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G = C23.17(S3×D5)  order 480 = 25·3·5

10th non-split extension by C23 of S3×D5 acting via S3×D5/C15=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C6×D5).16D4, C6.156(D4×D5), C6.D4⋊8D5, C30.228(C2×D4), C23.17(S3×D5), C6.82(C4○D20), C6.Dic10⋊32C2, Dic15⋊5C4⋊32C2, (C2×Dic5).59D6, (C22×D5).61D6, (C22×C6).29D10, (C22×C10).45D6, C30.143(C4○D4), D10⋊Dic3⋊30C2, C6.81(D4⋊2D5), C30.38D4⋊21C2, D10.21(C3⋊D4), C3⋊7(D10.12D4), (C2×C30).190C23, (C2×Dic3).59D10, C5⋊3(C23.23D6), C10.54(D4⋊2S3), C15⋊22(C22.D4), (C22×C30).52C22, C2.26(C30.C23), C2.27(Dic5.D6), (C6×Dic5).109C22, (C2×Dic15).130C22, (C10×Dic3).109C22, (C2×D5×Dic3)⋊14C2, (C2×C5⋊D4).5S3, (C6×C5⋊D4).5C2, C2.38(D5×C3⋊D4), C10.60(C2×C3⋊D4), (D5×C2×C6).48C22, C22.227(C2×S3×D5), (C5×C6.D4)⋊9C2, (C2×C6).202(C22×D5), (C2×C10).202(C22×S3), SmallGroup(480,624)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C30 — C23.17(S3×D5)
Chief series
C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — C23.17(S3×D5)
Lower central
C15 — C2×C30 — C23.17(S3×D5)
Upper central
C1 — C22 — C23

Generators and relations for C23.17(S3×D5)
 G = < a,b,c,d,e,f,g | a2=b2=c2=d3=f5=g2=1, e2=b, gag=ab=ba, eae-1=ac=ca, ad=da, 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=f-1 >

Subgroups: 732 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C22.D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, Dic3⋊C4, C6.D4, C6.D4, C22×Dic3, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, C6×D5, C2×C30, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C23.23D6, D5×Dic3, C6×Dic5, C3×C5⋊D4, C10×Dic3, C2×Dic15, D5×C2×C6, C22×C30, D10.12D4, D10⋊Dic3, Dic15⋊5C4, C6.Dic10, C5×C6.D4, C30.38D4, C2×D5×Dic3, C6×C5⋊D4, C23.17(S3×D5)
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C22.D4, C22×D5, D4⋊2S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, D4⋊2D5, C23.23D6, C2×S3×D5, D10.12D4, Dic5.D6, C30.C23, D5×C3⋊D4, C23.17(S3×D5)

Smallest permutation representation of C23.17(S3×D5)
►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)(121 196)(122 197)(123 198)(124 199)(125 200)(126 201)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 208)(134 209)(135 210)(136 181)(137 182)(138 183)(139 184)(140 185)(141 186)(142 187)(143 188)(144 189)(145 190)(146 191)(147 192)(148 193)(149 194)(150 195)(151 226)(152 227)(153 228)(154 229)(155 230)(156 231)(157 232)(158 233)(159 234)(160 235)(161 236)(162 237)(163 238)(164 239)(165 240)(166 211)(167 212)(168 213)(169 214)(170 215)(171 216)(172 217)(173 218)(174 219)(175 220)(176 221)(177 222)(178 223)(179 224)(180 225)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J12A12B15A15B20A···20H30A···30N
order12222223444444455666666610···10101010101212151520···2030···30
size1111410102661220303060222224420202···2444420204412···124···4

60 irreducible representations

dim111111112222222222244444444
type++++++++++++++++-++-+-
imageC1C2C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10C3⋊D4C4○D20D4⋊2S3S3×D5D4×D5D4⋊2D5C2×S3×D5Dic5.D6C30.C23D5×C3⋊D4
kernelC23.17(S3×D5)D10⋊Dic3Dic15⋊5C4C6.Dic10C5×C6.D4C30.38D4C2×D5×Dic3C6×C5⋊D4C2×C5⋊D4C6×D5C6.D4C2×Dic5C22×D5C22×C10C30C2×Dic3C22×C6D10C6C10C23C6C6C22C2C2C2
# reps111111111221114424822222444

Matrix representation of C23.17(S3×D5) ►in GL6(𝔽61)

6010000
010000
0060000
0006000
0000943
00001852
,
6000000
0600000
001000
000100
000010
000001
,
6000000
0600000
001000
000100
0000600
0000060
,
100000
010000
001000
000100
0000601
0000600
,
1100000
22500000
001000
000100
00004249
00003019
,
100000
010000
00606000
00191800
000010
000001
,
100000
2600000
0001700
0018000
000010
000001

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,1,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,9,18,0,0,0,0,43,52],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[11,22,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,30,0,0,0,0,49,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,19,0,0,0,0,60,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,60,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C23.17(S3×D5) in GAP, Magma, Sage, TeX 

C_2^3._{17}(S_3\times D_5)
 
% in TeX
 
G:=Group("C2^3.17(S3xD5)");
 
// GroupNames label
 
G:=SmallGroup(480,624);
 
// by ID
 
G=gap.SmallGroup(480,624);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,254,219,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^3=f^5=g^2=1,e^2=b,g*a*g=a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*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=f^-1>;
 
// generators/relations
 

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