, C15⋊Q8, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, S3×C2×C10, C22×C30, C20.17D4, Dic3×Dic5, D6⋊Dic5, C3×C23.D5, C30.38D4, C2×C15⋊Q8, C10×C3⋊D4, C23.D5⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4.4D4, C5⋊D4, C22×D5, C4○D12, S3×D4, D4⋊2S3, S3×D5, D4⋊2D5, C2×C5⋊D4, C23.11D6, C2×S3×D5, C20.17D4, C30.C23, Dic3.D10, S3×C5⋊D4, C23.D5⋊S3

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

(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 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)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(121 166)(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 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)

(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(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)(121 126 131)(122 127 132)(123 128 133)(124 129 134)(125 130 135)(136 141 146)(137 142 147)(138 143 148)(139 144 149)(140 145 150)(151 156 161)(152 157 162)(153 158 163)(154 159 164)(155 160 165)(166 171 176)(167 172 177)(168 173 178)(169 174 179)(170 175 180)(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)

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F10G10H10I10J10K10L10M10N12A12B12C12D15A15B20A20B20C20D30A···30N
order122222344444444556666610···1010101010101010101212121215152020202030···30
size111141226610102030306022222442···24444121212122020202044121212124···4

60 irreducible representations

dim11111112222222222244444444
type++++++++++++++++-+-+-
imageC1C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10D10C5⋊D4C4○D12S3×D4D4⋊2S3S3×D5D4⋊2D5C2×S3×D5C30.C23Dic3.D10S3×C5⋊D4
kernelC23.D5⋊S3Dic3×Dic5D6⋊Dic5C3×C23.D5C30.38D4C2×C15⋊Q8C10×C3⋊D4C23.D5C5×Dic3C2×C3⋊D4C2×Dic5C22×C10C30C2×Dic3C22×S3C22×C6Dic3C10C10C10C23C6C22C2C2C2
# reps11211111222142228411242444

Matrix representation of C23.D5⋊S3 ►in GL6(𝔽61)

100000
010000
001700
0006000
000010
00004760
,
100000
010000
0060000
0006000
0000600
0000060
,
100000
010000
001000
000100
0000600
0000060
,
100000
010000
001000
000100
0000200
00002258
,
6000000
0600000
0050000
0005000
00002521
00003736
,
60600000
100000
001000
000100
000010
000001
,
100000
60600000
001000
00526000
000010
00004760

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,7,60,0,0,0,0,0,0,1,47,0,0,0,0,0,60],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,22,0,0,0,0,0,58],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,25,37,0,0,0,0,21,36],[60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,52,0,0,0,0,0,60,0,0,0,0,0,0,1,47,0,0,0,0,0,60] >;
 

C23.D5⋊S3 in GAP, Magma, Sage, TeX 

C_2^3.D_5\rtimes S_3
 
% in TeX
 
G:=Group("C2^3.D5:S3");
 
// GroupNames label
 
G:=SmallGroup(480,601);
 
// by ID
 
G=gap.SmallGroup(480,601);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,590,135,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^5=f^3=g^2=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,g*a*g=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,g*e*g=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,g*f*g=f^-1>;
 
// generators/relations
 

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