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G = (S3×C10).D4  order 480 = 25·3·5

16th non-split extension by S3×C10 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.D5⋊7S3, D6⋊Dic5⋊31C2, (S3×C10).16D4, C30.235(C2×D4), C10.159(S3×D4), C5⋊7(C23.9D6), C23.20(S3×D5), D6.10(C5⋊D4), Dic15⋊5C4⋊35C2, C30.Q8⋊34C2, (C22×C6).34D10, (C22×C10).50D6, C30.149(C4○D4), C10.83(C4○D12), C30.38D4⋊26C2, C6.56(D4⋊2D5), (C2×C30).197C23, (C2×Dic3).61D10, (C2×Dic5).129D6, (C22×S3).53D10, C10.82(D4⋊2S3), C15⋊23(C22.D4), C3⋊3(C23.18D10), (C22×C30).59C22, C2.26(Dic3.D10), C2.28(C30.C23), (C6×Dic5).113C22, (C10×Dic3).114C22, (C2×Dic15).136C22, (C2×S3×Dic5)⋊15C2, (C2×C3⋊D4).5D5, C2.38(S3×C5⋊D4), C6.62(C2×C5⋊D4), (C10×C3⋊D4).6C2, (C3×C23.D5)⋊8C2, C22.230(C2×S3×D5), (S3×C2×C10).49C22, (C2×C6).209(C22×D5), (C2×C10).209(C22×S3), SmallGroup(480,631)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C30 — (S3×C10).D4
Chief series
C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×S3×Dic5 — (S3×C10).D4
Lower central
C15 — C2×C30 — (S3×C10).D4
Upper central
C1 — C22 — C23

Generators and relations for (S3×C10).D4
 G = < a,b,c,d,e | a10=b3=c2=d4=e2=1, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=a5c, ede=a5d-1 >

Subgroups: 684 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C22.D4, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, S3×C10, S3×C10, C2×C30, C2×C30, C10.D4, C23.D5, C23.D5, C22×Dic5, D4×C10, C23.9D6, S3×Dic5, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, S3×C2×C10, C22×C30, C23.18D10, D6⋊Dic5, C30.Q8, Dic15⋊5C4, C3×C23.D5, C30.38D4, C2×S3×Dic5, C10×C3⋊D4, (S3×C10).D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C22.D4, C5⋊D4, C22×D5, C4○D12, S3×D4, D4⋊2S3, S3×D5, D4⋊2D5, C2×C5⋊D4, C23.9D6, C2×S3×D5, C23.18D10, C30.C23, Dic3.D10, S3×C5⋊D4, (S3×C10).D4

Smallest permutation representation of (S3×C10).D4
►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 35 23)(2 36 24)(3 37 25)(4 38 26)(5 39 27)(6 40 28)(7 31 29)(8 32 30)(9 33 21)(10 34 22)(11 221 233)(12 222 234)(13 223 235)(14 224 236)(15 225 237)(16 226 238)(17 227 239)(18 228 240)(19 229 231)(20 230 232)(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)(101 114 127)(102 115 128)(103 116 129)(104 117 130)(105 118 121)(106 119 122)(107 120 123)(108 111 124)(109 112 125)(110 113 126)(131 143 156)(132 144 157)(133 145 158)(134 146 159)(135 147 160)(136 148 151)(137 149 152)(138 150 153)(139 141 154)(140 142 155)(161 173 186)(162 174 187)(163 175 188)(164 176 189)(165 177 190)(166 178 181)(167 179 182)(168 180 183)(169 171 184)(170 172 185)(191 203 216)(192 204 217)(193 205 218)(194 206 219)(195 207 220)(196 208 211)(197 209 212)(198 210 213)(199 201 214)(200 202 215)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E10A···10F10G10H10I10J10K10L10M10N12A12B12C12D15A15B20A20B20C20D30A···30N
order122222234444444556666610···1010101010101010101212121215152020202030···30
size111146621010122030306022222442···24444121212122020202044121212124···4

60 irreducible representations

dim111111112222222222244444444
type+++++++++++++++++-+-+-
imageC1C2C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10D10C5⋊D4C4○D12S3×D4D4⋊2S3S3×D5D4⋊2D5C2×S3×D5C30.C23Dic3.D10S3×C5⋊D4
kernel(S3×C10).D4D6⋊Dic5C30.Q8Dic15⋊5C4C3×C23.D5C30.38D4C2×S3×Dic5C10×C3⋊D4C23.D5S3×C10C2×C3⋊D4C2×Dic5C22×C10C30C2×Dic3C22×S3C22×C6D6C10C10C10C23C6C22C2C2C2
# reps111111111222142228411242444

Matrix representation of (S3×C10).D4 ►in GL8(𝔽61)

10000000
01000000
0018180000
0043600000
00001000
00000100
000000600
000000060
,
01000000
6060000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
6060000000
00100000
00010000
000060000
000006000
000000600
000000401
,
600000000
060000000
0053230000
004580000
0000592800
00002200
0000002450
0000001937
,
600000000
060000000
006000000
000600000
0000605900
00000100
0000004160
0000003320

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,43,0,0,0,0,0,0,18,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,40,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,53,45,0,0,0,0,0,0,23,8,0,0,0,0,0,0,0,0,59,2,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,24,19,0,0,0,0,0,0,50,37],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,59,1,0,0,0,0,0,0,0,0,41,33,0,0,0,0,0,0,60,20] >;
 

(S3×C10).D4 in GAP, Magma, Sage, TeX 

(S_3\times C_{10}).D_4
 
% in TeX
 
G:=Group("(S3xC10).D4");
 
// GroupNames label
 
G:=SmallGroup(480,631);
 
// by ID
 
G=gap.SmallGroup(480,631);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^10=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=a^5*d^-1>;
 
// generators/relations
 

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