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G = C3×C23.11D10order 480 = 25·3·5

Direct product of C3 and C23.11D10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3×C23.11D10, (C4×Dic5)⋊9C6, (C2×Dic5)⋊4C12, (C6×Dic5)⋊10C4, C10.D47C6, C23.15(C6×D5), C22.6(D5×C12), C23.D5.1C6, (C12×Dic5)⋊27C2, (C2×C12).270D10, (C22×C6).71D10, C1525(C42⋊C2), C30.227(C4○D4), C10.18(C22×C12), (C2×C30).335C23, C30.176(C22×C4), (C2×C60).393C22, Dic5.20(C2×C12), C6.106(D42D5), (C22×Dic5).4C6, (C22×C30).93C22, (C6×Dic5).282C22, C2.7(D5×C2×C12), C6.101(C2×C4×D5), C53(C3×C42⋊C2), (C2×C6).40(C4×D5), (C2×C4).21(C6×D5), C22.12(D5×C2×C6), (C2×C20).47(C2×C6), C10.18(C3×C4○D4), C2.1(C3×D42D5), (C5×C22⋊C4).3C6, C22⋊C4.3(C3×D5), (C3×C22⋊C4).6D5, (C2×C6×Dic5).10C2, (C2×C30).121(C2×C4), (C2×C10).24(C2×C12), (C15×C22⋊C4).8C2, (C3×C23.D5).7C2, (C3×C10.D4)⋊23C2, (C22×C10).12(C2×C6), (C2×C10).18(C22×C6), (C2×Dic5).26(C2×C6), (C3×Dic5).62(C2×C4), (C2×C6).331(C22×D5), SmallGroup(480,670)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C23.11D10
C1C5C10C2×C10C2×C30C6×Dic5C2×C6×Dic5 — C3×C23.11D10
C5C10 — C3×C23.11D10
C1C2×C6C3×C22⋊C4

Generators and relations for C3×C23.11D10
 G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e10=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 368 in 152 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C12 [×8], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C22×C4, Dic5 [×4], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×8], C22×C6, C30, C30 [×2], C30 [×2], C42⋊C2, C2×Dic5 [×2], C2×Dic5 [×6], C2×C20 [×2], C22×C10, C4×C12 [×2], C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4 [×2], C22×C12, C3×Dic5 [×4], C3×Dic5 [×2], C60 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5 [×2], C10.D4 [×2], C23.D5, C5×C22⋊C4, C22×Dic5, C3×C42⋊C2, C6×Dic5 [×2], C6×Dic5 [×6], C2×C60 [×2], C22×C30, C23.11D10, C12×Dic5 [×2], C3×C10.D4 [×2], C3×C23.D5, C15×C22⋊C4, C2×C6×Dic5, C3×C23.11D10
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, D5, C12 [×4], C2×C6 [×7], C22×C4, C4○D4 [×2], D10 [×3], C2×C12 [×6], C22×C6, C3×D5, C42⋊C2, C4×D5 [×2], C22×D5, C22×C12, C3×C4○D4 [×2], C6×D5 [×3], C2×C4×D5, D42D5 [×2], C3×C42⋊C2, D5×C12 [×2], D5×C2×C6, C23.11D10, D5×C2×C12, C3×D42D5 [×2], C3×C23.11D10

