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G = C3×C23.21D10order 480 = 25·3·5

Direct product of C3 and C23.21D10

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

Aliases: C3×C23.21D10, (C2×C60)⋊23C4, (C2×C20)⋊11C12, C4⋊Dic517C6, (C2×C12)⋊9Dic5, C60.252(C2×C4), C20.59(C2×C12), (C4×Dic5)⋊15C6, C4.15(C6×Dic5), C23.26(C6×D5), C23.D5.5C6, (C12×Dic5)⋊33C2, (C2×C12).453D10, (C22×C60).21C2, (C22×C20).14C6, C12.54(C2×Dic5), (C22×C12).17D5, C1531(C42⋊C2), C30.193(C4○D4), C6.123(C4○D20), C22.6(C6×Dic5), C10.37(C22×C12), (C2×C60).513C22, (C2×C30).361C23, C30.222(C22×C4), (C22×C6).106D10, C6.34(C22×Dic5), (C22×C30).157C22, (C6×Dic5).247C22, C2.5(C2×C6×Dic5), C55(C3×C42⋊C2), (C2×C4)⋊4(C3×Dic5), C2.4(C3×C4○D20), C22.22(D5×C2×C6), (C2×C20).96(C2×C6), (C3×C4⋊Dic5)⋊35C2, C10.14(C3×C4○D4), (C2×C4).103(C6×D5), (C22×C4).9(C3×D5), (C2×C30).191(C2×C4), (C2×C10).55(C2×C12), (C2×C6).24(C2×Dic5), (C2×C10).44(C22×C6), (C22×C10).44(C2×C6), (C2×Dic5).37(C2×C6), (C2×C6).357(C22×D5), (C3×C23.D5).11C2, SmallGroup(480,719)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C23.21D10
C1C5C10C2×C10C2×C30C6×Dic5C12×Dic5 — C3×C23.21D10
C5C10 — C3×C23.21D10
C1C2×C12C22×C12

Generators and relations for C3×C23.21D10
 G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e10=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 336 in 152 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C12 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×4], C22×C6, C30, C30 [×2], C30 [×2], C42⋊C2, C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C22×C12, C3×Dic5 [×4], C60 [×4], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×2], C22×C20, C3×C42⋊C2, C6×Dic5 [×4], C2×C60 [×2], C2×C60 [×4], C22×C30, C23.21D10, C12×Dic5 [×2], C3×C4⋊Dic5 [×2], C3×C23.D5 [×2], C22×C60, C3×C23.21D10
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], Dic5 [×4], D10 [×3], C2×C12 [×6], C22×C6, C3×D5, C42⋊C2, C2×Dic5 [×6], C22×D5, C22×C12, C3×C4○D4 [×2], C3×Dic5 [×4], C6×D5 [×3], C4○D20 [×2], C22×Dic5, C3×C42⋊C2, C6×Dic5 [×6], D5×C2×C6, C23.21D10, C3×C4○D20 [×2], C2×C6×Dic5, C3×C23.21D10

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

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

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

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

156 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F4G···4N5A5B6A···6F6G6H6I6J10A···10N12A···12H12I12J12K12L12M···12AB15A15B15C15D20A···20P30A···30AB60A···60AF
order122222334444444···4556···6666610···1012···121212121212···121515151520···2030···3060···60
size1111221111112210···10221···122222···21···1222210···1022222···22···22···2

156 irreducible representations

dim111111111111222222222222
type++++++-++
imageC1C2C2C2C2C3C4C6C6C6C6C12D5C4○D4Dic5D10D10C3×D5C3×C4○D4C3×Dic5C6×D5C6×D5C4○D20C3×C4○D20
kernelC3×C23.21D10C12×Dic5C3×C4⋊Dic5C3×C23.D5C22×C60C23.21D10C2×C60C4×Dic5C4⋊Dic5C23.D5C22×C20C2×C20C22×C12C30C2×C12C2×C12C22×C6C22×C4C10C2×C4C2×C4C23C6C2
# reps1222128444216248424816841632

Matrix representation of C3×C23.21D10 in GL3(𝔽61) generated by

100
0130
0013
,
100
010
0060
,
6000
010
001
,
100
0600
0060
,
6000
0280
0037
,
1100
0037
0280
G:=sub<GL(3,GF(61))| [1,0,0,0,13,0,0,0,13],[1,0,0,0,1,0,0,0,60],[60,0,0,0,1,0,0,0,1],[1,0,0,0,60,0,0,0,60],[60,0,0,0,28,0,0,0,37],[11,0,0,0,0,28,0,37,0] >;

C3×C23.21D10 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{21}D_{10}
% in TeX

G:=Group("C3xC2^3.21D10");
// GroupNames label

G:=SmallGroup(480,719);
// by ID

G=gap.SmallGroup(480,719);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,344,1094,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^10=d,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,f*b*f^-1=b*d=d*b,b*e=e*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