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G = D10.19(C4×S3)  order 480 = 25·3·5

4th non-split extension by D10 of C4×S3 acting via C4×S3/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D10.19(C4×S3)
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — D10.19(C4×S3)
 Lower central C15 — C30 — D10.19(C4×S3)
 Upper central C1 — C22 — C2×C4

Generators and relations for D10.19(C4×S3)
G = < a,b,c,d,e | a10=b2=c4=d3=1, e2=a5, bab=a-1, ac=ca, ad=da, ae=ea, cbc-1=a5b, bd=db, be=eb, cd=dc, ece-1=a5c, ede-1=d-1 >

Subgroups: 652 in 152 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×8], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], C10 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×4], C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×Dic3 [×2], C2×Dic3 [×6], C2×C12, C2×C12, C22×C6, C3×D5 [×2], C30 [×3], C42⋊C2, C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C5×Dic3 [×2], C5×Dic3, C3×Dic5, Dic15 [×2], Dic15, C60, C6×D5 [×2], C6×D5 [×2], C2×C30, C4×Dic5 [×2], C4⋊Dic5, D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C23.16D6, D5×Dic3 [×4], C6×Dic5, C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, C4⋊C47D5, Dic3×Dic5, D10⋊Dic3, C6.Dic10, C3×D10⋊C4, C5×Dic3⋊C4, C4×Dic15, C2×D5×Dic3, D10.19(C4×S3)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C4×D5 [×2], C22×D5, S3×C2×C4, D42S3 [×2], S3×D5, C2×C4×D5, D42D5, Q82D5, C23.16D6, C2×S3×D5, C4⋊C47D5, D20⋊S3, C4×S3×D5, C30.C23, D10.19(C4×S3)