Smallest permutation representation of C3×C23.11D10
On 240 points
Generators in S240
(1 68 127)(2 69 128)(3 70 129)(4 71 130)(5 72 131)(6 73 132)(7 74 133)(8 75 134)(9 76 135)(10 77 136)(11 78 137)(12 79 138)(13 80 139)(14 61 140)(15 62 121)(16 63 122)(17 64 123)(18 65 124)(19 66 125)(20 67 126)(21 174 181)(22 175 182)(23 176 183)(24 177 184)(25 178 185)(26 179 186)(27 180 187)(28 161 188)(29 162 189)(30 163 190)(31 164 191)(32 165 192)(33 166 193)(34 167 194)(35 168 195)(36 169 196)(37 170 197)(38 171 198)(39 172 199)(40 173 200)(41 234 101)(42 235 102)(43 236 103)(44 237 104)(45 238 105)(46 239 106)(47 240 107)(48 221 108)(49 222 109)(50 223 110)(51 224 111)(52 225 112)(53 226 113)(54 227 114)(55 228 115)(56 229 116)(57 230 117)(58 231 118)(59 232 119)(60 233 120)(81 150 211)(82 151 212)(83 152 213)(84 153 214)(85 154 215)(86 155 216)(87 156 217)(88 157 218)(89 158 219)(90 159 220)(91 160 201)(92 141 202)(93 142 203)(94 143 204)(95 144 205)(96 145 206)(97 146 207)(98 147 208)(99 148 209)(100 149 210)
(1 11)(2 156)(3 13)(4 158)(5 15)(6 160)(7 17)(8 142)(9 19)(10 144)(12 146)(14 148)(16 150)(18 152)(20 154)(21 31)(22 46)(23 33)(24 48)(25 35)(26 50)(27 37)(28 52)(29 39)(30 54)(32 56)(34 58)(36 60)(38 42)(40 44)(41 51)(43 53)(45 55)(47 57)(49 59)(61 209)(62 72)(63 211)(64 74)(65 213)(66 76)(67 215)(68 78)(69 217)(70 80)(71 219)(73 201)(75 203)(77 205)(79 207)(81 122)(82 92)(83 124)(84 94)(85 126)(86 96)(87 128)(88 98)(89 130)(90 100)(91 132)(93 134)(95 136)(97 138)(99 140)(101 111)(102 198)(103 113)(104 200)(105 115)(106 182)(107 117)(108 184)(109 119)(110 186)(112 188)(114 190)(116 192)(118 194)(120 196)(121 131)(123 133)(125 135)(127 137)(129 139)(141 151)(143 153)(145 155)(147 157)(149 159)(161 225)(162 172)(163 227)(164 174)(165 229)(166 176)(167 231)(168 178)(169 233)(170 180)(171 235)(173 237)(175 239)(177 221)(179 223)(181 191)(183 193)(185 195)(187 197)(189 199)(202 212)(204 214)(206 216)(208 218)(210 220)(222 232)(224 234)(226 236)(228 238)(230 240)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)(169 179)(170 180)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)(201 211)(202 212)(203 213)(204 214)(205 215)(206 216)(207 217)(208 218)(209 219)(210 220)(221 231)(222 232)(223 233)(224 234)(225 235)(226 236)(227 237)(228 238)(229 239)(230 240)
(1 145)(2 146)(3 147)(4 148)(5 149)(6 150)(7 151)(8 152)(9 153)(10 154)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 141)(18 142)(19 143)(20 144)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(61 219)(62 220)(63 201)(64 202)(65 203)(66 204)(67 205)(68 206)(69 207)(70 208)(71 209)(72 210)(73 211)(74 212)(75 213)(76 214)(77 215)(78 216)(79 217)(80 218)(81 132)(82 133)(83 134)(84 135)(85 136)(86 137)(87 138)(88 139)(89 140)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)(100 131)(101 187)(102 188)(103 189)(104 190)(105 191)(106 192)(107 193)(108 194)(109 195)(110 196)(111 197)(112 198)(113 199)(114 200)(115 181)(116 182)(117 183)(118 184)(119 185)(120 186)(161 235)(162 236)(163 237)(164 238)(165 239)(166 240)(167 221)(168 222)(169 223)(170 224)(171 225)(172 226)(173 227)(174 228)(175 229)(176 230)(177 231)(178 232)(179 233)(180 234)
(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 50 155 26)(2 59 156 35)(3 48 157 24)(4 57 158 33)(5 46 159 22)(6 55 160 31)(7 44 141 40)(8 53 142 29)(9 42 143 38)(10 51 144 27)(11 60 145 36)(12 49 146 25)(13 58 147 34)(14 47 148 23)(15 56 149 32)(16 45 150 21)(17 54 151 30)(18 43 152 39)(19 52 153 28)(20 41 154 37)(61 240 209 176)(62 229 210 165)(63 238 211 174)(64 227 212 163)(65 236 213 172)(66 225 214 161)(67 234 215 170)(68 223 216 179)(69 232 217 168)(70 221 218 177)(71 230 219 166)(72 239 220 175)(73 228 201 164)(74 237 202 173)(75 226 203 162)(76 235 204 171)(77 224 205 180)(78 233 206 169)(79 222 207 178)(80 231 208 167)(81 181 122 105)(82 190 123 114)(83 199 124 103)(84 188 125 112)(85 197 126 101)(86 186 127 110)(87 195 128 119)(88 184 129 108)(89 193 130 117)(90 182 131 106)(91 191 132 115)(92 200 133 104)(93 189 134 113)(94 198 135 102)(95 187 136 111)(96 196 137 120)(97 185 138 109)(98 194 139 118)(99 183 140 107)(100 192 121 116)

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F4G4H4I···4N5A5B6A···6F6G6H6I6J10A···10F10G10H10I10J12A···12H12I···12P12Q···12AB15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order12222233444444444···4556···6666610···101010101012···1212···1212···121515151520···2030···3030···3060···60
size111122112222555510···10221···122222···244442···25···510···1022224···42···24···44···4

120 irreducible representations

dim11111111111111222222222244
type+++++++++-
imageC1C2C2C2C2C2C3C4C6C6C6C6C6C12D5C4○D4D10D10C3×D5C4×D5C3×C4○D4C6×D5C6×D5D5×C12D42D5C3×D42D5
kernelC3×C23.11D10C12×Dic5C3×C10.D4C3×C23.D5C15×C22⋊C4C2×C6×Dic5C23.11D10C6×Dic5C4×Dic5C10.D4C23.D5C5×C22⋊C4C22×Dic5C2×Dic5C3×C22⋊C4C30C2×C12C22×C6C22⋊C4C2×C6C10C2×C4C23C22C6C2
# reps1221112844222162442488841648

Matrix representation of C3×C23.11D10 in GL4(𝔽61) generated by

47000
04700
0010
0001
,
60000
06000
0010
006060
,
60000
06000
0010
0001
,
1000
0100
00600
00060
,
57400
46000
006059
0001
,
343700
52700
001122
00050
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,1,60,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[57,46,0,0,4,0,0,0,0,0,60,0,0,0,59,1],[34,5,0,0,37,27,0,0,0,0,11,0,0,0,22,50] >;

C3×C23.11D10 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{11}D_{10}
% in TeX

G:=Group("C3xC2^3.11D10");
// GroupNames label

G:=SmallGroup(480,670);
// by ID

G=gap.SmallGroup(480,670);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,555,142,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^10=c,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
// generators/relations

׿
×
𝔽