Smallest permutation representation of D10.19(C4×S3)
On 240 points
Generators in S240
(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 52)(2 51)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 219)(12 218)(13 217)(14 216)(15 215)(16 214)(17 213)(18 212)(19 211)(20 220)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 50)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 70)(40 69)(71 103)(72 102)(73 101)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 120)(91 123)(92 122)(93 121)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(131 168)(132 167)(133 166)(134 165)(135 164)(136 163)(137 162)(138 161)(139 170)(140 169)(141 174)(142 173)(143 172)(144 171)(145 180)(146 179)(147 178)(148 177)(149 176)(150 175)(151 188)(152 187)(153 186)(154 185)(155 184)(156 183)(157 182)(158 181)(159 190)(160 189)(191 223)(192 222)(193 221)(194 230)(195 229)(196 228)(197 227)(198 226)(199 225)(200 224)(201 239)(202 238)(203 237)(204 236)(205 235)(206 234)(207 233)(208 232)(209 231)(210 240)
(1 113 53 83)(2 114 54 84)(3 115 55 85)(4 116 56 86)(5 117 57 87)(6 118 58 88)(7 119 59 89)(8 120 60 90)(9 111 51 81)(10 112 52 82)(11 190 215 160)(12 181 216 151)(13 182 217 152)(14 183 218 153)(15 184 219 154)(16 185 220 155)(17 186 211 156)(18 187 212 157)(19 188 213 158)(20 189 214 159)(21 110 50 80)(22 101 41 71)(23 102 42 72)(24 103 43 73)(25 104 44 74)(26 105 45 75)(27 106 46 76)(28 107 47 77)(29 108 48 78)(30 109 49 79)(31 121 61 91)(32 122 62 92)(33 123 63 93)(34 124 64 94)(35 125 65 95)(36 126 66 96)(37 127 67 97)(38 128 68 98)(39 129 69 99)(40 130 70 100)(131 226 161 196)(132 227 162 197)(133 228 163 198)(134 229 164 199)(135 230 165 200)(136 221 166 191)(137 222 167 192)(138 223 168 193)(139 224 169 194)(140 225 170 195)(141 236 171 206)(142 237 172 207)(143 238 173 208)(144 239 174 209)(145 240 175 210)(146 231 176 201)(147 232 177 202)(148 233 178 203)(149 234 179 204)(150 235 180 205)
(1 35 21)(2 36 22)(3 37 23)(4 38 24)(5 39 25)(6 40 26)(7 31 27)(8 32 28)(9 33 29)(10 34 30)(11 233 230)(12 234 221)(13 235 222)(14 236 223)(15 237 224)(16 238 225)(17 239 226)(18 240 227)(19 231 228)(20 232 229)(41 54 66)(42 55 67)(43 56 68)(44 57 69)(45 58 70)(46 59 61)(47 60 62)(48 51 63)(49 52 64)(50 53 65)(71 84 96)(72 85 97)(73 86 98)(74 87 99)(75 88 100)(76 89 91)(77 90 92)(78 81 93)(79 82 94)(80 83 95)(101 114 126)(102 115 127)(103 116 128)(104 117 129)(105 118 130)(106 119 121)(107 120 122)(108 111 123)(109 112 124)(110 113 125)(131 156 144)(132 157 145)(133 158 146)(134 159 147)(135 160 148)(136 151 149)(137 152 150)(138 153 141)(139 154 142)(140 155 143)(161 186 174)(162 187 175)(163 188 176)(164 189 177)(165 190 178)(166 181 179)(167 182 180)(168 183 171)(169 184 172)(170 185 173)(191 216 204)(192 217 205)(193 218 206)(194 219 207)(195 220 208)(196 211 209)(197 212 210)(198 213 201)(199 214 202)(200 215 203)
(1 148 6 143)(2 149 7 144)(3 150 8 145)(4 141 9 146)(5 142 10 147)(11 125 16 130)(12 126 17 121)(13 127 18 122)(14 128 19 123)(15 129 20 124)(21 135 26 140)(22 136 27 131)(23 137 28 132)(24 138 29 133)(25 139 30 134)(31 156 36 151)(32 157 37 152)(33 158 38 153)(34 159 39 154)(35 160 40 155)(41 166 46 161)(42 167 47 162)(43 168 48 163)(44 169 49 164)(45 170 50 165)(51 176 56 171)(52 177 57 172)(53 178 58 173)(54 179 59 174)(55 180 60 175)(61 186 66 181)(62 187 67 182)(63 188 68 183)(64 189 69 184)(65 190 70 185)(71 196 76 191)(72 197 77 192)(73 198 78 193)(74 199 79 194)(75 200 80 195)(81 206 86 201)(82 207 87 202)(83 208 88 203)(84 209 89 204)(85 210 90 205)(91 216 96 211)(92 217 97 212)(93 218 98 213)(94 219 99 214)(95 220 100 215)(101 226 106 221)(102 227 107 222)(103 228 108 223)(104 229 109 224)(105 230 110 225)(111 236 116 231)(112 237 117 232)(113 238 118 233)(114 239 119 234)(115 240 120 235)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 6A 6B 6C 6D 6E 10A ··· 10F 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 10 ··· 10 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 2 2 2 6 6 6 6 10 10 15 15 15 15 30 30 2 2 2 2 2 20 20 2 ··· 2 4 4 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D5 D6 D6 D6 C4○D4 D10 D10 C4×S3 C4×D5 D4⋊2S3 S3×D5 D4⋊2D5 Q8⋊2D5 C2×S3×D5 D20⋊S3 C4×S3×D5 C30.C23 kernel D10.19(C4×S3) Dic3×Dic5 D10⋊Dic3 C6.Dic10 C3×D10⋊C4 C5×Dic3⋊C4 C4×Dic15 C2×D5×Dic3 D5×Dic3 D10⋊C4 Dic3⋊C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 D10 Dic3 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 4 4 2 4 8 2 2 2 2 2 4 4 4

Matrix representation of D10.19(C4×S3) in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 17 17 0 0 0 0 43 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 60 0 0 0 0 0 1 0 0 0 0 0 0 17 16 0 0 0 0 43 44 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 59 60 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 60 0
,
 50 50 0 0 0 0 0 11 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 1

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,17,43,0,0,0,0,17,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,60,1,0,0,0,0,0,0,17,43,0,0,0,0,16,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,59,0,0,0,0,0,60,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[50,0,0,0,0,0,50,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,1,1] >;

D10.19(C4×S3) in GAP, Magma, Sage, TeX

D_{10}._{19}(C_4\times S_3)
% in TeX

G:=Group("D10.19(C4xS3)");
// GroupNames label

G:=SmallGroup(480,470);
// by ID

G=gap.SmallGroup(480,470);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^2=c^4=d^3=1,e^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=a^5*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^5*c,e*d*e^-1=d^-1>;
// generators/relations

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